Hecke algebra

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## Idea

**Hecke algebra** is a term for a class of [[algebras]]. They often appear as [[convolution algebras]] or as double [[coset spaces]]. For p-adic algebraic groups Hecke algebras often play a role similar a [[Lie algebra]] plays in the  complex case (the Lie algebra still exists, but is too small).

Typically the term refers to an algebra which is the [[endomorphisms]] of a [[permutation representation]] of a [[topological group]], though some liberties have been taken with this definition, and often the term means some modification of such an algebra.

For example:

* If we consider the [[general linear group]] $GL_n(\mathbb{F}_q)$ acting on the set of [[flag|complete flags]] in $\mathbb{F}_q^n$, then we obtain an algebra generated by the endomorphism $\sigma_i$ which sends the characteristic function of one flag $\mathbf{F}=\{F_1\subset \cdots \subset F_{n-1}\subset \mathbb{F}_q^n\}$ to the characteristic function of the set of flags $\mathbf{F}'$ with $F_j'=F_j$ for all $j\neq i$ and $F_i'\neq F_i$.  These elements satisfy the relations
  $$\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$$
  $$\sigma_i\sigma_j=\sigma_j\sigma_i \qquad (|i-j|\gt1)$$
  $$\sigma_i^2=(q-1)\sigma_i+q$$
* If we look at $GL_n(\mathbb{F}_q(\!(t)\!))$ acting on the set of $\mathbb{F}_q[\![t]\!]$ [[lattices]] in $\mathbb{F}_q(\!(t)\!)^n$, then we will obtain the __spherical Hecke algebra__.
* A variant of the Hecke algebra is the **degenerate affine Hecke algebra** of type $A$; this is a deformation of the semidirect product of the symmetric group $S_n$ with the polynomial ring in $n$ variables.  The generators are $S_n$ and $y_1, \dots, y_n$, with relations $\sigma y_i \sigma^{-1} = y_{\sigma(i)}$ and $[y_i,y_j] = \sum_{k \neq i,j} (k i j)-(k j i)$; one can replace the $y_i$'s with commuting $x_i$'s with slightly messier relations.  As [[George Lusztig]] showed, the representation theory of the affine Hecke algebra is related to the graded or degenerate case.
* There is a geometric construction of the representations of Weyl algebras when realized as certain Hecke convolution algebras by [[Victor Ginzburg]].

## Generalized Hecke algebras

To each [[Coxeter group]] $W$ one may associate a Hecke algebra, a certain [[deformation]] of the [[group algebra]] $k[W]$ over a [[field]] $k$, as follows. $W$ is [[group presentation|presented]] by generators $\langle s_i \rangle_{i \in I}$ and relations

$$(s_i s_j)^{m_{i j}} = 1$$

where $m_{i j} = m_{j i}$ and $m_{i i} = 1$ for all $i, j \in I$. The relations may be rewritten:

$$s_{i}^{2} = 1, \qquad s_i s_j \ldots = s_j s_i \ldots$$

where each of the words in the second equation alternate in the letters $s_i$, $s_j$ and has length $m_{i j}$, provided that $m_{i j} \lt \infty$. The corresponding Hecke algebra has basis $W$, and is presented by

$$s_{i}^{2} = \frac{q-1}{q} s_i + \frac1{q}, \qquad s_i s_j \ldots = s_j s_i \ldots$$

These relations may be interpreted structurally as follows (for simplicity, we will consider only finite, aka spherical Coxeter groups). A Coxeter group $W$ may be associated with a suitable [[BN-pair]]; the classical example is where $G$ is an [[algebraic group]], $B$ is a [[Borel subgroup]] (maximal solvable subgroup), and $N$ is the normalizer of a [[maximal torus]] $T$ in $G$. Such $G$ typically arise as automorphism groups of thick $W$-buildings, where $B$ is a stabilizer of a point of the building. The coset space $G/B$ may then be interpreted as a space of flags for a suitable geometry. The Coxeter group itself arises as the quotient $W \cong N/T$, and under the BN-pair axioms there is a well-defined map

$$W \to B\backslash G/B: w \mapsto B w B$$

which is a bijection to the set of double cosets of $B$. (In particular, the double cosets do not depend on the coefficient ring $R$ in which the points $G(R)$ are instantiated.)

When one takes points of the algebraic group $G$ over the coefficient ring $\mathbb{F}_q$, a finite field with $q$ elements, the flag manifold $G_q/B_q \coloneqq G(\mathbb{F}_q)/B(\mathbb{F}_q)$ is also finite. One may calculate

$$\array{
\hom_{k[G_q]}(k[G_q/B_q], k[G_q/B_q]) & \cong & k[G_q/B_q]^\ast \otimes_{k[G_q]} k[G_q/B_q] \\
 & \cong & k[B_q\backslash G_q] \otimes_{k[G_q]} k[G_q/B_q] \\
 & \cong & k[B_q\backslash G_q \otimes_{G_q} G_q/B_q] \\
 & \cong & k[B_q \backslash G_q / B_q]
}$$

so that the double cosets form a linear basis of the algebra of $G_q$-equivariant operators on the space of functions $k[G_q/B_q]$. This algebra is in fact the Hecke algebra.

It is a matter of interest to interpret the double cosets directly as operators on $k[G_q/B_q]$, and in particular the cosets $B s_i B$ where $s_i$ is a Coxeter generator.

To be continued...

## Related concepts

* [[double Hecke algebra]]

* [[double affine Hecke algebra]]

* [[Hecke correspondence]]

* [[Hecke category]]

* [[Iwahori–Hecke algebra]]

## References

In relation to the [[Knizhnik-Zamolodchikov equation]]:

* [[Ivan Cherednik]], *Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras*, Mathematical Society of Japan Memoirs **1998** (1998) 1-96 $[$[doi:10.2969/msjmemoirs/00101C010](https://doi.org/10.2969/msjmemoirs/00101C010)$]$

See also:

* Secret Blogging Seminar: [Interpreting the Hecke algebra](https://sbseminar.wordpress.com/2008/02/25/interpreting-the-hecke-algebra/)

* Secret Blogging Seminar: [Interpreting the Hecke Algebra II: the sheafification](https://sbseminar.wordpress.com/2009/04/09/interpreting-the-hecke-algebra-ii-the-sheafification/)

For the representation theory of the degenerate affine Hecke algebra see

* Takeshi Suzuki, _Rogawski's conjecture on the Jantzen filtration for the degenerate affine Hecke algebra of type A_, [math.QA/9805035](http://arxiv.org/abs/math/9805035)

[[!redirects Hecke algebras]]


Hecke category

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## Idea

Generally, for $X \longrightarrow Y$ a suitable map of [[schemes]] or [[algebraic stacks]], then the corresponding _Hecke category_ is the [[category]] ([[derived category]], [[(infinity,1)-category]]) $\mathcal{D}(X \times_Y X)$ of [[D-modules]] on the ([[homotopy fiber product|homotopy]]) [[fiber product]], regarded as a [[monoidal category]] by regarding its objects as specifying [[integral transforms]] in $Fun_{\mathcal{D}(Y)}(\mathcal{D}(X), \mathcal{D}(X))$ ([Ben-Zvi & Nadler 09, section 5.1](#BenZviNadler09)).

For $X\to Y$ the inclusion $\mathbf{B}B \to \mathbf{B}G$ of the [[delooping]] of a [[Borel subgroup]] of a complex [[reductive group]], then (generally for maps of delooped groups like this, see [here](homotopy+limit#DeloopedSubgroupOverGroup) at _[[homotopy limit]]_) the [[homotopy fiber product]] is $X \times_Y X \simeq B \backslash \backslash G // B $. The Hecke category for this case is the default case of Hecke categories used in [[geometric representation theory]].

The concept of Hecke category is a [[categorification]] of that of _[[Hecke algebra]]_.

## Related concepts

* [[Hecke correspondence]]

* [[Hecke eigensheaf]]

## References

* {#BenZviNadler09} [[David Ben-Zvi]], [[David Nadler]], section 5.1 of _The Character Theory of a Complex Group_ ([arXiv:0904.1247](http://arxiv.org/abs/0904.1247))

A sruvey is around slide 15 (40 of 77) in

* [[David Ben-Zvi]], _[[Loop Groups, Characters and Elliptic Curves]]_, talk (2012)
[[!redirects Hecke categories]]


Hecke correspondence

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## Idea

A _Hecke correspondence_ is a certain [[correspondence]] between [[moduli stacks of bundles]]. The [[integral transform]] induced by a [[Hecke correspondence]] is called a _Hecke transform_. The objects preserves by such a transform up to a tensor multiple are called [[Hecke eigensheaves]].

These are central objects of interest in [[geometric Langlands duality]].

## For modular curves

The reference for this is section 1.4 of [Calegari13](#Calegari13).

Let $X_{0}(p)$ be the moduli space of pairs $(E,E')$ where $E$ and $E'$ are $p$-isogenous elliptic curves. This has maps $\pi_{1}$ and $\pi_{2}$ to the moduli space of elliptic curves $X$, sending $(E,E')$ to $E$ and $E'$ respectively:

$$
  \array{
     && X_{0}(p)     \\
     & {}_{\mathllap{\pi_{1}}}\swarrow && \searrow_{\mathrlap{\pi_{2}}}
     \\
     X && && X
  }
$$

We can think of the Hecke correspondence as the multivalued function $\pi_{2 *}\circ\pi_{1}^{*}$. Although this is a multivalued function, upon linearizing (by, say, passing to [[modular forms]] i.e. global sections of the sheaf $\omega^{k}$) we get a legitimate function

$$pT_{p}:H^{0}(X(\Gamma),\omega^{k})\to H^{0}(X(\Gamma),\omega^{k})$$

where $T_{p}$ is the $p$-th _Hecke operator_.

Alternatively, we can embed $X_{0}(p)$ inside $X\times X$ as the graph of the above-mentioned multivalued function. In this way it is more properly seen as a correspondence.

## For moduli stacks of bundles

A reference for this is section 3.7 of [Frenkel05](#Frenkel05). A discussion can also be found in [Lafforgue18](#Lafforgue18).

Hecke correspondences for moduli stacks of bundles follow roughly the same idea but instead of $X$ we have $Bun_{G}$ and instead of $X_{0}$ we have the Hecke stack $Hck$, which parametrizes pairs of bundles $\mathcal{E}_{1}$, $\mathcal{E}_{2}$ together with a _modification_ which is an isomorphism between $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ away from some points:

$$
  \array{
     && Hck     \\
     & {}_{\mathllap{\pi_{1}}}\swarrow && \searrow_{\mathrlap{\pi_{2}}}
     \\
     Bun_{G} && && Bun_{G}
  }
$$

## Properties

### Relation to S-duality in super Yang-Mills theory

[[!include geometric Langlands QFT -- table]]

## Related concepts

* [[modular form]]

* [[Hecke algebra]]

* [[Hecke eigensheaf]]

* [[Hecke category]]

## References

* {#Frenkel05} [[Edward Frenkel]], section 3.7 in _Lectures on the Langlands Program and Conformal Field Theory_ ([arXiv:hep-th/0512172](http://arxiv.org/abs/hep-th/0512172))

* {#Calegari13} Frank Calegari, _Congruences between modular forms_, 2013 [pdf](https://swc-math.github.io/aws/2013/2013CalegariLectureNotes.pdf)

* {#Lafforgue18} [[Vincent Lafforgue]], _Shtukas for reductive groups and Langlands correspondence for function fields_, [arXiv:1803.03791](https://arxiv.org/abs/1803.03791)

[[!redirects Hecke correspondences]]

[[!redirects Hecke transformation]]
[[!redirects Hecke transformations]]

[[!redirects Hecke transform]]
[[!redirects Hecke transforms]]


Hecke eigensheaf

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## Idea

A _Hecke eigensheaf_ is an object that is taken to a tensor-multiple of itself under the [[integral transform]] induced by a [[Hecke correspondence]].

Hecke eigensheaves play a central role in the [[geometric Langlands correspondence]].

## Properties

### Relation to magnetic branes in super Yang-Mills theory

[[!include geometric Langlands QFT -- table]]

## Related concepts

## References

* {#Frenkel05} [[Edward Frenkel]], section 3.8 in _Lectures on the Langlands Program and Conformal Field Theory_ ([arXiv:hep-th/0512172](http://arxiv.org/abs/hep-th/0512172))

[[!redirects Hecke eigensheaves]]


Hecke L-function

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## Idea

A kind of _[[L-function]]_ for [[number fields]].

## Related concepts

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

## References

* E. Kowalski, sections 1.5, 2.2 of  _Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures_ ([pdf](http://www.math.ethz.ch/~kowalski/lectures.pdf))

* {#Snyder02} [[Noah Snyder]], section 1.4.7 of _Artin L-Functions: A Historical Approach_, 2002 ([pdf](http://www.math.columbia.edu/~nsnyder/thesismain.pdf))

[[!redirects Hecke L-functions]]


Hecke theta function

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## Idea

A [[theta function]] for [[number fields]] which generalizes the [[Jacobi theta function]]. As the latter is related to the [[Riemann zeta function]], so Hasse theta functions are related to the [[Dedekind zeta function]].

([Kowalski (2.4)](#Kowalski))

## Properties

### Relation to Dedekind zeta function

A certain [[integral]] over a kernel involving the Hecke theta function gives the [[Dedekind zeta function]]. See at _[Dedekind zeta function -- Relation to Hecke theta function](Dedekind+zeta+function#RelationToThetaFunctions)_.

### Function field analogy

[[!include function field analogy -- table]]

## Related concepts

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

## References

* {#Kowalski} E. Kowalski, first part of _Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures_ ([pdf](http://www.math.ethz.ch/~kowalski/lectures.pdf))


Hector Zenil

* [webpage](http://www.mathrix.org/zenil/)

## Related $n$Lab entries

* [[computable physics]]

category: people


Hedberg's theorem

\tableofcontents

## Idea

Hedberg's theorem in [[dependent type theory]] says that given a type $A$ with [[untruncated decidable equality]], then $A$ it is an [[h-set]]. This was first proven in ([Hedberg](#Hedberg))

## Theorem

+-- {: .num_theorem #DecidableIsSet}
###### Theorem
(**[[Hedberg's theorem]]**)

Suppose that $A$ is a [[type]] which has [[untruncated decidable equality]]. In other words, the projection
$$\array{Paths_A + (0\to A\times A)^{(Paths_A\to A\times A)}\\
\downarrow\\
A\times A}
$$

(where $Paths_A$ is the [[path object|path type]] of $A$, "+" forms the [[sum type]], and on the right we have the $A \times A$-[[dependent type|dependent]] [[function type]] into the [[empty type]]), has a section.

Then $A$ is a h-set.

=--

+-- {: .proof}
###### Proof
Let $d$ be the given [[section]].  Thus, for any $x,y\colon A$, $d(x,y)$ is either a path from $x$ to $y$ or a function from $Paths(x,y)$ to the empty type (implying that $Paths(x,y)$ is also empty).

It suffices to exhibit an operation connecting any endo-path $p \in Paths(x,x)$ to the identity path $1_x$.  Given such a path, define $q = d(x,x)$.  If $d(x,x)$ lies in the second case, then $Paths(x,x)$ is empty, a contradiction since we know it contains $1_x$; hence we may assume $q\in Paths(x,x)$ as well.

Let $r$ be the image of $(1_x,p) \in Paths_{A\times A}((x,x),(x,x))$ under the section $d$.  This is a path in the total space $Paths_A$ lying over the path $(1_x,p)$ in $A\times A$.  Equivalently, it is a path in the fiber over $x$ from $(1_x,p)_*(d(x,x))$ to $d(x,x)$, where $(1_x,p)_*$ denotes transport in the fibration $Paths_A \to A\times A$ along the path $(1_x,p)$.  However, we have defined $d(x,x) = q$, and transport in a path-space is just composition, so $r$ may be regarded as a path from $q p$ to $q$.  Canceling $q$, we obtain a path from $p$ to $1_x$.
=--

## See also

* [[h-set]]
* [[decidable equality]]

## References

* {#Hedberg} Michael Hedberg, *A coherence theorem for Martin-Löf's type theory*, J. Functional Programming, (1998)

* Nicolai Kraus, *A direct proof of Hedberg's theorem*, [blog post](http://homotopytypetheory.org/2012/03/30/a-direct-proof-of-hedbergs-theorem/)

* [[Nicolai Kraus]], [[Martin Escardo]], [[Thierry Coquand]] , [[Thorsten Altenkirch]], _Generalizations of Hedberg's theorem_, in M. Hasegawa (Ed.): TLCA 2013, LNCS 7941, pp. 173-188. Springer, Heidelberg 2013. ([pdf](http://www.cs.bham.ac.uk/~mhe/papers/hedberg.pdf))

* *Homotopy Type Theory: Univalent Foundations of Mathematics*, The [[Univalent Foundations Project]], Institute for Advanced Study, 2013. ([web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf))

[[!redirects Hedberg's theorem]]
[[!redirects Hedberg theorem]]


Hee-Cheol Kim

* [arXiv page](https://arxiv.org/search/hep-th?searchtype=author&query=Kim,+H)

## Selected writings

On [[quantum Seiberg-Witten curves]] in relation to [[class S-theories]] and "[[M3]]"-[[defect branes]] inside [[M5-branes]]:

* [[Jin Chen]], [[Babak Haghighat]], [[Hee-Cheol Kim]], [[Marcus Sperling]], *Elliptic Quantum Curves of Class $\mathcal{S}_k$*,  J. High Energ. Phys. **2021** 28 (2021) [[arXiv:2008.05155](https://arxiv.org/abs/2008.05155), doi:10.1007/JHEP03(2021)028]

On [[quantum Seiberg-Witten curves]] of [[E-string]]-theories in relation to [[D6-D8-brane bound states]]:

* [[Jin Chen]], [[Babak Haghighat]], [[Hee-Cheol Kim]], [[Marcus Sperling]], [[Xin Wang]], *E-string Quantum Curve*, Nuclear Physics B **973** (2021) 115602 [[arXiv:2103.16996](https://arxiv.org/abs/2103.16996), [doi:10.1016/j.nuclphysb.2021.115602](https://doi.org/10.1016/j.nuclphysb.2021.115602)]

category: people


Hega Baum

* [webpage](http://www2.mathematik.hu-berlin.de/~diffgeo/index.php?seite=baum)

## related $n$Lab entries

* [[conformal geometry]]

category: people


Hegel's "Logic" as Modal Type Theory

> under construction

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**Abstract** While [[analytic philosophy]] famously rejected the
speculative [[metaphysics]] of [[Hegel]] in favor of the analysis of concepts by means of [[mathematical logic]], in particular [[predicate logic]], recent developments in the [[foundations of mathematics]]  via [[homotopy type theory]] offer a way to re-read [[Hegel]] as having useful formal meaning not in [[predicate logic]], but in '[[modal type theory]]'. The essence of this suggestion has been made by [[Lawvere]] [[Some Thoughts on the Future of Category Theory|in 1991]], which however remains largely unnoticed. Here we aim to give a transparent account of this perspective both [[philosophy|philosophically]] as well as [[category theory|category-theoretically]]. We then further expand on Lawvere's formalization of Hegel's "[[Science of Logic]]" in terms of the [[categorical semantics]] given by [[cohesive (infinity,1)-topos|cohesive higher toposes]]. We discuss how there is a _useful_ formalization of a fair bit of modern fundamental [[physics]], in fact of [[local quantum field theory|local]] [[gauge field theory|gauge]] [[quantum field theory]], to be found here.

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## Introduction

With the rise of [[analytic philosophy]] in Britain, and its adoption and development of the new logic forged by [[Frege]] and [[Peano]], there came a radical rejection of the existing philosophical paradigm, the [[idealism]] of [[T. H. Green]] and [[F. H. Bradley]], and those considered to have provided its primary source, notably [[Hegel]]. [[Bertrand Russell]] was adamant that the new logic would thoroughly transform philosophy, as he made clear at the end of his chapter _Logic As The Essence Of Philosophy_:

> The old logic put thought in fetters, while the new logic gives it wings. It has, in my opinion, introduced the same kind of advance into philosophy as Galileo introduced into physics, making it possible at last to see what kinds of problems may be capable of solution, and what kinds are beyond human powers. And where a solution appears possible, the new logic provides a method which enables us to obtain results that do not merely embody personal idiosyncrasies, but must command the assent of all who are competent to form an opinion. (Logic As The Essence Of Philosophy, 1914)

In the same chapter, Hegel was taken to task for his attempt to widen logic to a form of [[metaphysics]]. Much of it appears to Russell to result from muddled thinking, for instance, a failure to distinguish the 'is' of predication from the 'is' of identity.

This set the tone for analytic philosophy's opinion of Hegel for several decades. In 1951, the logical empiricist [[Hans Reichenbach]] wrote

>Hegel has been called the successor of Kant; that is a serious misunderstanding of Kant and an unjustified elevation of Hegel. Kant's system, though proved untenable by later developments, was the attempt of a great mind to establish rationalism on a scientific basis. Hegel's system is the poor construction of a fanatic who has seen one empirical truth and attempts to make it a logical law within the most unscientific of all logics. Whereas Kant's system marks the peak of the historical line of rationalism, Hegel's system belongs in the period of decay of speculative philosophy which characterizes the nineteenth century. (Reichenbach 1951, p. 72)

However, despite the initial lengthy rejection of Hegelian ideas by analytic philosophy, there have recently been increasing signs of a willingness to re-engage with Hegel's writings, in particular on the part of [[John MacDowell]] and [[Robert Brandom]] (Redding, 2007). Interestingly, one feature of analytic philosophy which needs reconsidering according to Redding is the primacy of [[first-order logic|first-order]] predicate logic, not, however, that he is promoting an alternative formalization based on Hegel's ideas.

It took a mathematician [[William Lawvere]] to find something formalizable of value in Hegel's _Logic_, in particular the concept of a '[[unity of opposites|Unity of Opposites]]'. In this paper, taking our lead from Lawvere in seeing common elements between [[categorical logic]] and Hegel's logic, we will pursue this connection right up to [[modal type theory|modal]] forms of [[homotopy type theory]].

Now, Hegel's understanding of the term 'Logik' was much broader than is normally the case. Indeed, the 'Objective logic' of [[Science of Logic]] is more akin to a kind of [[metaphysics]].  While the [[Vienna Circle]], admirers of Russell, had adopted the [[positivism|positivist]] attitude towards [[metaphysics]], dismissing it as meaningless ([Carnap 32](#Carnap32)), later in the twentieth century there was a resurgence of analytic metaphysics. One important source of this resurgence was the rise of [[modal logic]] as a tool for philosophy, especially due to [[Saul Kripke]] in _Naming and Necessity_. If the necessity of a proposition consists in its holding in all neighboring possible worlds, sense had to be made of these possible worlds. To what would have been Carnap's amazement, philosophers such as David Lewis proposed the reality of all possible worlds, the actual world being in no way special, except for its being our world.

Today, standard topics for analytic metaphysics today include causality, necessity, space and time, identity, and mental states. Little attention is paid to the question of explaining the kind of physical principles which operate in our universe. It is noteworthy, however, that the need for some such questioning seems to be as strongly felt among modern physicists as it must have been to Hegel and his predecessors, as witnessed by ... debates such as critically reviewed in ([Albert 12](#Albert12)). (...) Attempts to re-install this kind of metaphysics are these days undertaken by [[particle physics|particle physicists]] and [[cosmology|cosmologists]] themselves, whose success at formally describing remote aspects of the [[observable universe]] has emboldened many to feel as superior to modern philosophy as modern philosophy typically feels towards Hegel' idealism.

(...)

## Modern foundations -- Types, Judgements and Deduction

(...) something like:

analytic philosophy was certainly right to ask for a formal basis of philosophical reasoning, in order to make genuine intellectual progress possible. Back then [[predicate logic]] and [[set theory]] was the state of the art concerning the [[foundations of mathematics]] and so this is what analytic philosophy adopted and developed. But since then there has been much progress in the foundations of maths. ... [[intuitionistic type theory]] ... [[homotopy type theory]]...

Accordingly, the original starting point of analytic philosophy needs to be re-examined. While predicate logic has no way of making formal sense of Hegel's "Logic", for type theory the situation is rather different...

## Categorical semantics and Categorical logic

... type theory proper is a formal system for symbol manipulation ... useful for computer encoding, and much more readable than full formalization in, say, ZFC, but still not really what a discussion such as our here would directly work with...

... instead, the [[syntax]] of type theory has [[semantics]] in [[categories]] -- [[categorical semantics]] ... [[relation between category theory and type theory]] -- and this semantic level is what we will be concerned with here

... originates with Lawvere's thesis, see also "[[Adjointness in Foundations]]"

## Modalities, moments and opposites

(... basics of [[modalities]], [[modal operators]], [[monads]], leading up to [[adjoint modality]]/[[unity of opposites]]...)

## Formal determinations of Being

... using this and building on Lawvere's lead, we are led to objectively investigate the following:

starting with bare homotopy type theory and adding [[adjoint modalities]] to it, what is the nature of the new propositions that can be proven with these modalities, hence what is the new "quality" of the types in the presence of these new axioms...

...

$$
   \array{
      reines\;Sein && Dasein && Realitaet
      \\
      being && existence && reality
      \\
       pure being && determinate being
      \\
      \\
      && && \Re
      \\
     && && \bot
      \\
      && \int & \subset & \Im
      \\
      && \bot && \bot
      \\
      \emptyset &\subset& \flat & \subset & \&
      \\
      \bot & & \bot &&
      \\
      \ast & \subset& \sharp
   }
$$

| symbol | $\;\;$ name |
|---|---|
| $\int$ | [[shape modality]] |
| $\flat$ | [[flat modality]] |
| $\sharp$ | [[sharp modality]] |
| $\Re$ | [[reduction modality]] |
| $\Im$ | [[infinitesimal shape modality]] |
| $\&$ | [[infinitesimal flat modality]] |

(...)

## References

* {#Carnap32} [[Rudolf Carnap]], _Überwindung der Metaphysik durch logische Analyse der Sprache_, Erkenntnis Vol II (1932). Translated by A. Pap as _The Elimination of Metaphysics Through Logical Analysis of Language_ ([pdf](http://www.calstatela.edu/sites/default/files/dept/phil/pdf/res/Carnap-Elimination-of-Metaphysics.pdf))

* [[Georg Hegel]], _[[Science of Logic]]_ (1812)

* [[William Lawvere]], _[[Some Thoughts on the Future of Category Theory]]_ (1991)

* [[William Lawvere]] _[[Cohesive Toposes and Cantor's "lauter Einsen"]]_ (1994)

* [[William Lawvere]] _[[Unity and Identity of Opposites in Calculus and Physics]]_ (1996)

* [[Higher toposes of laws of motion]]

* [[schreiber:Homotopy-type semantics for quantization]]

* Paul Redding, _Analytic Philosophy and the Return of Hegelian Thought_, Cambridge University Press, 2007 ([review](http://ndpr.nd.edu/news/23465-analytic-philosophy-and-the-return-of-hegelian-thought/) by Willem A. deVries)

* [[Bertrand Russell]], 'Logic As The Essence Of Philosophy', chap. 2 of _Our Knowledge of the External World_, (1914)

* [[Hans Reichenbach]], 'The Rise of Scientific Philosophy', 1951.

* {#Albert12} [[David Albert]], _On the Origin of Everything_, book review of
"A Universe From Nothing", by Lawrence M. Krauss, in the New York Times from March 25, 2012, on page BR20 of the _Sunday Book Review_ ([web](http://www.nytimes.com/2012/03/25/books/review/a-universe-from-nothing-by-lawrence-m-krauss.html?))

[[!redirects Lawverian formalization of Hegelian metaphysics]]


Hegelian taco

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}

### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

[[!redirects taco]]
[[!redirects hegelian taco]]

## Idea

The **Hegelian taco** is food for thought from [[William Lawvere| William Lawvere's]] kitchen. More concretely, it is an eight-element [[graphic category|graphic monoid]] that intends to capture diagrammatically the essence of a certain configuration of stacking subcategories of a given category that occurs in Lawvere's mathematical rendering of [[Georg Hegel|Hegel's]] dialectical logic.

## Construction

The situation we start with is the following sequence of functors and adjunctions

$$
\mathcal{A}\overset{\hookleftarrow}{\underset{\hookleftarrow}{\to}} \mathcal{B}\overset{\hookleftarrow}{\underset{\hookleftarrow}{\to}}\mathcal{C}\overset{\hookleftarrow}{\underset{\hookleftarrow}{\to}} 1
$$

Here every column of arrows, numbered descendingly $2, 1, 0$ from left to right describes an essential localization $j_n\dashv f_n\dashv i_n$ with $j_n,i_n$ fully faithful. These yield a pair of idempotent [[adjoint modality|adjoint modalities]] whose functor parts are $j_n f_n\dashv i_n f_n$. Importantly, all the junctures vanish in the sense $f_n i_n=1$, since this composition corresponds to the inclusion of a subcategory followed by a projection back to the subcategory.

If we compose the functors appropriately we obtain four such adjunctions on $\mathcal{A}$ which correspond to reflective and coreflective embeddings of descending complexity of the chain of subcategories $\mathcal{A}\supset\mathcal{B}\supset\mathcal{C}\supset 1$:

$$
\begin{aligned}
id_\mathcal{A}&\dashv id_\mathcal{A} \\
L &\dashv R \\
l &\dashv r \\
0 &\dashv 1
\end{aligned}
$$

The respective endofunctors on $\mathcal{A}$ are defined as

$$
\begin{aligned}
1&:=i_2 i_1 i_0 f_0 f_1 f_2 \\
0&:=j_2 j_1 j_0 f_0 f_1 f_2 \\
r&:=i_2 i_1 f_1 f_2\\
l&:=j_2 j_1 f_1 f_2\\
R&:=i_2 f_2\\
L&:=j_2 f_2
\end{aligned}
$$

They are supposed to provide the elements of the taco monoid $M$. So far we have seven of them including $id_\mathcal{A}$. The eighth will show up when we set up the multiplication table, so let's do that:

1. The first case we consider is when we multiply two elements that appear on the same side of $\dashv$ like e.g. $L$ and $l$ that are both left adjoints: here $l$ is of lower complexity than $L$ as it corresponds to a smaller subcategory, namely $\mathcal{C}$. When we multiply, ${l}L$ or $L{l}$, at the junctures will arise in both cases $f_2 j_2$ which is just the 'splitting' of $L$ and therefore cancels and we retain the 'lower' functor $l={l}L=L{l}$. This happens in all cases where we multiply elements of the same 'laterality': the lower element absorbs the higher. This can be viewed as capturing the facet of 'preservation' present in Hegelian [[Aufhebung]]: the higher concept $L$ is just $l$ when restricted to the smaller subcategory.

2. Next we come to the cases where we multiply elements of opposite laterality of the same complexity e.g. like $L$ and $R$. Here the factors arising at the junctures cancel again and in these cases we retain always the left factor e.g. $L{R}=L$ and $R{L}=R$. One can view this as capturing the destructive nature of a Hegelian 'dialogue': 'objecting' $L$ to 'proposal' $R$ cancels $R$ and vice versa.

3. Now we come to the cases that express the constructive-progressive nature of Hegelian dialectic: namely, when we multiply elements of opposite laterality of different complexity e.g. like $R$ and $l$. Let us concentrate on the case that the difference in complexity is just a single step. First we note, that the dialogue situation is conceived of as being asymmetrical: the right adjoint acts as proponent whereas the left adjoint opponent contradicts. So in this particular formalization, the proponent tries to assimilate the opponent's view but not vice versa: in a 'situation' $l{r}=l$ where the 'proponent' $r$ is faced with 'opposition' $l$, she might just insist $r{l}r=r{l}=r$ but she might as well try to take $l$ into account i.e. change her view $r$ to $R$, with the 'higher order refinement' $R$ now 'accepting' $l$ in the sense that $R{l}=l$. This relation that $L_i$ not only factors through $L_{i+1}$ but through $R_{i+1}$, is called **resolution** of (the contradiction of) level $n$ by level $n+1$  (For further details see at [[Aufhebung]]). It gives us the equations ${r}0=0$ , $R{l}=l$, and (trivially) $id {L}=L$ for the three steps of sublations.

As we've said, the situation is handled somewhat asymmetrical in that we systematically care only that we can incorporate the negative view into the positive 'right adjoint' view but not the reverse. For the reverse cases ${l}1$ and ${L}r$ we _stipulate_: 1) ${l}1=1$ (this would hold, as often occurs in the applications, that there is an additional left adjoint $\pi\dashv j_2j_1\dashv f_1f_2$ since in that case $l$ as a composition of right adjoints would preserve the limit $1$)  and 2.) $q:={L}r$ which give us our eighth element!

So much for the 'philosophical' motivation behind the multiplication table that is going to follow in the next section!

Some might feel more comfortable with the information that, technically, the taco monoid intends to abstract the relations occuring between three essential localizations such that, starting from $0\dashv 1$, a higher localization resolves the preceding lower level in the sense that $R_{n+1} L_n= L_n$.

## Definition

The _Hegelian taco_ is the monoid on the set $\{id,0,1,l,r,L,R,q\}$ with the following multiplication table:

$$
\array{
  && 0 & 1 & l & q & r & L & R & id \\
  &&&&&&&&&\\
0 && 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\
1 && 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1  \\
l && 0 & 1 & l & l & l & l & l & l  \\
q && 0 & 1 & q & q & q & q & q & q  \\
r && 0 & 1 & r & r & r & r & r & r  \\
L && 0 & 1 & l & q & q & L & L & L  \\
R && 0 & 1 & l & r & r & R & R & R  \\
id&& 0 & 1 & l & q & r & L & R & id
}
$$

## Properties

$M$ is a [[graphic category|graphic monoid]] i.e. for all $x,y\in M$ we have $x y x=x y$, in particular all elements are [[idempotent]].

The poset of left ideals in $M$ is:

$$
\emptyset\subset \{ 0,1 \} \subset \{ 0,1,l,q,r \} \subset \{ 0,1,l,q,r,L,R \} \subset \{ 0,1,l,q,r,L,R,id \}
$$

They parametrize the essential localizations of the associated presheaf topos $Set^{M^{op}}$. The dimension theory that comes with this is detailed in [Kelly-Lawvere (1989)](#Kl89) and [Lawvere (1989)](#Law89b). In the present case, the result is that $Set^{M^{op}}$ is three-dimensional.

This motivates also the culinary allusion in the name, where the essential localization corresponding to $\mathcal{B}$ is seen as providing the two-dimensional 'sides' via its reflective and coreflective inclusions around the three-dimensional 'filling' $\mathcal{A}$. Of course, the one-dimensional pieces $l,q,r$ corresponding to $\mathcal{C}$ find their place on the sandwich as well. The following picture of the taco appears in [Lawvere (1989)](#Law89b), where further details may be found.

[[Hegel_taco.png:pic]]

Let us point out that $M$ can be viewed as 'cubical generalization' of the three-element graphic monoid $\Delta_1$ that underlies the (one-dimensional) presheaf topos of [[reflexive graph|reflexive graphs]] (for a detailed description of $\Delta_1$ and its topos see at [[graphic category]]). $\Delta_1$ encapsulates the diagram

$$
\mathcal{A}\overset{\hookleftarrow}{\underset{\hookleftarrow}{\to}} \mathcal{B}
$$

that expresses abstractly a [[adjoint cylinder|cylinder configuration]] corresponding to 'opposition' or 'conflict' from the dialogical point of view.[^Law]

[^Law]: Lawvere expands a bit on the diagrammatic perspective on $\Delta_1$ in [Lawvere (1996)](#Law96).

## Remarks

* If one takes into account also the natural transformations arising from the adjunctions one can turn $M$ into a finite [[monoidal category]].

* The main reference for the taco is [Lawvere (1989, pp.70-73)](#Law89b). It is mentioned in Lawvere ([1991](#Law91),[2003](#Law02)) as well.

## Related entries

* [[graphic category]]

* [[graphic topos]]

* [[shelf]]

* [[Aufhebung]]

* [[graph]]

* [[Science of Logic]]

* [[adjoint cylinder]]

* [[2-category of adjunctions]]

* [[Adj]]

## References

* {#KL89}[[G. M. Kelly]], [[F. W. Lawvere]], _On the complete lattice of essential localizations_ , Bull. Soc. Math. de Belgique **XLI** (1989) 289-319 [[[Kelly-Lawvere_EssentialLocalizations.pdf:file]]]

* [[F. W. Lawvere]], _Display of graphics and their applications, as exemplified by 2-categories and the Hegelian "taco"_  Proceedings of the first international conference on algebraic methodology and software technology University of Iowa, May 22-24 1989, Iowa City, pp. 51-74. {#Law89b}  ([pdf](https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1989-display-of-graphics-and-their-applications-exemplifed-by-2-categories-and-the-hegelian-taco.pdf))

* [[F. W. Lawvere]], _More on graphic toposes_, Cah. Top. Géom. Diff. Cat. **XXXII** no. 1 (1991) pp.5-10. ([pdf](http://archive.numdam.org/article/CTGDC_1991__32_1_5_0.pdf)) {#Law91}

* {#Law96}[[F. W. Lawvere]], _Unity and identity of opposites in calculus and physics_ , App. Cat. Struc **4** (1996) pp.167-174.  ([pdf](https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1996-unity-and-identity-of-opposites-in-calculus-and-physics.pdf))

* [[F. W. Lawvere]], _Linearization of graphic toposes via Coxeter groups_, JPAA **168** (2002) pp. 425-436. ([pdf](http://www.sciencedirect.com/science/article/pii/S0022404901001074/pdfft?md5=a4ca9bc67df6ae63ddf53c559bd71315&pid=1-s2.0-S0022404901001074-main.pdf)) {#Law02}


Heidegger's topology

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### Context
#### Philosophy
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[[!include philosophy - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

_But poetry that thinks is in truth the topology of Being..._

## Related concepts

* [[Heidegger]]

* [[topology]]

## References

* [[Jeff Malpas]], _Heidegger's topology_, Cambridge, Mass.: MIT Press, 2006.

* [[Jeff Malpas]], _From the transcendental to the 'topological': Heidegger on Ground, Unity, and Limit._ In Jeff Malpas (ed.), From Kant to Davidson: Philosophy and the Idea of the Transcendental (London: Routledge, 2002).

* [[Jeff Malpas]], _Heidegger's topology of being._ In Transcendental Heidegger, ed. Steven Galt Crowel and Jeff Malpas, ed., 119-134. Stanford: Stanford University Press, 2006.

* [[Edward Casey]], _The fate of Place_ (Berkeley: University of California Press, 1996).


height of a formal group

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###Context###
#### $\infty$-Lie theory
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[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[moduli stack of formal groups]] $\mathcal{M}_{FG}$ admits a natural [[stratification]] whose open [[strata]] are labeled by a [[natural number]] called the _height_ of formal groups.

The [[complex oriented cohomology theories]] associated to these [[formal groups]] by the [[Landweber exact functor theorem]] accordingly also inherit such an integer label, called _[[chromatic filtration]]_. Studying this is the topic of [[chromatic homotopy theory]].

## Definition

Let $R$ be a [[commutative ring]] and fix

$$
  f(x,y) \in R [ [ x,y ] ]
$$

a [[formal group law]] over $R$.

+-- {: .num_defn}
###### Definition

For every $n \in \mathbb{N}$ the **$n$-series** of $f$

$$
  [n](t) \in R[ [ t ] ]
$$

is defined [[recursion|recursively]] by

1. if $n = 0$ then $[n](t) = 0$;

1. if $n \gt 0$ then $[n](t) = f([n-1](t),t)$.

=--

Now fix $p \in \mathbb{N}$ a [[prime number]],

+-- {: .num_defn}
###### Definition

Write $v_n$ for the [[coefficient]] of $t^{p^n}$ in the $p$-series $[p]$ of $f$.

=--

+-- {: .num_defn}
###### Definition

Say that $f$

* has **height** $\geq n$ if $v_i = 0$ for $i \lt n$;

* has **height exactly** $n$ if it has height $\geq n$ and $v_n \in R$ is invertible.

=--

For instance ([Lurie 10, lecture 12, def. 13](#Lurie10)).

## Examples

+-- {: .num_example}
###### Example

For $f(x,y) = x + y + x y$ the [[formal multiplicative group]] the $n$-series is

$$
  [n](t) = (1+t)^n - 1
  \,.
$$

If $p = 0$ in $R$ then

$$
  [p](t) = (1+t)^p - 1 = t^p
$$

and thus $f$ has height exactly 1.

=--

For instance ([Lurie 10, lecture 12, example 16](#Lurie10)).

+-- {: .num_example}
###### Example

An [[elliptic curve]] over a [[field]] of [[positive number|positive]] [[characteristic]] whose [[formal group law]] has [[height of a formal group]] equal to 2 is called a _[[supersingular elliptic curve]]_. Otherwise the height equals 1 and the elliptic curve is called _ordinary_.

=--

## Properties

### Chromatic height filtation

The height of formal groups induces the [[height filtration]] on the [[moduli stack of formal groups]].

[[!include Lurie spectral sequences -- table]]

## Related pages

* [[moduli stack of formal groups]]

* [[chromatic homotopy theory]]

* [[Morava K-theory]]

## References

* {#Lurie10} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series (2010) ([lecture notes](http://www.math.harvard.edu/~lurie/252x.html)), Lecture 12 _Heights and formal groups_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture12.pdf))

[[!redirects height of a formal group law]]

[[!redirects height of formal groups]]


height of a variety

#Contents#
* table of contents
{:toc}

## Idea

The _height_ of a [[variety]] should reflect how close the variety is to being ordinary and other arithmetic properties.

## Definition

Let $X$ be a [[smooth scheme|smooth]] [[proper scheme|proper]] $n$ [[dimension|dimensional]] [[variety]] over an [[algebraically closed field]] $k$ of [[characteristic]] $p$. Then one can define the [[Artin-Mazur formal group]] $\Phi$. Since it is a one-dimensional [[formal group]], it is completely determined up to [[isomorphism]] by its [[height]]. This is the **height of the variety** . The height could be infinite if $\Phi\simeq \widehat{\mathbb{G}_a}$, otherwise $\Phi$ is a [[p-divisible group]].

### Examples

For an [[elliptic curve]], the height is either $1$ in which case it is ordinary, or $2$ in which case it is supersingular.

A [[Calabi-Yau variety]] of any dimension is ordinary if and only if it has height $1$. For [[K3 surfaces]] the height is less than or equal to $10$ or infinite, but for all higher dimensional Calabi-Yau varieties the height has no known bound. Infinite height Calabi-Yau varieties are known as supersingular.

The height of an [[abelian variety]] depends on its $p$-rank, but must be $1$, $2$, or infinite.

## Relation to Witt Cohomology

Let $\mathcal{W}$ be the sheaf of Witt vectors on a variety $X$ satisfying the conditions above. If $X$ has finite height, then the [[Dieudonne module]] of the [[Artin-Mazur formal group]] is isomorphic to $H^n(X, \mathcal{W})$. By standard Dieudonne theory, $D(\Phi)$ is a free of rank $ht(X)$ module over $W$, so $ht(X)=\dim_K H^n(X, \mathcal{W})\otimes K$ where $K$ is the fraction field of $W$.

One consequence of the above is that $X$ is supersingular (of infinite height) if $H^n(X, \mathcal{W})$ is not a finite-type $W$-module. It is possible also that it is a torsion module in which case $H^n(X, \mathcal{W})\otimes K=0$ and again we can conclude that $X$ is of infinite height (since if $X$ were of finite height it would be a free module).

## Relation to Crystalline Cohomology

Suppose that $X$ is a variety with the above hypotheses, then the torsion free part of the [[crystalline cohomology]] $H^n_{crys}(X/W)$ is a [[Cartier module]] under the action of Frobenius. We can consider the part with slopes less than $1$, i.e. $H^n_{crys}(X/W)\otimes_W K_{[0,1)}$. The dimension of this is the height of $X$.

## Related concepts

* [[formal group]]

* [[Artin-Mazur formal group]]

## References

* [[Michael Artin]], [[Barry Mazur]], _Formal Groups Arising from Algebraic Varieties_, [numdam](http://www.numdam.org/item?id=ASENS_1977_4_10_1_87_0), [MR56:15663](http://www.ams.org/mathscinet-getitem?mr=56:15663)
 {#ArtinMazur}


Heine-Borel theorem

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### Context
#### Analysis
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[[!include analysis - contents]]
=--
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The classical _Heine--Borel theorem_ identifies those [[topological subspaces]] of [[Cartesian spaces]] ($\mathbb{R}^n$s) that are [[compact space|compact]] in terms of simpler properties.  A generalisation applies to all [[metric spaces]] and even to [[uniform spaces]].

## Versions

This is the classical theorem:

+-- {: .num_theorem #classical}
###### Theorem

Let $S$ be a [[topological subspace]] of a [[Cartesian space]] ($S \subset \mathbb{R}^n$).  Then $S$ is a [[compact topological space]] (with the [[induced topology]]) precisely if it is [[closed subset|closed]] and [[bounded subset|bounded]] in $\mathbb{R}^n$.
=--

It\'s easy to prove that $S$ is closed precisely if it is a [[complete metric space]] as with the induced [[metric]], and similarly $S$ is bounded precisely if it is [[totally bounded metric space|totally bounded]].  This gives the next version:

+-- {: .num_theorem #cartesian}
###### Theorem

Let $S$ be a [[topological subspace]] of a [[Cartesian space]] ($S \subset \mathbb{R}^n$).  Then $S$ is a [[compact topological space]] (with the [[induced topology]]) precisely if it is [[complete space|complete]] and [[totally bounded space|totally bounded]] (with the [[induced metric]]).
=--

This refers entirely to $S$ as a metric space in its own right.  In fact it holds much more generally than for subspaces of a cartesian space:

+-- {: .num_theorem #metric}
###### Theorem

Let $S$ be a [[metric space]].  Then $S$ is [[compact space|compact]] precisely if it is [[complete space|complete]] and [[totally bounded space|totally bounded]].
=--

This theorem refers only to uniform properties of $S$, and in fact a further generalistion is true:

+-- {: .num_theorem #uniform}
###### Theorem

Let $S$ be a [[uniform space]].  Then $S$ is [[compact space|compact]] precisely if it is [[complete space|complete]] and [[totally bounded space|totally bounded]].
=--

The hard part is proving that a complete totally bounded space is compact; the converse is easy.

We could also try to generalise Theorem \ref{classical} to subspaces of other metric spaces, but this fails: every compact subspace of a metric space is closed and bounded (which is the easy direction), but not conversely.

Another variant of these statements is:

* _[[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]]_

## Logical status

In the old days, one called a closed and bounded [[interval]] in the [[real line]] 'compact'; once closedness and boundedness were generalised from intervals to arbitrary [[subsets]] (of the real line), the definition of 'compact' also generalised.  The content of Theorem \ref{classical}, then, is that this condition is equivalent to the modern definition of 'compact' using [[open covers]], and indeed the modern definition was only derived afterwards as a name for the conclusion of the theorem.

In [[constructive mathematics]], one sees several definitions of 'compact', which may make the theorem provable, refutable, or undecidable in various constructive systems.  Using the open-cover definition of 'compact', Theorems \ref{classical} and \ref{cartesian} are equivalent to the [[fan theorem]] (and so hold in [[intuitionism]]), but Theorems \ref{metric} and \ref{uniform} are stronger and [[Jan Brouwer|Brouwer]] could not prove them, leading him to *define* 'compact' (for a metric space) to mean complete and totally bounded.  In other literature, one sometimes sees the abbreviation 'CTB' used instead.  In [[Russian constructivism]], already Theorems \ref{classical} and \ref{cartesian} can be refuted using the open-cover definition, but CTB spaces are still important.

In [[locale theory]] and other approaches to [[pointless topology]], the open-cover definition of 'compact' is clearly correct, and the failure of CTB spaces to be compact (constructively) may be seen as a consequence of working with points.  Already in Bishop\'s weak system of constructivism, every CTB metric space $X$ gives rise to a compact locale, which classically (assuming [[excluded middle]] and [[dependent choice]]) is the [[locale of open subsets]] of $X$ but constructively requires a more nuanced construction; see [Vickers](#CTBlocale).

{#InDTT} In [[dependent type theory]], the open-cover definition of 'compact' holds as well, but one has to use the [[higher inductive type|higher]] [[inductive family]] of [[inductive covers]] instead of the usual pointwise definitions of covers ([UFP13, §11.5](#UFP13)).

## Proofs
 {#Proofs}

For definiteness, we restate the versions which we prove:

+-- {: .num_lemma #CompactClosedInterval}
###### Lemma
**(closed interval is compact)**

In [[classical mathematics]]:

For any $a \lt b \in \mathbb{R}$ the [[closed interval]]

$$
  [a,b] \subset \mathbb{R}
$$

regarded with its [[topological subspace|subspace topology]] is a [[compact topological space]].

=--

+-- {: .proof}
###### Proof

Since all the closed intervals are [[homeomorphism|homeomorphic]]
it is sufficient to show the statement for $[0,1]$. Hence let $\{U_i \subset [0,1]\}_{i \in I}$ be an
open cover. We need to show that it has an open subcover.

Say that an element $x \in [0,1]$ is _admissible_ if the closed sub-interval $[0,x]$ is covered by
finitely many of the $U_i$. In this terminlogy, what we need to show is that $1$ is admissible.

Observe from the definition that

1. 0 is admissible,

1. if $y \lt x \in [0,1]$ and $x$ is admissible, then also $y$ is admissible.

This means that the set of admissible $x$ forms either an [[open interval]] $[0,g)$ or a [[closed interval]]
$[0,g]$, for some $g \in [0,1]$. We need to show that the latter is true, and for $g = 1$.
We do so by observing that the alternatives lead to contradictions:

1. Assume that the set of admissible values were an open interval $[0,g)$. By assumption there would be
   a finite subset $J \subset I$ such that $\{U_i \subset [0,1] \}_{i \in J \subset I}$ were a finite
   open cover of $[0,g)$. Accordingly, since there is some $i_g \in I$ such that $g \in U_{i_g}$, the union
   $\{U_i\}_{i \in J } \sqcup \{U_{i_g}\}$ were a finite cover of the closed interval $[0,g]$,
   contradicting the assumption that $g$ itself is not admissible (since it is not contained in $[0,g)$).

1. Assume that the set of admissible values were a closed interval $[0,g]$ for $g \lt 1$.
   By assumption there would then be a finite set $J \subset I$ such that $\{U_i \subset [0,1]\}_{i \in J \subset I}$
   were a finite cover of $[0,g]$. Hence there would be an index $i_g \in J$ such that $g \in U_{i_g}$.
   But then by the nature of open subsets in the Euclidean space $\mathbb{R}$, this $U_{i_g}$ would also
   contain an open ball $B^\circ_g(\epsilon) = (g-\epsilon, g + \epsilon)$. This would mean that
   the set of admissible values includes the open interval $[0,g+ \epsilon)$, contradicting the assumption.

This gives a [[proof by contradiction]].

=--

+-- {: .num_prop #BorelHeine}
###### Proposition
**([[Heine-Borel theorem]] classically)**

For $n \in \mathbb{N}$, regard $\mathbb{R}^n$ as the $n$-dimensional [[Euclidean space]],
regarded as a [[topological space]] via its [[metric topology]].

Then in [[classical mathematics]] for a [[topological subspace]] $S \subset \mathbb{R}^n$ the following are equivalent:

1. $S$ is  [[compact topological space|compact]],

1. $S$ is [[closed subset|closed]] and [[bounded subset|bounded]].

=--

+-- {: .proof #BorelHeineProof}
###### Proof

First consider a [[subset]] $S \subset \mathbb{R}^n$ which is closed and bounded. We need to show that
regarded as a [[topological subspace]] it is [[compact topological space|compact]].

The assumption that $S$ is bounded by (hence contained in) some [[open ball]] $B^\circ_x(\epsilon)$ in $\mathbb{R}^n$
implies that it is contained in $\{ (x_i)_{i = 1}^n \in \mathbb{R}^n \,\vert\, -\epsilon \leq x_i \leq \epsilon \}$.
This topological subspace is homeomorphic to the $n$-cube
$[-\epsilon, \epsilon]^n$. Since the closed interval $[-\epsilon, \epsilon]$
is compact by lemma \ref{CompactClosedInterval}, the [[Tychonoff theorem]]
implies that this $n$-cube is compact.

Since [[subsets are closed in a closed subspace precisely if they are closed in the ambient space]] the closed subset $S \subset \mathbb{R}^n$ is still closed as a subset $S \subset [-\epsilon, \epsilon]^n$.
Since [[closed subspaces of compact spaces are compact]] this implies that $S$ is compact.

Conversely, assume that $S \subset \mathbb{R}^n$ is a compact subspace. We need to show that it is closed and bounded.

The first statement follows since
the [[Euclidean space]] $\mathbb{R}^n$ is [[Hausdorff topological space|Hausdorff]]
and since [[compact subspaces of Hausdorff spaces are closed]].

Hence what remains is to show that $S$ is bounded.

To that end, choose any [[positive number|positive]] [[real number]] $\epsilon \in \mathbb{R}_{\gt 0}$
and consider the [[open cover]] of all of $\mathbb{R}^n$ by the open [[n-cubes]]

$$
  (k_1-\epsilon, k_1+1+\epsilon)
    \times
  (k_2-\epsilon, k_2+1+\epsilon)
    \times
      \cdots
    \times
  (k_n-\epsilon, k_n+1+\epsilon)
$$

for [[n-tuples]] of [[integers]] $(k_1, k_2 , \cdots, k_n ) \in \mathbb{Z}^n$.
The restrictions of these to $S$ hence form an open cover of the subspace $S$. By the assumption that
$S$ is compact, there is then a finite subset of $n$-tuples of integers such that the
corresponding $n$-cubes still cover $S$. But the union of any finite number of bounded closed $n$-cubes
in $\mathbb{R}^n$ is clearly a bounded subset, and hence so is $S$.

=--

## Related theorem

* [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]]

* [[Tietze extension theorem]]

* [[Tychonoff theorem]]

* [[topological invariance of dimension]]

* [[Brouwer's fixed point theorem]]

* [[Jordan curve theorem]]

## References

[According to Wikipedia](https://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem#History_and_motivation), the theorem was first proved by [[Pierre Cousin]] in 1895.  It is named after [[Eduard Heine]] (who used it but did not prove it) and [[Émile Borel]] (who proved a limited version of it), an instance of [[Stigler's law of eponymy]].

A proof is spelled out for instance at

* [ProofWiki](http://www.proofwiki.org/wiki/Main_Page),  _Heine-Borel Theorem [(General case)](http://www.proofwiki.org/wiki/Heine-Borel_Theorem_%28General_Case%29),  [(Special case)](http://www.proofwiki.org/wiki/Heine-Borel_Theorem_%28Special_Case%29)_

On constructing a compact locale from a CTB metric space:

*  {#CTBlocale} [[Steve Vickers]], [Localic completion of generalized metric spaces](http://www.tac.mta.ca/tac/volumes/14/15/14-15abs.html)

On the Heine-Borel theorem in [[dependent type theory]] using [[inductive covers]], see section 11.5 of:

* {#UFP13} [[Univalent Foundations Project]]: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]*, The Institute for Advanced Study (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

[[!redirects Heine-Borel theorem]]
[[!redirects Heine-Borel theorems]]
[[!redirects Heine--Borel theorem]]
[[!redirects Heine--Borel theorems]]
[[!redirects Heine–Borel theorem]]
[[!redirects Heine–Borel theorems]]


Heinrich Behnke

* [Wikipedia entry](https://en.wikipedia.org/wiki/Heinrich_Behnke)

## Selected writings

On open [[Riemann surfaces]] as [[Stein manifolds]]:

* {#BehnkeStein47} [[Heinrich Behnke]], [[Karl Stein]], *Entwicklung analytischer Funktionen auf Riemannschen Flächen*, Mathematische Annalen volume 120, pages 430–461 (1947) ([doi:10.1007/BF01447838](https://doi.org/10.1007/BF01447838))

category: people


Heinrich Brandt

* [Wikipedia entry](https://en.wikipedia.org/wiki/Heinrich_Brandt) (de)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=47792)

## Selected writings

Introducing the notion of (connected) [[groupoids]] ([[Brandt groupoids]]):

* {#Brandt27} [[Heinrich Brandt]], *Über eine Verallgemeinerung des Gruppenbegriffes*, Mathematische Annalen **96** 1 (1927) 360-366 [[doi:10.1007/BF01209171](https://doi.org/10.1007%2FBF01209171)]

category: people


Heinrich Kleisli

A Swiss mathematician, pronounced /ˈklaɪsli/.

* [Wikipedia entry](https://en.wikipedia.org/wiki/Heinrich_Kleisli)

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=18361)

* [[Kleisli category]], [[Kleisli 2-category]], [[Kleisli object]], [[Kleisli category of a comonad]]

## Selected writings

Introducing the notion now known as thew *[[Kleisli category]]*:

* {#Kleisli65} [[Heinrich Kleisli]], *Every standard construction is induced by a pair of adjoint functors*, Proc. Amer. Math. Soc. **16**, AMS (1965) 544-546 [[ISSN0002-9939](http://www.ams.org/journals/proc/1965-016-03/S0002-9939-1965-0177024-4/), [jstor:2034693](https://www.jstor.org/stable/2034693)]

On [[star-autonomous categories]] of [[unit balls]] in [[Banach spaces]]:

* [[Michael Barr]], [[Heinrich Kleisli]], *Topological balls*, [[Cahiers|Cahiers Topologie Géométrie Différentielle Catégorique]], **40** 1 (1999) 3–20 [[numdam:CTGDC_1999__40_1_3_0](http://www.numdam.org/item/?id=CTGDC_1999__40_1_3_0)]

category: people


Heinrich Leutwyler

* [webpage](http://www.leutwyler.itp.unibe.ch/)

* [Wikipedia entry](https://de.m.wikipedia.org/wiki/Heinrich_Leutwyler)

## Selected writings

On [[chiral perturbation theory]]:

* [[Jürg Gasser]], [[Heinrich Leutwyler]], _Quark masses_, Physics Reports Volume 87, Issue 3, July 1982, Pages 77-169 (doi:10.1016/0370-1573(82)90035-7)

* [[Jürg Gasser]], [[Heinrich Leutwyler]], _Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark_, Nucl. Phys. B 250 (1985) 465-516, ([spire:200027](https://inspirehep.net/literature/200027), doi:10.1016/0550-3213(85)90492-4))

* [[Jürg Gasser]], [[Heinrich Leutwyler]], _Chiral perturbation theory to one loop_, Annals of Physics
Volume 158, Issue 1, November 1984, Pages 142-210 (doi:10.1016/0003-4916(84)90242-2)

* [[Heinrich Leutwyler]], _On the Foundations of Chiral Perturbation Theory_, Annals Phys. 235 (1994) 165-203 ([arXiv:hep-ph/9311274](https://arxiv.org/abs/hep-ph/9311274))

category: people


Heinrich Tietze

* [Wikipedia entry](https://en.wikipedia.org/wiki/Heinrich_Franz_Friedrich_Tietze) ([deutsch](https://de.wikipedia.org/wiki/Heinrich_Tietze))

## Selected writings

Introducing the [[Tietze extension theorem]]:

* [[Heinrich Tietze]], *Über Funktionen, die auf einer abgeschlossenen Menge stetig sind*, Journal für die reine und angewandte Mathematik, **145** (1915) [[doi:10.1515/crll.1915.145.9](https://doi.org/10.1515/crll.1915.145.9)]

category: people


Heinrich Weber

Heinrich Martin Weber was a German mathematician.

* [Wikipedia page](https://en.wikipedia.org/wiki/Heinrich_Martin_Weber).

## Selected writings

The original article that gives a definition equivalent to the modern definition of a *[[group]]*

* [[Heinrich Weber]], *Beweis des Satzes, dass jede eigentlich primitive quadratische Form unendlich viele Primzahlen darzustellen fähig ist*, Mathematische Annalen 20:3 (1882), 301–329 ([doi:10.1007/bf01443599](http://dx.doi.org/10.1007/bf01443599))

For the original article that gives a definition of an arbitrary [[field]] (as opposed to subfields of the complex numbers):

* [[Heinrich Weber]] (1893), "Die allgemeinen Grundlagen der Galois'schen Gleichungstheorie", Mathematische Annalen (in German), 43 (4): 521–549, $[$[doi:10.1007/BF01446451](https://doi.org/10.1007%2FBF01446451), [ISSN:0025-5831](https://www.worldcat.org/issn/0025-5831), [JFM:25.0137.01](https://zbmath.org/?format=complete&q=an:25.0137.01), [S2CID:120528969](https://api.semanticscholar.org/CorpusID:120528969)$]$

category: people


Heinz Hopf

* [Wikipedia entry](http://en.wikipedia.org/wiki/Heinz_Hopf)

## Selected writings

On [[Clifford-Klein space forms]]:

* [[Heinz Hopf]], *Zum Clifford-Kleinschen Raumproblem*,  Math. Ann. 95, 313–339 (1926).  ([doi:10.1007/BF01206614](https://doi.org/10.1007/BF01206614), [pdf](https://link.springer.com/content/pdf/10.1007/BF01206614.pdf))

Introducing what came to be called the [[Hopf fibrations]]:

* {#Hopf31} [[Heinz Hopf]], *Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche*, Mathematische Annalen **104**  (1931) 637–665 [[doi:10.1007/BF01457962](https://doi.org/10.1007/BF01457962)]

* {#Hopf35} [[Heinz Hopf]], *Über die Abbildungen von Sphären auf Sphäre niedrigerer Dimension*, Fundamenta Mathematicae **25** 1 (1935) 427-440 [[eudml:212801](https://eudml.org/doc/212801)]

## Related entries

* [[Chern-Weil theory]]

* [[H-space]]

* [[Hopf construction]]

* [[Hopf fibration]]

* [[Hopf degree theorem]]

* [[Hopf invariant]], [[Hopf invariant one]]

* [[Poincaré–Hopf theorem]]

* [[Hopf algebra]]

category: people


Heinz-Peter Breuer

* [InSpire page](https://inspirehep.net/authors/2126702)

## Selected writings

On [[open quantum system|open]] [[quantum mechanics|quantum mechanical]] [[quantum systems|systems]]:

* [[Heinz-Peter Breuer]], [[Francesco Petruccione]], *The Theory of Open Quantum Systems*, Oxford University Press (2007) [[doi:10.1093/acprof:oso/9780199213900.001.0001](https://doi.org/10.1093/acprof:oso/9780199213900.001.0001)]

category: people


Heisenberg double

Given a field $k$ and two $k$-[[bialgebra]]s $A$ and $B$ with [[Hopf pairing]] $\lt, \gt : A\otimes B\to k$, one defines a left [[Hopf action]] $\blacktriangleright$ of $B$ on $A$ by formulas
$$
b\blacktriangleright a = \sum \lt b, a_{(2)}\gt a_{(1)}= (\lt,\gt \otimes \id)(b\otimes \tau\Delta_A(a))
$$
one forms the __Heisenberg double__ corresponding to these data as the crossed product algebra ("smash product") $A\sharp B$ associated to the Hopf action $\blacktriangleright$. If $H$ is finite-dimensional then the Heisenberg double has a structure of a [[scalar extension bialgebroid]].

For example if $A = S(V)$ is the symmetric (Hopf) algebra on a finite-dimensional vector space $V$, and $B$ its algebraic dual $(S(V))^*\cong \hat{S}(V^*)$, considered as its dual topological Hopf algebra, the result is the [[Weyl algebra]] of [[regular differential operators]], completed with respect to the filtration corresponding to the degree of differential operator. If $B$ is just the finite dual of $S(V)$ which is a usual Hopf algebra, then there is no completion, of course.

* [[Jiang-Hua Lu]], _On the Drinfeld double and the Heisenberg double of a Hopf algebra_, Duke Math. J. __74__ (1994) 763–776.

In the following paper there is an example showing that the Heisenberg double $A^*\sharp A$ has a structure of a Hopf algebroid over $A^*$; moreover $A^*$ can be replaced by any module algebra over the [[Drinfel'd double]] $D(A)$:

* [[Jiang-Hua Lu]], _Hopf algebroids and quantum groupoids_, Int. J. Math. __7__, 1 (1996) pp. 47-70, [q-alg/9505024](http://arxiv.org/abs/q-alg/9505024), [MR95e:16037](http://www.ams.org/mathscinet-getitem?mr=95e:16037), [doi](https://doi.org/10.1142/S0129167X96000050)

An example of an infinite dimensional analogue coming from Lie algebras is treated in

* S. Meljanac, [[Z. Škoda]], _Lie algebra type noncommutative phase spaces are Hopf algebroids_, [arxiv:1409.8188](https://arxiv.org/abs/1409.8188)

which partly refers to previous paper (which however neglects the issues related to completions, and has some expositional errors)

* [[Zoran Škoda]], _Heisenberg double versus deformed derivatives_, Int. J. of Modern Physics A __26__, Nos. 27 & 28 (2011) 4845--4854, [arXiv:0909.3769](https://arxiv.org/abs/0909.3769), [doi](https://doi.org/10.1142/S0217751X11054772)

The canonical element in the Heisenberg double satisfies a [[pentagon relation]], which is a version of pentagon relation for [[multiplicative unitaries]] of Baaj-Skandalis.  Kashaev has explained the pentagon relations for quantum [[dilogarithm]] as coming from the pentagon for the canonical element in the double.

* R.M. Kashaev, _Heisenberg double and the pentagon relation_, St. Petersburg Math. J. 8 (1997) 585--592 [q-alg/9503005](https://arxiv.org/abs/q-alg/9503005).
* [[Gigel Militaru]], _Heisenberg double, pentagon equation, structure and classification of finite-dimensional Hopf algebras_, J. London Math. Soc. (2) 69 (2004) 44--64 ([doi](https://doi.org/10.1112/S0024610703004897)).

Miscellaneous articles on Heisenberg doubles

* F. Panaite, _Doubles of (quasi) Hopf algebras and some examples of quantum groupoids and vertex groups related to them_, [math.QA/0101039](https://arxiv.org/abs/math/0101039)

* A.M. Semikhatov, _A Heisenberg double addition to the logarithmic Kazhdan–Lusztig duality_, [arXiv:0905.2215](http://arxiv.org/abs/0905.2215).

* A.M. Semikhatov, _Yetter--Drinfeld structures on Heisenberg doubles and chains_, [arXiv:0908.3105](https://arxiv.org/abs/0908.3105)

There are some generalizations or Heisenberg doubles in different setups

* [[Mikhail Kapranov]], _Heisenberg doubles and derived categories_, J. Alg. 202, 712--744 (1998), arXiv:[q-alg/9701009](https://arxiv.org/abs/q-alg/9701009).

* Daniele Rosso, Alistair Savage, _Twisted Heisenberg doubles_, [arxiv/1405.7889](http://arxiv.org/abs/1405.7889)

* Robert Laugwitz, _Braided Drinfeld and Heisenberg doubles_, J. Pure Appl. Alg. __219__:10 (2015) 4541-4596 [doi](https://doi.org/10.1016/j.jpaa.2015.02.031)

category: algebra
[[!redirects Heisenberg doubles]]


Heisenberg group

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###Context###
#### Group Theory
+-- {: .hide}
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=--
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Heisenberg group_ (or _Weyl-Heisenberg group_) is a [[Lie group]] [[Lie integration|integrating]] a _[[Heisenberg Lie algebra]]_.

There are several such, and so the conventions in the literature vary slightly as to which one to pick by default.

The Heisenberg group historically originates in and still has its strongest ties to [[quantum physics]]: there it is a group of [[unitary operators]] acting on the [[space of states]] induced from those [[observables]] on a linear [[phase space]] -- a [[symplectic vector space]] -- which are given by linear or by constant functions.
So any Heisenberg group is a subgroup of a group of observables in certain simple examples of quantum mechanical systems.

## Definition

A **Heisenberg group** is a [[Lie group]] whose [[Lie algebra]] is a [[Heisenberg Lie algebra]].

We spell out some special cases in detail.

### $H_3$ in components

The simplest non-trivial example of a Heisenberg group is the unique [[simply connected topological space|simply connected]] Lie integration of the [[Heisenberg Lie algebra]] $Heis(\mathbb{R}^2, \omega = d p \wedge dq)$ on the the canonical 2-dimensional [[symplectic vector space]] $\mathbb{R}^2$ with canonical [[coordinates]] $(p,q)$.

This Heisenberg Lie algebra is [[generators and relations|generated]] from 3 elements, here to be denoted $\mathbf{p}, \mathbf{q},\mathbf{e}$, subject to the single non-trivial [[Lie bracket]]

$$
  [\mathbf{q}, \mathbf{p}] = \mathbf{e}
  \,.
$$

The corresponding Heisenberg group is usually denoted $H_3$:

$$
  H_3 \coloneqq \exp Heis(\mathbb{R}^2 , d p \wedge d q)
  \,.
$$

The underlying [[smooth manifold]] of this [[Lie group]] is the [[Cartesian space]] $\mathbb{R}^3$. A general element may be written as

$$
  g_{a,b,c}
   =
  \exp(a \mathbf{q} + b \mathbf{p} )\exp(c \mathbf{e})
$$

with $a, b , c\in \mathbb{R}$. In terms of this notation the product in the group is given (by the [[Baker-Campbell-Hausdorff formula]]) by

$$
  \exp(a_1 \mathbf{q} + b_1 \mathbf{p})
  \exp(c_1 \mathbf{e})
  \cdot
  \exp(a_2 \mathbf{q} + b_2 \mathbf{p})
  \exp(c_2 \mathbf{e})
  =
  \exp((a_1 + b_1) \mathbf{q} + (b_1 + c_1) \mathbf{p})
  \exp(c_1 + c_2 - \frac{1}{2}(a_1 b_2 - a_2 b_1) \mathbf{e})
  \,.
$$

A discrete [[quotient]] group of this, which still has the same [[Lie algebra]], has as underlying manifold $\mathbb{R}^2 \times U(1)$ (the second factor being the [[circle group]]), with the projection

$$
  \mathbb{R}^2 \times \mathbb{R}
  \to
  \mathbb{R}^2 \times U(1)
$$

being quotienting by $\mathbb{Z}$: $U(1) \simeq \mathbb{R} / \mathbb{Z}$.

If the circle group is instead thought of as the unit circle in the complex plane, then this quotient is thought of as the exponential map $\exp(2 \pi i (-)) : \mathbb{R} \to U(1)$. In terms of this the group elements in the quotient read

$$
  g_{a, b, c}
   =
  \exp(a \mathbf{q} + b \mathbf{p} )\exp(2 \pi i c )
$$

and their product is

$$
  \exp(a_1 \mathbf{q} + b_1 \mathbf{p})
  \exp(2 \pi i c_1 )
  \cdot
  \exp(a_2 \mathbf{q} + b_2 \mathbf{p})
  \exp(2 \pi i c_2 )
  =
  \exp((a_1 + b_1) \mathbf{q} + (b_1 + c_1) \mathbf{p})
  \exp(2 \pi i(c_1 + c_2 - \frac{1}{2}(a_1 b_2 - a_2 b_1)))
  \,.
$$

While the Lie algebra is still the same real [[Heisenberg Lie algebra]] as before, it is now suggestive to write the Lie bracket as

$$
  [q,p] = i
  \,.
$$

This is the way the relation appears in texts on [[quantum physics]].
### For a symplectic vector space

Generally, there is a Heisenberg group $H(V, \omega)$ associated to
any [[symplectic vector space]] $(V, \omega)$.

Regard $V$ with its [[abelian group]] structure underlying its [[vector space]] structure.

The Heisenberg group $H(V,\omega)$ is the space $V \times U(1)$ (for $U(1)$ the [[circle group]]) equipped with the group product

$$
  (v_1, c_1) \cdot (v_2, c_2)
  =
  (v_1 + v_2, c_1 c_2 \exp(2 \pi i \omega(v,w)))
  \,.
$$

## Properties

### Relation to Poisson algebra
 {#RelationToPoissonBracket}

A [[symplectic vector space]] $(V, \omega)$ is in particular a [[symplectic manifold]]. Accordingly its algebra of [[smooth functions]] $C^\infty(V)$ is a [[Poisson algebra]]. The [[Lie algebra]] underlying the Poisson algebra contains the [[Heisenberg Lie algebra]] as the subspace with is the [[direct sum]] of the [[linear functions]] $V^* \hookrightarrow C^\infty(V)$ and the constant functions $\mathbb{R} \hookrightarrow C^\infty(V)$.

For more details in this at _[[Heisenberg Lie algebra]]_ the section _[Relation to Poisson algebra](#http://ncatlab.org/nlab/show/Heisenberg%20Lie%20algebra#RelationToPoissonAlgebra)_.

### Relation to symplectomorphisms
 {#RelationToSymplectomorphisms}

By the [above](#RelationToPoissonBracket), the Heisenberg group is a [[subgroup]] of the group that integrates the [[Poisson bracket]]. The latter is a [[central extension]] of the group of [[Hamiltonian symplectomorphisms]].

(Of course, on a contractible [[symplectic manifold]] such as a [[symplectic vector space]], every [[symplectomorphism]] is automatically a [[Hamiltonian symplectomorphism]].)

### Cocycle and extension

The additive group on the [[Cartesian space]] $\mathbb{R}^2$ with group operation

$$
  (a,b) + (a',b') = (a + a' , b + b')
$$

carries a degree-2 [[group cocycle]] $\omega$ with values in $\mathbb{R}$ given by

$$
  \omega : ((a_1,b_1), (a_2,b_2)) \mapsto a_1 \cdot b_2
  \,.
$$

The cocycle condition for this is the identity

$$
  a_1 \cdot (b_2 + b_3) + a_2 \cdot b_3
  =
  a_1 \cdot b_2 + (a_1 + a_2) \cdot b_3
$$

The Heisenberg group $H_3$ is the [[group extension]] of $\mathbb{R}^2$ by this cocycle.

### Unitary representations

See _[[Stone-von Neumann theorem]]_.

### Automorphism group

The [[automorphism group]] of the Heisenberg group is the [[symplectic group]].

## Related concepts

* [[Heisenberg Lie algebra]]

* [[quantomorphism group]]

* [[spherical CR manifold]]

* [[Heisenberg n-group]]

## References

### General

An original account:

* [[Bertram Kostant]], _Quantization and unitary representations_, in _Lectures in modern analysis and applications III_. Lecture Notes in Math. 170 (1970), Springer Verlag, 87—208

Monographs:

* [[Jean-Luc Brylinski]], section II.3 of: _Loop spaces, characteristic classes and geometric quantization_, Birkhäuser (1993) [([doi:10.1007/978-0-8176-4731-5](https://www.springer.com/gp/book/9780817647308)]

* [[Ernst Binz]], Sonja Pods, *The geometry of Heisenberg groups --- With Applications in Signal Theory, Optics, Quantization, and Field Quantization*, Mathematical Surveys and Monographs **151**, American Mathematical Society (2008) [[ams:surv-151](https://bookstore.ams.org/surv-151)]

Discussion in the context of [[geometric quantization]] is in

* _Geometric quantization II, Prequantization and the Heisenberg group_ ([pdf](http://www.maths.ed.ac.uk/~jthomas7/GeomQuant/Lecture2.pdf))

Discussion of the [[representation theory]] of the infinite-dimensional Heisenberg group is in

* Günther Hörmann, _Representations of the infinite dimensional Heisenberg group_, PhD thesis, Vienna 1993 ([pdf](http://www.mat.univie.ac.at/~michor/hoermann_diss.pdf))

On [[group algebras]] of ([[underlying]] [[discrete group|discrete]]) [[Heisenberg groups]] as [[strict deformation quantizations]] of [[presymplectic manifold|pre-]][[symplectic vector spaces|symplectic]] [[topological vector spaces]] by [[continuous field of C-star algebras|continuous fields of]] [[Weyl algebras]]:

* [[Ernst Binz]], [[Reinhard Honegger]], [[Alfred Rieckers]], *Infinite dimensional Heisenberg group algebra and field-theoretic strict deformation quantization*, International Journal of Pure and Applied Mathematics **38** 1 (2007) [[ijpam:2007-38-1/6](https://ijpam.eu/contents/2007-38-1/6/index.html), [pdf](https://www.ijpam.eu/contents/2007-38-1/6/6.pdf)]

* [[Reinhard Honegger]], [[Alfred Rieckers]], *Heisenberg Group Algebra and Strict Weyl Quantization*, Chapter 23 in: *Photons in Fock Space and Beyond, Volume I: From Classical to Quantized Radiation Systems*, World Scientific (2015) [chapter:[doi;10.1142/9789814696586_0023](https://doi.org/10.1142/9789814696586_0023), book:[doi:10.1142/9251-vol1](https://doi.org/10.1142/9251-vol1)]

### For Chern-Simons theory

The Heisenberg group for the [[phase space]] $U(1)$-[[Chern-Simons theory]] on an arbitrary [[Riemann surface]] (and its relation to [[skein relations]] and [[theta functions]]) is discussed in

* {#GelcaUribe10a} [[Razvan Gelca]], [[Alejandro Uribe]], _From classical theta functions to topological quantum field theory_  ([arXiv:1006.3252](http://arxiv.org/abs/1006.3252), [slides pdf](http://www.math.ttu.edu/~rgelca/berk.pdf))

* {#GelcaHamilton12} [[Razvan Gelca]], [[Alastair Hamilton]], _Classical theta functions from a quantum group perspective_ ([arXiv:1209.1135](http://arxiv.org/abs/1209.1135))

[[!redirects Heisenberg groups]]

[[!redirects Heisenberg Lie group]]
[[!redirects Heisenberg Lie groups]]

[[!redirects Weyl-Heisenberg group]]
[[!redirects Weyl-Heisenberg groups]]


Heisenberg Lie algebra

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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

### General

Suppose we are given a commutative unital [[ring]] $k$ and a [[module]] $V$ over $k$ equipped with a skew-symmetric [[bilinear form]]

$$
  \omega: V\otimes_k V\to k
  \,.
$$

(Typically, one requires $\omega$ to be non-degenerate, see [below](InSymplecticGeometry), but this is not needed for the following definition).

The __Heisenberg Lie algebra__ $Heis(V, \omega)$ corresponding to $(V,\omega)$ is the [[Lie algebra]] given by the $k$-module $V\oplus k$ together with the unit $k \hookrightarrow V\oplus k$, $s\mapsto (0,s) =: s 1$ and Lie $k$-algebra bracket

$$
  [(m,s),(m',s')] := (0, \omega(m,m')1)
  \,.
$$

### In symplectic geometry
 {#InSymplecticGeometry}

The notion of Heisenberg algebra arose in the study of [[quantization]] by tools of [[symplectic geometry]]:

A special case of the above definition is that where $(V,\omega)$ a [[symplectic vector space]] (hence $k$ a [[field]] and $\omega$ non-degenerate).

In this case the Heisenberg algebra is a sub-Lie algebra of the Lie algebra underlying the [[Poisson algebra]] of $(V,\omega)$. For more on this see [below](#RelationToPoissonAlgebra).

### Heisenberg Lie $n$-algebras in $n$-plectic geometry
 {#InnplecticGeometry}

We discuss a generalization of the notion of Heisenberg Lie algebra from ordinary [[symplectic geometry]] to a notion of _Heisenberg [[Lie n-algebra]]_  in [[higher geometric quantization]] of [[n-plectic geometry]]. See
at _[[Heisenberg Lie n-algebra]]_ for more.

The following definition is naturally motivated from the fact that:

1. The ordinary Heisenberg Lie algebra is the sub-Lie algebra of the
[[Poisson bracket]] Lie algebra, the one underlying the corresponding [[Poisson algebra]] (see [below](#RelationToPoissonAlgebra)) on the constant and linear functions.

1. The generalization of [[Poisson bracket|Poisson brackets]] to [[Poisson Lie n-algebras]] in [[n-plectic geometry]] for all $n$ is established (see there).

In view of this, the following definition takes the Heisenberg Lie $n$-algebra to be the sub-Lie $n$-algebra of the [[Poisson Lie n-algebra]] on the linear and constant differential forms.

First we need the following definition, which is elementary, but nevertheless worth making explicit once.

+-- {: .num_defn}
###### Definition

Let $n \in \mathbb{N}$, let $(V, \omega)$ be an [[n-plectic vector space]].

The **corresponding $n$-plectic manifold** is the [[n-plectic manifold]] $(V, \mathbf{\omega})$, with $V$ now the canonical [[smooth manifold]] structure on the given vector space, and with

$$
  \mathbf{\omega} \in \Omega^{n+1}(V)
$$

the [[differential form]] obtained by left (right) translating $\omega$ along $V$.

Explicitly, for all [[vector fields]] $\{v_i \in \Gamma(T V)\}_{i = 1}^n$ and all points $x \in V$ we set

$$
  \mathbf{\omega}_x(v_1, \cdots, v_n)
  :=
  \omega(v_1(x), \cdots, v_n(x))
  \,.
$$

Here on the right --  and in all of the following -- we are using that every [[tangent space]] $T_x V$ of $V$ is naturally identified with $V$ itself

$$
  T_x V \stackrel{\simeq}{\to} V
  \,.
$$

=--

+-- {: .num_defn}
###### Definition

Let $n \in \mathbb{N}$,
let $(V, \omega)$ be an [[n-plectic vector space]]
and let $(V, \mathbf{\omega})$ be the corresponding
[[n-plectic manifold]].

The **Heisenberg Lie $n$-algebra** $Heis(V,\omega)$ is the sub-[[Lie n-algebra]] of the [[Poisson Lie n-algebra]] $\mathcal{P}(V, \omega)$ on those [[differential forms]] which are either linear or constant (with respect to left/right translation on $V$).

=--

All one has to observe is:

+-- {: .num_prop}
###### Proposition

This is indeed a sub-Lie $n$-algebra.

=--

+-- {: .proof}
###### Proof

We need to check that the linear and constant forms are closed under the [[L-infinity algebra]] brackets of $\mathcal{P}(V, \omega)$.

The only non-trivial such brackets are the unary one, and the ones on elements all of degree 0.

The unary bracket is given by the de Rham differential. Since this sends a linear form to a constant form and a constant form to 0, our sub-complex is closed under this.

Similarly, the brackets on elements all in degree 0 is given by contraction of $\mathbf{\omega}$ with the Hamiltonian vector fields of linear or constant forms. Since $\mathbf{\omega}$ is a constant form, and since the de Rham differential of a linear or constant form is constant (or even 0), these Hamiltonian vector fields are necessarily constant. Hence their contraction with $\mathbf{\omega}$ gives a constant form.

=--

## Properties

### Relation to Poisson algebra
 {#RelationToPoissonAlgebra}

We discuss how the notion of Heisenberg Lie algebra relates to that of _[[Poisson algebra]]_.

+-- {: .num_prop}
###### Proposition

For $(X, \omega)$ a [[symplectic vector space]], there is a [[natural transformation|natural]]
[[Lie algebra]] [[homomorphism]]

$$
  Heis(V, \omega) \hookrightarrow \mathcal{P}(V,\omega)
$$

exhibiting the Heisenberg Lie algebra as a [[subobject|sub]]-Lie algebra of that underlying the [[Poisson algebra]] $\mathcal{P}(V,\omega)$ of $V$.

=--

Namely, it is the sub-Lie algebra on the [[linear functions]] and the constant functions.

+-- {: .proof}
###### Proof

Let $(V, \omega)$ be a [[symplectic vector space]] over the [[real numbers]]. Using the canonical [[isomorphism]]
$\phi : T V \simeq V \times V$ of the [[tangent bundle]] of $V$ with the [[projection]] $p_1 : V \times V \to V$, we obtain from the [[bilinear form]] $\omega$ a [[differential form|differential 2-form]] $\mathbf{\omega} \in \Omega^2(V)$ by the assignment

$$
  \mathbf{\omega}(\mathbf{v}, \mathbf{w})
   :=
  \omega(p_2 \phi(\mathbf{v}), p_2 \phi(\mathbf{w}))
$$

for all $\mathbf{v}, \mathbf{w} \in \Gamma(T V)$.

This way $(V, \mathbf{\omega})$ is a [[symplectic manifold]] and thus comes with a [[Poisson algebra]].
Write $\mathcal{P}(V,\mathbf{\omega})$ for the [[Lie algebra]] underlying the [[Poisson algebra]] of $(V, \mathbf{\omega})$.

Its underlying [[vector space]] is the space $C^\infty(V)$ of [[smooth functions]] $V \to \mathbb{R}$. To every element $f \in C^\infty(V)$ is associated its _[[Hamiltonian vector field]]_ $\mathbf{v}_f \in \Gamma(T X)$, defined (uniquely, due to the non-degeneracy of $\omega$) by the equation

$$
  d_{dR} f = \mathbf{\omega}(\mathbf{v}_f, -)
  \,.
$$

In terms of this, the [[Lie bracket]] of the Poisson algebra is defined to be

$$
  [f,g] := \mathbf{\omega}(\mathbf{v}_f, \mathbf{v}_g)
  \,.
$$

Inside all smooth functions sit the [[linear functions]] $V \to \mathbb{R}$, which form the [[dual vector space]] to $V$:

$$
  V^* \hookrightarrow C^\infty(V)
$$
$$
  \alpha \mapsto \alpha(-)
  \,.
$$

By the non-degeneracy of $\omega$, for every $f \in V^*$ there is an element $v_f \in V$ such that

$$
  f = \omega(v_f,-) \in C^\infty(V)
  \,.
$$

Moreover, the canonical extension $\mathbf{v}_f$ of $v_f$ to a vector field on $V$ is a [[Hamiltonian vector field]] for $f$

$$
  d_{dR} f = \mathbf{\omega}(\mathbf{v}_f,-)
  \,.
$$

It follows that the Lie bracket of two [[linear functions]] $f,g$ in the Poisson algebra is

$$
  \begin{aligned}
    [f,g]
      & =
    \mathbf{\omega}(\mathbf{v}_f, \mathbf{v}_g)
     \\
    & =
     \omega(v_f, v_g)
  \end{aligned}
  \,.
$$

Notice that on the right we have a _constant_ function on $V$.

Write $\rho_2 : \mathbb{R} \hookrightarrow C^\infty(V)$ for the inclusion of the constant functions, and write

$$
  \rho_1
  :
  V
    \stackrel{\omega}{\to}
  V^*
    \hookrightarrow
  C^\infty(V)
  \,.
$$

Then, by the above, the inclusion

$$
  \rho : V \oplus \mathbb{R} \stackrel{(\rho_1, \rho_2)}{\to} C^\infty(V)
$$

induces a Lie algebra [[homomorphism]]

$$
  \rho
  :
  Heis(V,\omega)
    \hookrightarrow
  \mathcal{P}(V, \mathbf{\omega})
$$

which exhibits the Heisenberg Lie algebra as a [[subobject|sub]]-Lie algebra of that underlying the Poisson algebra.

=--

### Integration to Heisenberg group

As for any [[Lie algebra]] one has [[Lie integration]] of the Heisenberg Lie algebra to a [[Lie group]]. This is called the _[[Heisenberg group]]_ (of the given [[symplectic vector space]]).

### Relation to the Weyl algebra
 {#RelationToTheWeylAlgebra}

In the case of standard [[symplectic form]] on the [[Cartesian space]] $\mathbb{R}^{2n}$, the
[[universal enveloping algebra]] of the Heisenberg Lie algebra is an [[associative algebra]] $\mathcal{U}\big(Heis(\mathbb{R}^{2n})\big)$. Depending on conventions, this either already is the [[Weyl algebra]] on $2n$ generators or else it becomes so after after forming the [[quotient algebra]] in which the central generator is identified with the [[unit element]] of the [[ground field]] -- whereas in the former case (considered eg. in [Kravchenko 2000, Def. 2.1](#Kravchenko00); [Bekaert 2005, p. 9](#Bekaert05)) the central generator plays the role of the formal [[Planck constant]] $\hbar$ with the Weyl algebra regarded as a [[formal deformation quantization]] of the [[symplectic manifold]] $\mathbb{R}^{2m}$.

Accordingly, given a [[Heisenberg Lie n-algebra|Heisenberg Lie $n$-algebra]] it makes sense to call its [[universal enveloping E-n algebra|universal enveloping $E_n$-algebra]] a _[[Weyl n-algebra|Weyl $n$-algebra]]_.

### Relation to the Heisenberg double

Given any [[Hopf algebra]], one can define its [[Heisenberg double]], which generalized the Heisenberg-Weyl algebra, which corresponds to the case when the Hopf algebra is the polynomial
algebra.

## References

Monograph:

* [[Ernst Binz]], Sonja Pods, *The geometry of Heisenberg groups --- With Applications in Signal Theory, Optics, Quantization, and Field Quantization*, Mathematical Surveys and Monographs **151**, American Mathematical Society (2008) [[ams:surv-151](https://bookstore.ams.org/surv-151)]

Lecture notes:

* (section 4 in) Gordon, _Infinite-dimensional Lie algebras_, Lecture notes, Edinburgh (2008) ([pdf](http://www.maths.ed.ac.uk/~igordon/LA1.pdf))

* Teruji Thomas, _Geometric quantization II: Prequantization and the Heisenberg group_ ([pdf](http://www.maths.ed.ac.uk/~jthomas7/GeomQuant/Lecture2.pdf)), section 4 (relating to [[geometric quantization]])

Relation to the [[Weyl algebra]]:

* {#Kravchenko00} Olga Kravchenko, *Deformation Quantization of Symplectic Fibrations*, Compositio Mathematica **123** (2000) 131–165 [[arXiv:math/9802070](https://arxiv.org/abs/math/9802070), [doi:10.1023/A:1002452002677](https://doi.org/10.1023/A:1002452002677)]

* {#Bekaert05} [[Xavier Bekaert]], *Universal enveloping algebras and some applications in physics* (2005) [[cds:904799](https://cds.cern.ch/record/904799), [pdf](https://cds.cern.ch/record/904799/files/cer-002575006.pdf)]

A [[categorification]] of the Heisenberg algebra:

* [[Mikhail Khovanov]], _Heisenberg algebra and a graphical calculus_ ([arXiv:1009.3295](http://arxiv.org/abs/1009.3295))

* [[Owen Gwilliam]], [[Rune Haugseng]], _Linear Batalin-Vilkovisky quantization as a functor of ∞-categories_ ([arXiv:1608.01290](https://arxiv.org/abs/1608.01290))

An $n$-fold categorification of the Lie algebra underlying the [[Poisson algebra]] (and hence including the Weil algebra) for all $n$ to a [[Lie n-algebra]] is considered in [[n-plectic geometry]],

* [[Chris Rogers]], _Higher symplectic geometry_ PhD thesis (2011) ([arXiv:1106.4068](http://arxiv.org/abs/1106.4068))

[[!redirects Heisenberg Lie algebras]]

[[!redirects Heisenberg algebra]]
[[!redirects Heisenberg algebras]]


Heisenberg Lie n-algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
#### Geometric quantization
+--{: .hide}
[[!include geometric quantization - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _Heisenberg Lie $n$-algebra_ is the generalization to [[n-plectic geometry]] of the notion of [[Heisenberg Lie algebra]] in [[symplectic geometry]].

The Heisenberg Lie $n$-algebra [[Lie integration|integrates]] to the _[[Heisenberg n-group]]_.

## Definition

We discuss a generalization of the notion of Heisenberg Lie algebra from ordinary [[symplectic geometry]] to a notion of _Heisenberg [[Lie n-algebra]]_  in [[higher geometric quantization]] of [[n-plectic geometry]].

The following definition is naturally motivated from the fact that:

1. The ordinary Heisenberg Lie algebra is the sub-Lie algebra of the
[[Poisson bracket]] Lie algebra, the one underlying the corresponding [[Poisson algebra]] (see [below](#RelationToPoissonAlgebra)) on the constant and linear functions.

1. The generalization of [[Poisson bracket|Poisson brackets]] to [[Poisson Lie n-algebras]] in [[n-plectic geometry]] for all $n$ is established (see there).

In view of this, the following definition takes the Heisenberg Lie $n$-algebra to be the sub-Lie $n$-algebra of the [[Poisson Lie n-algebra]] on the linear and constant differential forms.

First we need the following definition, which is elementary, but nevertheless worth making explicit once.

+-- {: .num_defn}
###### Definition

Let $n \in \mathbb{N}$, let $(V, \omega)$ be an [[n-plectic vector space]].

The **corresponding $n$-plectic manifold** is the [[n-plectic manifold]] $(V, \mathbf{\omega})$, with $V$ now the canonical [[smooth manifold]] structure on the given vector space, and with

$$
  \mathbf{\omega} \in \Omega^{n+1}(V)
$$

the [[differential form]] obtained by left (right) translating $\omega$ along $V$.

Explicitly, for all [[vector fields]] $\{v_i \in \Gamma(T V)\}_{i = 1}^n$ and all points $x \in V$ we set

$$
  \mathbf{\omega}_x(v_1, \cdots, v_n)
  :=
  \omega(v_1(x), \cdots, v_n(x))
  \,.
$$

Here on the right --  and in all of the following -- we are using that every [[tangent space]] $T_x V$ of $V$ is naturally identified with $V$ itself

$$
  T_x V \stackrel{\simeq}{\to} V
  \,.
$$

=--

+-- {: .num_defn}
###### Definition

Let $n \in \mathbb{N}$,
let $(V, \omega)$ be an [[n-plectic vector space]]
and let $(V, \mathbf{\omega})$ be the corresponding
[[n-plectic manifold]].

The **Heisenberg Lie $n$-algebra** $Heis(V,\omega)$ is the sub-[[Lie n-algebra]] of the [[Poisson Lie n-algebra]] $\mathcal{P}(V, \omega)$ on those [[differential forms]] which are either linear or constant (with respect to left/right translation on $V$).

=--

All one has to observe is:

+-- {: .num_prop}
###### Proposition

This is indeed a sub-Lie $n$-algebra.

=--

+-- {: .proof}
###### Proof

We need to check that the linear and constant forms are closed under the [[L-infinity algebra]] brackets of $\mathcal{P}(V, \omega)$.

The only non-trivial such brackets are the unary one, and the ones on elements all of degree 0.

The unary bracket is given by the de Rham differential. Since this sends a linear form to a constant form and a constant form to 0, our sub-complex is closed under this.

Similarly, the brackets on elements all in degree 0 is given by contraction of $\mathbf{\omega}$ with the Hamiltonian vector fields of linear or constant forms. Since $\mathbf{\omega}$ is a constant form, and since the de Rham differential of a linear or constant form is constant (or even 0), these Hamiltonian vector fields are necessarily constant. Hence their contraction with $\mathbf{\omega}$ gives a constant form.

=--

## Related concepts

* the [[universal enveloping E-n algebra]] of a Heisenberg Lie n-algebra deserves to be called a _[[Weyl n-algebra]]_

[[!include slice automorphism groups in higher prequantum geometry - table]]

[[!include geometric quantization extensions - table]]

## References

The topological part of the Heisenberg Lie 2-algebra of the [[string]] [[sigma-model]] called the [[WZW model]] has been discussed (not under this name, though) in

* [[John Baez]], [[Chris Rogers]], _Categorified Symplectic Geometry and the String Lie 2-Algebra_. [arXiv:0901.4721](http://arxiv.org/abs/0901.4721).

and shown to be the [[string Lie 2-algebra]].

General discussion in the broader context of [[higher differential geometry]] and [[higher prequantum geometry]] is in

* {#FRS13a} [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:Higher geometric prequantum theory]]_ ([arXiv:1304.0236](http://arxiv.org/abs/1304.0236))

* {#FRS13b} [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:L-∞ algebras of local observables from higher prequantum bundles]]_, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107 – 142 ([arXiv:1304.6292](http://arxiv.org/abs/1304.6292))

[[!redirects Heisenberg Lie n-algebras]]

[[!redirects Heisenberg Lie 2-algebra]]
[[!redirects Heisenberg Lie 2-algebras]]

[[!redirects Heisenberg L-∞ algebra]]
[[!redirects Heisenberg L-∞ algebras]]
[[!redirects Heisenberg L-infinity algebra]]
[[!redirects Heisenberg L-infinity algebras]]


Heisenberg model

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### In solid state physics

In [[solid state physics]], the *Heisenberg model*
is  [[spin chain]]-[[model (in theoretical physics)|model]] for [[magnetism|magnetic]] [[quantum materials]].

The Heisenberg model is akin to the [[Ising model]], but where the latter only has [[internal symmetry]]-group [[cyclic group of order 2|$\mathbb{Z}/2$]] (spin flips along one axis), the Heisenberg model has the full [[Spin(3)]] $\simeq$ [[SU(2)]]-group of [[internal symmetries]] [[group action|acting]] on [[spin representations]] located at each of the "sites" of the model.

There are generalizations of the model with [[SU(N)|$SU(N)$]]-[[internal symmetry]] for any $N \in \mathbb{N}$.

### In SYM theory and AdS/CFT

As an abstract [[model (in theoretical physics)|model]], the Heisenberg spin chain has a curious re-incarnation as describing the [[dynamics]] of [[BMN operators]] in [[D=4 N=4 super Yang-Mills theory]] under the [[dilatation operator]] in the [[large-N limit]] of the [[AdS/CFT correspondence]]. See [here](single+trace+operator#ReferencesRelationToStringExcitations) for more.

## Properties

### Symmetry protected topological phase

The Heisenberg model has a curious [[internal symmetry|internal]]-[[symmetry protected topological phase]],  observed in [Haldane 1983](#Haldane83) and hence known as the *Haldane phase*.

## Related entries

* [[spin chain]]

* [[Ising model]]

## References

### General

See also:

* Wikipedia, *[Quantum Heisenberg model](https://en.wikipedia.org/wiki/Quantum_Heisenberg_model)*

### SPTs

On [[symmetry protected topological phases]] of the Heisenberg model:

* {#Haldane83} [[Duncan Haldane]], *Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the $O(3)$ nonlinear sigma model*, Physics Letters A **93** 9 (1983) 464-468 $[$doi:10.1016/0375-9601(83)90631-X $]$

* Pierre Fromholz, Philippe Lecheminant, *Symmetry-protected topological phases in the $SU(N)$ Heisenberg spin chain: a Majorana-fermion approach*, Phys. Rev. B **102** 094410 (2020) $[$[arXiv:2006.13553](https://arxiv.org/abs/2006.13553), [doi:10.1103/PhysRevB.102.094410](https://doi.org/10.1103/PhysRevB.102.094410)$]$

[[!redirects Heisenberg spin chain]]
[[!redirects Heisenberg spin chains]]

[[!redirects Haldane phase]]
[[!redirects Haldane phases]]


Heisenberg n-group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
#### Geometric quantization
+--{: .hide}
[[!include geometric quantization - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Heisenberg $n$-group_ is a sub-[[∞-group]] of a [[quantomorphism n-group]] of a [[n-plectic geometry|n-plectic space]] $(G, \omega)$ equipped with [[∞-group]] structure, on those whose underlying [[n-plectomorphisms]] act  by lect [[∞-action]] of $G$ on itself. This is the higher refinement of the traditional notion of _[[Heisenberg group]]_.

## Definition

For $(G, \omega)$ an [[n-plectic geometry]] with [[higher geometric quantization|higher geometric prequantization]] $(G, \nabla)$ and for $G$ equipped with [[∞-group]] structure, the corresponding _Heisenberg $\infty$-group_ is the sub-[[∞-group]] of the [[quantomorphism n-group]] on those elements whose corresponding [[n-plectomorphism]] is given by such a left action.

The corresponding [[Lie n-algebra]] is the _[[Heisenberg Lie n-algebra]]_.

## Properties

### Relation to parameterized WZW models

* [[parameterized WZW model]]

## Examples

### String 2-group is Heisenberg 2-group of WZW gerbe

For $G$ a compact simply connected simple [[Lie group]], there is the "[[WZW gerbe]]", hence the [[circle n-bundle with connection|circle 2-bundle with connection]] on $G$ whose [[curvature]] 3-form is the [[left invariant differential form|left invariant]] extension $\langle \theta \wedge [\theta \wedge \theta]\rangle$ of the canonical Lie algebra 3-cocycle to the group

$$
  \mathcal{L}_{WZW} \;\colon\; G \longrightarrow \mathbf{B}^2
  \,.
$$

+-- {: .num_prop }
###### Proposition

The string 2-group is the [[smooth infinity-group|smooth 2-groupo]] of [[automorphism infinity-group|automorphism]] of $\mathcal{L}_{WZW}$ which cover the left [[action]] of $G$ on itself (hence the "[[Heisenberg 2-group]]" of $\mathcal{L}_{WZW}$ regarded as a [[prequantum 2-bundle]])

$$
  \mathbf{Aut}(\mathcal{L}_{WZW}) \simeq String(G)
  \,,
$$

=--

This is due to ([Fiorenza-Rogers-Schreiber 13, section 6.2.1](#FiorenzaRogersSchreiber13)).

## Related concepts

[[!include geometric quantization extensions - table]]

## References

* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:Higher geometric prequantum theory]]_, 2013 ([arXiv:1304.0236](http://arxiv.org/abs/1304.0236))
  {#FiorenzaRogersSchreiber13}

[[!redirects Heisenberg ∞-group]]
[[!redirects Heisenberg infinity-group]]

[[!redirects Heisenberg n-groups]]
[[!redirects Heisenberg ∞-groups]]
[[!redirects Heisenberg infinity-groups]]

[[!redirects Heisenberg 2-group]]
[[!redirects Heisenberg 2-groups]]


Heisuke Hironaka

* [Wikipedia entry](https://en.wikipedia.org/wiki/Heisuke_Hironaka)

## selected writings

* {#Hironaka64} [[Heisuke Hironaka]], _Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I_, Annals of Mathematics Second Series, Vol. 79, No. 1 (Jan., 1964), pp. 109-203 (95 pages) ([jstor:1970486](https://www.jstor.org/stable/1970486))

## related $n$Lab entries

* [[resolution of singularities]]

category: people


Helga Baum

* [webpage](http://www2.mathematik.hu-berlin.de/~diffgeo/index.php?seite=baum)

category: people


helicity

## Idea

for the moment see [[unitary irrep of the Poincare group]]

## Related concepts

* [[angular momentum]]

* [[Pauli-Lubanski vector]]

* [[twistor]]

[[!redirects helicities]]


Hellman-Feynman theorem

## Idea

Hellman-Feynman theorem relates the time derivative of energy in [[quantum mechanics|quantum mechanical]] system with the expectation value of the time derivative of the Hamiltonian operator.

It is a consequence of the variational principle, hence it is valid not only to true eigenstates but also for some other wave functions (e.g. Hartree-Fock) steming from the variational principle.

## References

* wikipedia: [Hellman-Feynman theorem](http://en.wikipedia.org/wiki/Hellmann%E2%80%93Feynman_theorem)

Historical references in physics are

* H. Hellmann, _Einfuehrung in die Quantenchemie_ (Franz Deuticke, Leipzig, 1937)

* [[Richard Feynman|R. P. Feynman]], _Forces in molecules_, Phys. Rev. 56, 340 (1939)

A strong version with careful functional analysis can be found in

* David Carfì, _The pointwise Hellman-Feynman theorem_, AAPP Physical, Mathematical, and Natural Sciences 88 (1). no. C1A1001004 [doi](http://dx.doi.org/10.1478/C1A1001004)

The theorem also holds for the [[WKB method|WKB approximations]]:

*  K. Banerjee, _W. K. B. approximation and scaling_, Proc. Royal Soc. London A 363, No. 1712 (Nov. 1, 1978) 147-151
[jstor](http://www.jstor.org/stable/79720)


Hellmut Baumgärtel

* [website](http://oldwww.math.uni-potsdam.de/mp1/baumg.htm)

## related $n$Lab entries

* [[AQFT]]

category: people