fuzzy sphere

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#### Noncommutative geometry
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#### Spheres
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## Idea

A _fuzzy sphere_ is a variant of an [[n-sphere]] in [[noncommutative geometry]]. Often the fuzzy _[[2-sphere]]_ is meant by default, but there are also fuzzy spheres of higher dimension.

## Definition

### Fuzzy 2-sphere
 {#Fuzzy2SphereDefinition}

For $N \in \mathbb{N}$, $N \geq 2$, the _fuzzy 2-sphere_ of $N$ bits is the [[formal duality|formal dual]] to the [[associative algebra]] which is the sub-algebra in the [[matrix algebra]] $Mat_{N \times N}$ generated from the elements of the $N$-dimensional [[complex numbers|complex]] [[irreducible representation|irreducible]] [[Lie algebra representation]] of [[su(2)]].

We now say this more in detail:

#### Conventions and Normalizations
 {#ConventionsAndNormalizations}

First to introduce relevant notation and to set out the proper choices of normalizations.

\linebreak

With $\mathfrak{su}(2)$ the [[Lie algebra]] [[su(2)]] of [[SU(2)]], with [[complexification]] the [[special linear Lie algebra]] [[sl(2)]] (see [here](su2#Complexification))

$$
  \mathfrak{su}(2) \otimes_{\mathbb{R}} \mathbb{C}
  \;\simeq\;
  \mathfrak{sl}(2,\mathbb{C})
$$

write

\[
  \label{CanonicalBasisOfsu2C}
  \sigma_i
    \;\in\;
  \mathfrak{su}(2) \otimes_{\mathbb{R}} \mathbb{C}
  ,
  \phantom{AAA}
  i \in \{1,2,3\}
\]

for a choice of [[linear basis]] such that the [[Lie bracket]] takes the form

\[
  \label{CommutationRelationsu2}
  [\sigma_i, \sigma_j]
  \;=\;
  i \underset{k}{\sum} \epsilon_{i j k} \sigma_k
  \,.
\]

(Here $\epsilon$ is the [[Levi-Civita symbol]], hence $\epsilon_{\sigma(1), \sigma(2)\sigma(3)} \in \{\pm 1\}$ is the [[signature of a permutation|signature]] of the [[permutation]] $\sigma \in Sym(3)$.)

Notice that the element

\[
  \label{TheUnrepresentedR2}
  \underset{i}{\sum}
  \sigma_i \cdot \sigma_i
  \;\in\;
  U(\mathfrak{su}(2))
\]

is a [[Casimir element]] in the [[universal enveloping algebra]].

Let then

\[
  \label{TheNDimensionalRepresentation}
  \rho
  \;\colon\;
  \mathfrak{su}(2)
  \longrightarrow
  Mat_{N \times N}
\]

denote [[generalized the|the]] [[complex numbers|complex]] $N$-[[dimension|dimensional]] [[complex numbers|complex]] [[irreducible representation|irreducible]] [[Lie algebra representation]] of [[su(2)]] (hence of $\mathfrak{sl}(2,\mathbb{C})$).

In the discussion of [[angular momentum]] in [[quantum mechanics]] the image

\[
  \label{TheUnnormalizedR2}
  \underset{i}{\sum}
  \rho_i \circ \rho_i
  \;\in\;
  Mat_{N \times N}
\]

of the [[Casimir element]] (eq:TheUnrepresentedR2) under the representation (eq:TheNDimensionalRepresentation) is traditionally denoted $J^2$, and the canonical [[linear basis]] for the N-dimensional representation $\rho$ (eq:TheNDimensionalRepresentation) is then traditionally denoted $\{\vert j, m \rangle\}_{m = -j}^{m = +j}$ with

\[
  \label{EigenvaluesOfCasimir}
  J^2 \vert j,m\rangle \;=\; j(j+1)
  \phantom{AAA}
  m \in \{-j, -j+1, \cdots, j-1, j\}
  \,.
\]

hence with

$$
  N(j) = 2 j + 1
  \;\;
  \Leftrightarrow
  \;\;
  j(N) = \tfrac{1}{2}(N - 1)
  \,.
$$

This means that the [[eigenvalues]] of (eq:TheUnnormalizedR2) as a function not of the [[angular momentum]] $j$ but of the [[dimension]] $N$ of the given [[irreducible representation]] are:

\[
  \label{EigenvaluesOfCasimir}
  \begin{aligned}
    j(N)\big(j(N)+1\big)
    & =
    \phantom{+\;}
    (j(N))^2 + j(N)
    \\
    & =
    \phantom{+\;}
    \tfrac{1}{4}(N-1)^2 + \tfrac{1}{2}(N-1)
    \\
    & =
    \phantom{+\;}
    \tfrac{1}{4}N^2 - \tfrac{1}{2}N + \tfrac{1}{4}
    \\
    &
    \phantom{=\;}
    + \tfrac{1}{2}N - \tfrac{1}{2}
    \\
    &
    =
    \phantom{+\;}
    \tfrac{1}{4}N^2 - \tfrac{1}{4}
    \\
    &
    =
    \phantom{+\;}
    \tfrac{1}{4}
    (N^2 - 1)
  \end{aligned}
\]

#### Algebra of functions

Write now

\[
  \label{BasisFunctionsOnFuzzy2Sphere}
  X_i
    \;\coloneqq\;
  \tfrac
    {2}
    {\sqrt{N^2-1}}
    \rho_N(\sigma_i)
\]

for the [[square matrices]] representing the generators $\sigma_i$ (eq:CanonicalBasisOfsu2C) in the $N$-dimensional complex irrep (eq:TheNDimensionalRepresentation), suitably normalized in view of (eq:EigenvaluesOfCasimir).

(It is here that we need the assumption $N \geq 2$, hence excluding the complex 1-dimensional [[trivial representation|trivial]] [[irrep]].)

Due to the normalization, the commutation relation (eq:CommutationRelationsu2) in this representation reads

\[
  \label{NormalizedCommutationforsu2}
  [X_i, X_j]
  \;=\;
  i
  \tfrac
    {2}
    {\sqrt{N^2-1}}
  \underset{k}{\sum} \epsilon_{i j k} X_k
\]

and, by (eq:EigenvaluesOfCasimir), the image of the [[Casimir element]] (eq:TheUnrepresentedR2) under this representation is the [[identity matrix]]:

\[
  \label{RadiusOfFuzzy2Sphere}
  \begin{aligned}
    R^2
    &
    \coloneqq
    \underset{i}{\sum}
    X_i \cdot X_i
    \\
    & =
    I
    \,.
  \end{aligned}
\]

Equation (eq:NormalizedCommutationforsu2) shows that in the [[large N limit]] $N \to \infty$, the algebra generated by the $X_i$ becomes commutative, and (eq:RadiusOfFuzzy2Sphere) says that for any $N$, the algebra generated by the $X_i$ satisfies the same relation as the [[smooth algebra]] on generators $x_i$ restricted to the actual [[2-sphere]] of unit [[radius]]:

$$
  C^\infty
  \big(
    S^2
  \big)
  \;\simeq\;
  SmoothAlg\big( \{x_1, x_2,x_3\}\big)
  \big/
  \Big(
     \underset{i}{\sum} x_i \cdot x_i   = 1
  \Big)
$$

#### Integration

With the above, the [[volume]] density of the fuzzy 2-sphere scales with
$\tfrac{1}{\sqrt{N^2-1}}$

$$
  \begin{aligned}
   \tfrac{4\pi i}{3!}
   \epsilon^{i j k }
    X_i \cdot X_j \cdot X_k
    & =
    \tfrac{4\pi i}{3!}
    \tfrac{1}{2}
    \epsilon^{i j k }
    [X_i, X_j] \cdot X_k
    \\
    & =
    \tfrac{4\pi i}{3!}
    \tfrac
      {i}
      {\sqrt{N^2-1}}
    \epsilon^{i j k }
    \epsilon_{i j l}
    X^l \cdot X_k
    \\
    & =
    \tfrac{4\pi i}{3!}
    \tfrac
      {2i}
      {\sqrt{N^2-1}}
    X^k \cdot X_k
    \\
    & =
    \tfrac{4\pi i}{3!}
    \tfrac
      {2i}
      {\sqrt{N^2-1}}
    \, I
    \\
    & =
    -
    \tfrac{4\pi}{3}
    \tfrac
      {1}
      {\sqrt{N^2-1}}
    \, I
  \end{aligned}
$$

In this vein, one defines the fuzzy refinement of the [[integral]] of functions over the [[2-sphere]], against its canonical [[volume form]], to be given by the matrix [[trace]], normalized as follows

\[
  \label{FuzzyS2Integration}
  \array{
    \langle X_1, X_2, X_3\rangle
    &\overset{\int_{S^2_N}}{\longrightarrow}&
    \mathbb{C}
    \\
    M &\mapsto& 4 \pi \tfrac{1}{ \sqrt{N^2 -1 } }  tr(M)
  }
\]

With this definition the [[volume]] of the fuzzy 2-sphere of $N$ bits comes out as

$$
  vol(S^2_N)
  \;=\;
  \int_{S^2_N} I
  \;=\;
  \tfrac{4 \pi}{ \sqrt{N^2 - 1 } }  \underset{N}{\underbrace{tr(I)}}
  \;=\;
  4 \pi \tfrac{ N }{ \sqrt{N^2 -1} }
  \,.
$$

This indeed goes to the volume of the actual [[2-sphere]] in the [[limit of a sequence|limit]] $N \to \infty$:

$$
  \underset{N \to \infty}{\lim}
  vol(S^2_N)
  \;=\;
  4 \pi
  \;=\;
  vol(S^2)
  \,.
$$

## Properties

### Shape observables as weight systems on chord diagrams
 {#ShapeObservablesAsVassilievInvariants}

We discuss how the "shape observables" on the fuzzy 2-sphere ([above](#Fuzzy2SphereDefinition)) are given by [[single trace observables]] which are [[Lie algebra weight systems]]
on [[chord diagrams]] (following [Ramgoolam-Spence-Thomas 04](#RamgoolamSpenceThomas04), [McNamara-Papageorgakis 05](#McNamaraPapageorgakis05), see [McNamara 06, Section 4](#McNamara06) for review).

For more see at _[[weight systems on chord diagrams in physics]]_.

\linebreak

While in the commutative [[large N limit]], all powers of the [[radius]] function are equal

$$
  \underset{N\to \infty}{\lim}
  \int_{S^2_N} R^{2 k}
  \;=\;
  4 \pi
  \,;
$$

for [[finite number|finite]] $N$ there is an ordering ambiguity: In fact, the number of functions on the fuzzy 2-sphere at finite $N$ that all go to the same function $R^{2k}$ in the [[large N limit]] grows rapidly with $k$.

At $k = 1$ there is the single radius observable (eq:RadiusOfFuzzy2Sphere)

$$
  \int_{S^2_N} R^2
  \;=\;
  \int_{S^2_N}
  \underset{i}{\sum} X_i \cdot X_i
  \;=\;
  4 \pi \tfrac{ N }{ \sqrt{N^2 -1} }
$$

At $k = 2$ there are, under the integral (eq:FuzzyS2Integration), two radius observables:

1. $ \int_{S^2_N} \underset{i,j}{\sum} X_i X_i X_j X_j$

1. $\int_{S^2_N} \underset{i,j}{\sum} X_i X_j X_j X_i$

(Here we are using that under the integral/trace, a [[cyclic permutation]] of the factors in the integrand does not change the result).

Similarly for higher $k$, where the number of possible orderings increases rapidly. The [[combinatorics]] that appears here is familiar in [[knot theory]]:

Every ordering of operators, up to cyclic permutation, in the [[single trace observable]] $Tr(R^2)^n$ is encoded in a [[chord diagram]] and the value of the corresponding [[single trace observable]] is the value of the [[su(2)]]-[[Lie algebra weight system]] on this chord diagram.

> from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

## Appearance in D-brane geometry

The [[fuzzy spheres]] appear in [[D-brane geometry]]:

1. the [[fuzzy funnels]] of [[Dp-D(p+2)-brane intersections]] have [[fuzzy 2-sphere]] slices

1. the [[fuzzy funnels]] of [[Dp-D(p+4)-brane intersections]] have [[fuzzy 4-sphere]] slices

1. the [[supersymmetry|supersymmetric]] [[classical field theory|classical]] solutions of the [[BMN matrix model]] are precisely [[fuzzy 2-sphere]] configurations ([BMN 02 (5.4)](BMN+matrix+model#BerensteinMaldacenaNastase02)).

## Related concepts

* [[fuzzy funnel]]

* [[noncommutative torus]]

## References

### General

Review in the context of [[D-brane geometry]], [[matrix models]] of [[string theory]]/[[M-theory]] ([[BFSS matrix model]], [[BMN matrix model]], [[IKKT matrix model]]):

* {#Ydri18} [[Badis Ydri]], _Review of M(atrix)-Theory, Type IIB Matrix Model and Matrix String Theory_ ([arXiv:1708.00734](https://arxiv.org/abs/1708.00734)), published as: _Matrix Models of String Theory_, IOP 2018 ([ISBN:978-0-7503-1726-9](https://iopscience.iop.org/book/978-0-7503-1726-9))

### Fuzzy 2-sphere

On the [[fuzzy 2-sphere]]:

#### General

The [[fuzzy 2-sphere]] was introduced in:

* [[John Madore]], _The Fuzzy sphere_, Class. Quant. Grav. 9 (1992) 69-88 ([spire:314358](http://inspirehep.net/record/314358))

Discussion of the [[spectral triple|spectral]] [[Riemannian geometry]] of the [[fuzzy 2-sphere]]:

* [[Francesco D'Andrea]], [[Fedele Lizzi]], [[Joseph Várilly]], _Metric Properties of the Fuzzy Sphere_, Lett. Math. Phys.103 (2013), 183-205 ([arXiv:1209.0108](https://arxiv.org/abs/1209.0108))

* Anwesha Chakraborty, Partha Nandi, Biswajit Chakraborty, *A note on spectral triple with real structure on fuzzy sphere* ([arXiv:2111.03012](https://arxiv.org/abs/2111.03012))

In the context of the [[IKKT matrix model]]:

* [[Harold Steinacker]], _Non-commutative geometry and matrix models_, PoS QGQGS2011 (2011) 004 ([arXiv:1109.5521](https://arxiv.org/abs/1109.5521))

See also:

* Wikipedia, _[Fuzzy sphere](https://en.wikipedia.org/wiki/Fuzzy_sphere)_

#### Observables via weight systems on chord diagrams

Relation of [[Dp-D(p+2)-brane bound states]] ([hence](Dp-Dp+2-brane+bound+states#ReferencesRelationToMonopoles)
[[Yang-Mills monopoles]]) to [[su(2)]]-[[Lie algebra weight systems]] on [[chord diagrams]] computing [[radii]] averages of [[fuzzy spheres]]:

* {#RamgoolamSpenceThomas04} [[Sanyaje Ramgoolam]], [[Bill Spence]], S. Thomas, Section 3.2 of: _Resolving brane collapse with $1/N$ corrections in non-Abelian DBI_, Nucl. Phys. B703 (2004) 236-276 ([arxiv:hep-th/0405256](https://arxiv.org/abs/hep-th/0405256))

* {#McNamaraPapageorgakis05} [[Simon McNamara]], [[Constantinos Papageorgakis]], [[Sanyaje Ramgoolam]], [[Bill Spence]], Appendix A of: _Finite $N$ effects on the collapse of fuzzy spheres_, JHEP 0605:060, 2006 ([arxiv:hep-th/0512145](https://arxiv.org/abs/hep-th/0512145))

* {#McNamara06} [[Simon McNamara]], Section 4 of: _Twistor Inspired Methods in Perturbative FieldTheory and Fuzzy Funnels_, 2006 ([spire:1351861](http://inspirehep.net/record/1351861), [pdf](https://strings.ph.qmul.ac.uk/sites/default/files/Mcnamaraphd.pdf), [[McNamara06.pdf:file]])

* [[Constantinos Papageorgakis]], p. 161-162 of: _On matrix D-brane dynamics and fuzzy spheres_, 2006 ([[Papageorgakis06.pdf:file]])

### Fuzzy 3-sphere

The [[fuzzy 3-sphere]] was first discussed (in the context of [[D0-brane]]-systems) in

* Z. Guralnik, [[Sanyaje Ramgoolam]], _On the Polarization of Unstable D0-Branes into Non-Commutative Odd Spheres_, JHEP 0102:032, 2001 ([arXiv:hep-th/0101001](https://arxiv.org/abs/hep-th/0101001))

Discussion in the context of [[M2-M5-brane bound states]]/[[E-strings]]:

* {#BasuHarvey05} [[Anirban Basu]], [[Jeffrey Harvey]], _The M2-M5 Brane System and a Generalized Nahm's Equation_, Nucl.Phys. B713 (2005) 136-150 ([arXiv:hep-th/0412310](https://arxiv.org/abs/hep-th/0412310))

See also:

* Samuel Kováčik, Juraj Tekel, *Fuzzy Onion as a Matrix Model* [[arXiv:2309.00576](https://arxiv.org/abs/2309.00576)]

### Fuzzy 4-sphere

The [[fuzzy 4-sphere]]:

* [[Harald Grosse]], [[Ctirad Klimcik]], P. Presnajder, _On Finite 4D Quantum Field Theory in Non-Commutative Geometry_, Commun. Math. Phys.180:429-438, 1996 ([arXiv:hep-th/9602115](https://arxiv.org/abs/hep-th/9602115))

* Judith Castelino, Sangmin Lee, [[Washington Taylor]], _Longitudinal 5-branes as 4-spheres in Matrix theory_, Nucl. Phys. B526:334-350, 1998 ([arXiv:hep-th/9712105](https://arxiv.org/abs/hep-th/9712105))

  (via [[D5-branes]])

### Fuzzy 6-sphere and higher

The [[fuzzy 6-sphere]] and higher:

* [[Sanjaye Ramgoolam]], Section 5 of: _On spherical harmonics for fuzzy spheres in diverse dimensions_, Nucl. Phys. B610: 461-488, 2001 ([arXiv:hep-th/0105006](https://arxiv.org/abs/hep-th/0105006))

* Yusuke Kimura, _On Higher Dimensional Fuzzy Spherical Branes_,  	Nucl. Phys. B664 (2003) 512-530 ([arXiv:hep-th/0301055](https://arxiv.org/abs/hep-th/0301055))

* Francesco Pisacane, _$O(D)$-equivariant fuzzy spheres_ ([arXiv:2002.01901](https://arxiv.org/abs/2002.01901))

See also:

* Denjoe O'Connor, Brian P. Dolan, *Exceptional fuzzy spaces and octonions* [[arXiv:2210.14754](https://arxiv.org/abs/2210.14754)]

[[!redirects fuzzy spheres]]

[[!redirects fuzzy sphere]]
[[!redirects fuzzy spheres]]

[[!redirects fuzzy 2-sphere]]
[[!redirects fuzzy 2-spheres]]

[[!redirects fuzzy 3-sphere]]
[[!redirects fuzzy 3-spheres]]

[[!redirects fuzzy 4-sphere]]
[[!redirects fuzzy 4-spheres]]

[[!redirects fuzzy 6-sphere]]
[[!redirects fuzzy 6-spheres]]


Fyodor Malikov

* [page at mathnet.ru](http://www.mathnet.ru/php/person.phtml?option_lang=eng&personid=21838)

## Selected writings

On [[fractional level WZW models]] (non-integrable admissible [[affine Lie algebra]] [[irreps]]) for [[sl(2)|$\mathfrak{sl}(2, \mathbb{C})$]]:

* [[Boris Feigin]], [[Feodor Malikov]], *Modular functor and representation theory of $\widehat{\mathfrak{sl}_2}$ at a rational level*, p. 357-405 in: Loday, Stasheff, Voronov (eds.) *Operads: Proceedings of Renaissance Conferences*,  Contemporary Mathematics **202**, AMS (1997) [[arXiv:q-alg/9511011](https://arxiv.org/abs/q-alg/9511011), [ams:conm-202](https://bookstore.ams.org/conm-202))]

  > (appearance of [[osp|$\mathfrak{osp}(1\vert2)$]]-contributions)

On [[chiral algebras]] in relation to [[quantum cohomology]]:

* [[Fyodor Malikov]], [[Vadim Schechtman]], *Deformations of chiral algebras and quantum cohomology of toric varieties* [[arXiv:math/0001170](https://arxiv.org/abs/math/0001170)]

## Related entries

* [[chiral de Rham complex]]

* [[sheaf of vertex operator algebras]]

* [[2d (2,0)-superconformal QFT]]

category: people

[[!redirects Fyodor Malikov]]
[[!redirects Fedor Malikov]]
[[!redirects Feodor Malikov]]
[[!redirects Fedor Georgievich Malikov]]
[[!redirects Fëdor Malikov]]
[[!redirects Фёдор Маликов]]
[[!redirects Федор Маликов]]


G(3)

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#### Exceptional structures
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{:toc}

##Idea

$G(3)$ is an [[exceptional structure|exceptional]] [[super Lie algebra]] (not to be confused with [[Dwyer-Wilkerson H-space]], the homotopy Lie group sometimes denoted $G_3$).

## References

* Boris Kruglikov, Andrea Santi, Dennis The, _G(3)-supergeometry and a supersymmetric extension of the Hilbert-Cartan equation_ ([arXiv:1908.09106](https://arxiv.org/abs/1908.09106))


G-crossed braided fusion category

[[!redirects $G$-crossed braided fusion category]]

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##### Fusion categories
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## Intuition

A [[fusion category]] over a field $k$ can be seen as a [[categorification]] of a semisimple $k$-algebra. For example, the group algebra of a finite group $G$ would categorify to the fusion category of $G$-graded vector spaces. On the other hand, the notion of a [[2-group]] is an [[internalisation]] of the notion of a group. A [[strict 2-group]] (which is essentially the same as a [[crossed module]]) in particular has an underlying group.

In this sense, a $G$-crossed braided fusion category can be seen as some kind of categorification of a crossed module $(G, H, \delta\colon H \to G, \alpha\colon G \to Aut(H))$, where the group $H$ is lifted to a fusion category, but $G$ is still a (finite) group. The boundary morphism $\delta$ is replaced by a $G$-grading, and the Peiffer rule ($G$-graded commutativity) is categorified to a _crossed braiding_ on $\mathcal{C}$.

$G$-crossed braided fusion category should also be related to monoidal bicategories.

## Name

There are various names for this particular flavour of fusion category,
involving permutations of the words "braided" and "crossed", or possibly trading "braided" for "$G$-braided".

The latter choice has its justification in the fact that in general, it is _not_ a [[braided category]], but the braiding is in a sense twisted by the grading, just as the second group $H$ in a crossed module need not be abelian, but up to a group action.

## Definition

A $G$-crossed braided fusion category consists of the following:

* A finite group $G$,
* a fusion category $\mathcal{C}$ over a field $k$,
* a $G$-[[graded fusion category|grading]] on $\mathcal{C}$,
* a monoidal $G$-action $\rho$ on $\mathcal{C}$ (i.e. a monoidal functor $(\rho, \rho^2, \rho^0)\colon \underline{G} \to Aut_\otimes(\mathcal{C})$ from $G$ viewed as a discrete monoidal category to the category of tensor automorphisms of $\mathcal{C}$)
such that $\rho(g_1)\mathcal{C}_{g_2} \subset \mathcal{C}_{g_1g_2g_1^{-1}}$,
* for each $g \in G$, a natural isomorphism $c_{g,X,Y}\colon X \otimes Y \to \rho(g)(Y) \otimes X$, where $X \in \mathcal{C}_g, Y \in \mathcal{C}$

satisfying

* a $G$-graded hexagon equation for $c$:

$$
  \array{
      &
        & (X \otimes Y) \otimes Z
    \\
      & \mathllap{{}^{\alpha_{X,Y,Z}}\swarrow}
        &
          & \mathrlap{\searrow^{c_{X,Y} \otimes 1_Z}}
    \\
    X \otimes (Y \otimes Z)
      &
        &
          &
            & (\rho(g)(Y) \otimes X) \otimes Z
    \\
    {}^{c_{X,Y \otimes Z}}\downarrow
      &
        &
          &
            & \downarrow^{\alpha_{\rho(g)(Y), X, Z}}
    \\
    \rho(g)(Y \otimes Z) \otimes X
      &
        &
          &
            & \rho(g)(Y) \otimes (X \otimes Z)
    \\
    {}^{\rho^2(g)_{Y, Z} \otimes 1_X}\downarrow
      &
        &
          &
            & \downarrow^{1_{\rho(g)(Y)} \otimes c_{X, Z}}
    \\
    (\rho(g)(Y) \otimes \rho(g)(Z)) \otimes X
      &
        & \mathclap{\underset{\alpha_{\rho(g)(Y), \rho(g)(Z), X}}{\longrightarrow}}
          &
            & \rho(g)(Y) \otimes (\rho(g)(Z) \otimes X)
  }
$$

* compatibility of the crossed braiding and the group action.

TODO Commutative diagrams

If $\mathcal{C}$ carries extra structure (e.g. a [[pivotal category|pivotal]] structure), the group action is typically required to preserve it.

## Examples

* Every braided fusion category can be trivially graded, with the trivial action.
* Every crossed module $(G, H, \delta, \alpha)$ makes $H$-graded vector spaces into a $G$-crossed category. (Check/reference!)

## (De-)Equivariantisation

See also [[equivariantisation]] for more details.

Let $G$ be a finite group, and $\mathcal{C}$ a braided fusion category.

A braided action of the (finite dimensional) representations $\operatorname{Rep}_G$ on $\mathcal{C}$ is the same as an inclusion of $\operatorname{Rep}_G$ in the symmetric centre $\mathcal{C}'$. By deequivariantisation, this is basically the same as a braided $G$-action on $\mathcal{C}_G$ (the category of internal $k[G]$-modules).
This makes $\mathcal{C}_G$ into a $G$-crossed braided fusion category, but most examples don't arise like this, e.g. the grading obtained this way will always be trivial.

The idea is then to generalise the full inclusion $\operatorname{Rep}_G \hookrightarrow \mathcal{C}' \hookrightarrow \mathcal{C}$ to a full inclusion $\operatorname{Rep}_G \hookrightarrow \mathcal{C}$ that need not factor over $\mathcal{C}'$.
One can still deequivariantise the underlying fusion category of $\mathcal{C}$ with respect to the $\operatorname{Rep}_G$-action, but the procedure will not respect the braiding. Naturally, the result is not a braided fusion category, but a $G$-crossed braided fusion category.

Vice versa, given a $G$-crossed braided fusion category $\mathcal{C}$, one can equivariantise it to a braided fusion category with a full inclusion of $\operatorname{Rep}_G$. When $c$ is the crossed braiding of $\mathcal{C}$, the braiding of two equivariant objects $(X \in \operatorname{ob} \mathcal{C}, u_g\colon \rho(g)(X) \to X)$ and $(Y \in \operatorname{ob} \mathcal{C}, v_g\colon \rho(g)(Y) \to Y)$ is given by $X \otimes Y \xrightarrow{c_{X,Y}} \rho(g)Y \otimes X \xrightarrow{v_g \otimes 1_X} Y \otimes X$.

## 4d extended TQFTs

TODO - Cui's thesis

## Higher Categories

TODO Cui's 2-category

## References

* Section 4.4 in _On braided fusion categories I_, Drinfeld, Gelaki, Nikshych, Ostrik. [ArXiv](http://arxiv.org/abs/0906.0620)
* Turaev
* Müger
* Cui

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G-CW approximation

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#### Representation theory
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#Contents#
* table of contents
{:toc}

## Idea

The generalization of [[CW-approximation]] from plain [[homotopy theory]] to $G$-[[equivariant homotopy theory]] is called _$G$-CW approximation_:

For suitable [[equivariance groups]] $G$, every [[topological G-space]] $X$ receives a $G$-[[equivariant function]] $Q X \overset{f}{\longrightarrow} X$ from a [[G-CW complex]] $Q X$, such that this restricts to a [[weak homotopy equivalence]] $f^H \;\colon (Q X)^H \to X^H$ on $H$-[[fixed loci]], for all suitable [[subgroups]] $H \subset G$.

This should hold for $G$ a [[compact Lie group]] (such as a [[finite group]]) and $H$ ranging over its [[closed subspace|closed]] [[subgroups]].

## Related concepts

* [[equivariant Whitehead theorem]]

* [[Elmendorf's theorem]]

* [[tom Dieck splitting]]

## References

* {#May96} [[Peter May]] et al., Thm 3.6 in: _Equivariant homotopy and cohomology theory_, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996 ([ISBN: 978-0-8218-0319-6](https://bookstore.ams.org/cbms-91/?startBookmarkIdx=200)  [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/alaska1.pdf), [[MayEtAlEquivariant96.pdf:file]])

[[!redirects G-CW approximations]]

[[!redirects G-CW-approximation]]
[[!redirects G-CW-approximations]]

[[!redirects G-CW approximation theorem]]
[[!redirects G-CW approximation theorems]]

[[!redirects G-CW-approximation theorem]]
[[!redirects G-CW-approximation theorems]]


G-CW complex

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#Contents#
* table of contents
{:toc}

## Idea

The concept of _$G$-CW  complex_ is to that of [[CW-complexes]] as [[topological G-spaces]] are to [[topological spaces]]: for $G$ a [[compact topological group]], the notion of _$G$-CW-complex_ is much like that of [[CW-complex]], only that where in the latter case one builds a [[topological space]] from gluing of [[disks]] $D^n$ ("[[cell complex|cells]]") for a $G$-CW-complex one glues [[products]] ([[product topological spaces]]) of [[disks]] with $G$-[[orbits]] $G/H$ ([[coset spaces]]) for [[compact Lie group|compact]] [[subgroups]] $H$.

These are [[cofibrant objects]] in the $G$-[[fine model structure on topological G-spaces]].

## Definition

(...)

\begin{remark}\label{FiniteGroupCWComplexesAreCWComplexesWithCellularGAction}
  For [[finite group|finite]] [[equivariance groups]] $G$, a $G$-CW-complex structure may be identified with

* a plain [[CW-complex]]-structure

* on which the $G$-[[group action|action]] is [[cell complex|cellular]] (sends $n$-cells onto $n$-cells, respecting their boundaries)

* which on cells that are sent to themselves actually restricts to the [[identity function|identity]] (i.e. cells that are fixed under the action are actually fixed point-wise).

This special case was the original definition on [Bredon 1967b, I.1](#Bredon67b).
\end{remark}

## Examples

### $G$-Manifolds
  {#GManifolds}

The _[[equivariant triangulation theorem]] says that
if a [[compact Lie group]] $G$ acts on a [[compact topological space|compact]] [[smooth manifold]] $X$, then the manifold admits an [[equivariant triangulation]]. In particular:

\begin{proposition}\label{GManifoldsAreGCWComplexes}
For $G$ a [[compact Lie group]], every [[closed manifold|closed]] [[smooth manifold|smooth]] [[G-manifold]] admits the structure of a G-CW complex.
\end{proposition}
This is due to [Matumoto 72, Prop. 4.4](#Matumoto72), [Illman 72a, Thm. 3.1, Cor. 4.1](#Illman72a), [Illman 72b, Thm. 2.6](#Illman72), [Illman 73, Thm. 2.1](#Illman73), [Illman 83, Thm. 7.1, Cor. 7.2](equivariant+triangulation+theorem#Illman83) -- review in [ALR 07, theorem 3.2](#ALR07), see also [Waner 80, p. 6](#Waner80).

Moreover, if the manifold does have a [[manifold with boundary|boundary]], then its G-CW complex may be chosen such that the boundary is a G-subcomplex. ([Illman 83, last sentence above theorem 7.1](equivariant+triangulation+theorem#Illman83))

In particular:

+-- {: .num_prop #RepresentationSpheresAreGCWComplexes}
###### Proposition
**([[G-representation spheres are G-CW-complexes]])**

For $G$ a [[compact Lie group]] (e.g. a [[finite group]]) and $V \in RO(G)$ a [[finite dimensional vector space|finite-dimensional]] [[orthogonal group|orthogonal]] $G$-[[linear representation]], the [[representation sphere]] $S^V$ admits the structure of a [[G-CW-complex]].

=--

## Properties

### Closure properties

\begin{proposition}\label{ClosureUnderProductsForFiniteGroups}
  For $G$ a [[finite group]] (at least), the [[product]] of two $G$-CW-complexes in [[compactly generated weak Hausdorff spaces]] is itself a $G$-CW-complex.
\end{proposition}
\begin{proof}
  Since for finite $G$, a $G$-CW complex is the same as a plain [[CW-complex]] equipped with a [[cell complex|cellular]] [[group action|action]] by $G$ (Rem. \ref{FiniteGroupCWComplexesAreCWComplexesWithCellularGAction}) it is clear that for this structure to be preserved by the product operation it is sufficient that the products of underlying cells constitute a CW-complex, hence that products preserve CW-complexes in compactly generated Hausdorff spaces. This is this case by [this Prop.](CW+complex#ClosureOfCWComplexesUnderCartesianProduct).
\end{proof}

### Equivariant cellular approximation

See at _[[equivariant cellular approximation theorem]]_.

### Equivariant CW-approximation

See at _[[G-CW approximation]]_.

### Equivariant Whitehead theorem

See at _[[equivariant Whitehead theorem]]_.

### Elmendorf's theorem

See at _[[Elmendorf's theorem]]_.

## References

The notion of G-CW complexes is, for the case of [[finite groups]] $G$, due to

* {#Bredon67b} [[Glen Bredon]], Section I.1. of _[[Equivariant cohomology theories]]_, Springer Lecture Notes in Mathematics Vol. 34. 1967 ([doi:10.1007/BFb0082690](https://link.springer.com/book/10.1007/BFb0082690))

announced in

* {#Bredon67a} [[Glen Bredon]], _Equivariant cohomology theories_, Bull. Amer. Math. Soc. Volume 73, Number 2 (1967), 266-268. ([educlid:1183528794](https://projecteuclid.org/euclid.bams/1183528794))

In the broader generality of general [[topological groups]] and specifically of [[compact Lie groups]], the notion of G-CW-complexes and their [[equivariant Whitehead theorem]] is due to:

* {#Matumoto71} [[Takao Matumoto]], _On $G$-CW complexes and a theorem of JHC Whitehead_, J. Fac. Sci. Univ. Tokyo Sect. IA **18** (1971) 363-374 [[irdb:00926/0001786419](https://irdb.nii.ac.jp/en/00926/0001786419), [[matumoto.pdf:file]]]

*  {#Matumoto72} [[Takao Matumoto]], _Equivariant K-theory and Fredholm operators_, J. Fac. Sci. Tokyo 18 (1971/72), 109-112 ([pdf](https://repository.dl.itc.u-tokyo.ac.jp/?action=repository_action_common_download&item_id=39826&item_no=1&attribute_id=19&file_no=1), [[MatumotoEquivariantKTheory.pdf:file]])

and, independently, due to:

* {#Illman72a} [[Sören Illman]], Chapter I of: *Equivariant algebraic topology*, Princeton University 1972 ([[Illman_EquivariantAlgebraicTopology1972.pdf:file]])

* {#Illman72b} [[Sören Illman]], Section 2 of: *Equivariant singular homology and cohomology for actions of compact lie groups* ([doi:10.1007/BFb0070055](https://doi.org/10.1007/BFb0070055)) In: H. T. Ku, L. N. Mann, J. L. Sicks, J. C. Su (eds.), *Proceedings of the Second Conference on Compact Transformation Groups* Lecture Notes in Mathematics, vol 298. Springer 1972 ([doi:10.1007/BFb0070029](https://link.springer.com/book/10.1007/BFb0070029))

* {#Illman73} [[Sören Illman]], Section 2 of: *Equivariant algebraic topology*, Annales de l'Institut Fourier, Tome 23 (1973) no. 2, pp. 87-91 ([doi:10.5802/aif.458](https://doi.org/10.5802/aif.458))

(Which, in hindsight and with [[Elmendorf's theorem]], gives a deeper justification for the parametrization over the [[orbit category]] already proposed in [Bredon 67a](#Bredon67a), [Bredon 67b](#Bredon67b).)

* {#Waner80} [[Stefan Waner]], _Equivariant Homotopy Theory and Milnor's Theorem_, Transactions of the American Mathematical Society Vol. 258, No. 2 (Apr., 1980), pp. 351-368 ([JSTOR](http://www.jstor.org/stable/1998061))

Proof that $G$-[[absolute neighbourhood retract|ANRs]] have the [[equivariant homotopy type]] of [[G-CW-complexes]] (for $G$ a [[compact Lie group]]):

* {#Kwasik81} [[Slawomir Kwasik]], _On the Equivariant Homotopy Type of $G$-ANR's_, Proceedings of the American Mathematical Society Vol. 83, No. 1 (Sep., 1981), pp. 193-194 (2 pages) ([jstor:2043921](https://www.jstor.org/stable/2043921))

Textbook accounts:

* {#tomDieck87} [[Tammo tom Dieck]], Sections I.1, I.2 of: _[[Transformation Groups]]_, de Gruyter 1987  ([doi:10.1515/9783110858372]( https://doi.org/10.1515/9783110858372))

* {#Lueck89} [[Wolfgang Lück]], Sections I.1, I.2 of: _Transformation Groups and Algebraic K-Theory_, Lecture Notes in Mathematics **1408** (Springer 1989) ([doi:10.1007/BFb0083681](https://doi.org/10.1007/BFb0083681))

* {#May96} [[Peter May]] et al., Section I.3 of:  _Equivariant homotopy and cohomology theory_, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996 ([ISBN: 978-0-8218-0319-6](https://bookstore.ams.org/cbms-91/?startBookmarkIdx=200)  [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/alaska1.pdf), [[MayEtAlEquivariant96.pdf:file]])

Lecture notes:

* {#Blumberg17} [[Andrew Blumberg]], _Equivariant homotopy theory_, 2017 ([pdf](https://www.ma.utexas.edu/users/a.debray/lecture_notes/m392c_EHT_notes.pdf), [GitHub](https://github.com/adebray/equivariant_homotopy_theory))

[[!redirects G-CW complexes]]

[[!redirects G-CW-complex]]
[[!redirects G-CW-complexes]]

[[!redirects G-cell complex]]
[[!redirects G-cell complexes]]


G-delta subspace

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#Contents#
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## Definition

+-- {: .num_defn #Gdelta}
###### Definition
A **G-delta**, $G_\delta$, subset of a [[topological space]] is a set that can be written as the intersection of a [[countable family]] of open sets.
=--

## Results

One place where $G_\delta$-subsets occur is when looking at [[continuous maps]] from an arbitrary [[topological space]] to a [[metric space]] (or, more generally, a [[first countable space]]).  In particular, when considering continuous real-valued functions.  Thus we have the following connections to the [[separation axioms]].

+-- {: .num_theorem #PerNorm}
###### Theorem
A [[normal space]] in which every closed set is a $G_\delta$-set is [[perfectly normal space|perfectly normal]].
=--

+-- {: .num_theorem #ComReg}
###### Theorem
In a [[completely regular space]], every singleton set that is a $G_\delta$-set is the unique global maximum of a continuous real-valued function.
=--

+-- {: .proof}
###### Proof
One direction is obvious.  For the other, let $v$ be a point in a completely regular space $X$ such that $\{v\}$ is a $G_\delta$-set.  Let $\{V_n\}$ be a sequence of open sets such that $\bigcap V_n = \{v\}$.  We now define a sequence of functions $(f_n)$ recursively with the properties:

1. $f_n \colon X \to [0,1]$ is a continuous function,
1. $f_n(v) = 1$,
1. $f_n^{-1}(1)$ is a neighbourhood of $v$,
1. for $n \gt 1$, $f_n$ has support in $V_n \cap f_{n-1}^{-1}(1)$ whilst $f_1$ has support in $V_1$,

Having defined $f_1, \dots, f_{n-1}$, we define $f_n$ as follows.  Since $V_n \cap f_{n-1}^{-1}(1)$ is a neighbourhood of $v$ and $X$ is completely regular, there is a continuous function $\tilde{f}_n \colon X \to [0,1]$ with support in this neighbourhood and such that $\tilde{f}_n(v) = 1$.  We then compose with a continuous, increasing surjection $[0,1] \to [0,1]$ which maps $[\frac12,1]$ to $1$.  The resulting function is the required $f_n$.

We then define a function $f \colon X \to [0,1]$ by

$$
f(x) \coloneqq \sum_{n=1}^\infty \frac1{2^n} f_n(x).
$$

By construction, $f^{-1}(1) = \{v\}$.

We need to prove that this is continuous.  First, note that if $f_n(x) \ne 0$ then $f_k(x) = 1$ for $k \lt n$ and if $f_n(x) \ne 1$ then $f_k(x) = 0$ for $k \gt n$.  Hence the preimage under $f$ of $(\frac{2^k-1}{2^k}, \frac{2^{k+1}-1}{2^{k+1}})$ is $f_n^{-1}(0,1)$ and $f$ restricted to this preimage is a scaled translate of $f_n$.
From this, we deduce that the preimage of any open set not containing $1$ is open.  Thus $f$ is continuous everywhere except possibly at $v$.  Continuity at $v$ is similarly simple: given a set of the form $(1 -\epsilon,1]$ then there is some $n$ such that $2^{-n} \lt \epsilon$, whence $f^{-1}(1-\epsilon,1]$ contains all points such that $f_k(x) = 1$ for $k \le n$, which by construction is a neighbourhood of $v$.  Hence $f$ is continuous and has a single global maximum at $v$.
=--

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G-manifold

For $G$ a [[Lie group]], by a $G$-manifold one may mean

* a [[smooth manifold]] with $G$-[[special holonomy]];

* a [[smooth manifold]] equipped with a [[smooth function|smooth]] [[group action]] by $G$

  (the smooth version of a [[topological G-space]], see also at *[[equivariant differential topology]]*)

[[!redirects G-manifolds]]

category: disambiguation


G-norm

# G-norms
* table of contents
{: toc}

## Idea

A G-norm is like a [[normed vector space|norm]] (or better, an [[F-norm]]) on a [[vector space]], but on an [[abelian group]] instead.  Just as the [[topological structure]] of any [[topological vector space]] may be specified by a family of F-pseudonorms, so the topological structure of any [[topological abelian group]] may be specified by a family of G-pseudonorms.  A G-norm on an abelian group is equivalent to a translation-invariant [[metric space|metric]] on it.

## Definitions

Let $G$ be an [[abelian group]].

A __G-pseudonorm__, or __G-seminorm__, on $G$ is a [[function]] $\rho\colon G \to \mathbb{R}$ that satisfies the following conditions:

*  $\rho(0) \leq 0$, where $0$ is the [[identity element]] of $G$;
*  $\rho(x + y) \leq \rho(x) + \rho(y)$ for all $x, y$ in $G$; and
*  $\rho(-x) \leq \rho(x)$ for all $x$ in $G$.

From these axioms, the following simple conditions (sometimes included in the definition) follow:

*  $\rho(x) \geq 0$ for all $x$ in $G$
*  $\rho(0) = 0$;
*  $\rho(-x) = \rho(x)$; and
*  $\rho(n x) \leq {|n|} \rho(x)$ for every [[integer]] $n$ and all $x$ in $G$.

A G-pseudonorm is __definite__ (or __postive-definite__, but we have positivity anyway) if in addition:

*  $\rho(x) \ne 0$ for all $x \ne 0$ in $G$.

And of course, from that follows:

*  $\rho(x) \gt 0$ for all $x \ne 0$ in $G$.

A __G-norm__ is a definite G-pseudonorm.

A G-(pseudo)norm is __homogeneous__ (or $\mathbb{Z}$-homogeneous) if we have:

*  $\rho(n x) \geq {|n|} \rho (x)$ for infinitely many integers $n$, for all $x$ in $G$.

It then follows that

*  $\rho(n x) = {|n|} \rho (x)$ for every integer $n$ and all $x$ in $G$.

While homogeneous G-norms are particularly nice, we need general G-pseudonorms if we wish to describe arbitrary [[topological abelian groups]].

## Examples

Given any translation-invariant [[metric space|(pseudo)metric]] $d$ on $G$, we get a G-(pseudo)norm $\rho$ on $G$ by $\rho(x) \coloneqq d(0,x)$.  Conversely, given any G-(pseudo)norm $\rho$, we get a translation-invariant (pseudo)metric $d$ by $d(x,y) \coloneqq \rho(y - x)$.  These operations are [[inverses]].  (We could add a 'quasi-' in here if we drop the rule that $\rho(-x) = \rho(x)$.  But note that G-quasinorms and translation-invariant quasimetrics on an [[abelian monoid]] do *not* correspond.)

If $G$ happens to be the underlying abelian group of a real (or complex) [[vector space]] $V$, then any [[F-norm|F-(pseudo)norm]] on $V$ is a G-(pseudo)norm on $G$, but not conversely.  Similarly, any [[normed vector space|(pseudo)norm]] on $V$ is a homogeneous G-(pseudo)norm on $G$, but not conversely.

The obvious norms on $\mathbb{Z}^n$, seen as a subset of $\mathbb{R}^n$ with one of its usual structures as a [[Banach space]], is a homogeneous G-norm.

Any abelian group has a G-norm given by $\rho(x) = 1$ whenever $x \ne 0$ (and of course $\rho(x) = 0$ otherwise).  Except on the [[trivial group]], this is not homogeneous.  It corresponds to the [[discrete metric]] on $G$.

Given any [[topological abelian group]] $G$ and any [[neighbourhood]] $N$ of $0$, let $B \coloneqq N \cap -N = \{ x \;|\; x \in N,\; -x \in N \}$; then $B$ is also a neighbourhood of $0$.  Let $B_0 \coloneqq G$, let $B_1 \coloneqq B$, and recursively choose $B_{n^+}$ so that $x + y + z \in B_n$ whenever $x, y, z \in B_{n^+}$ (which is possible since addition is continuous).  Then
$$ \rho (x) \coloneqq \inf \Big\{ \sum_{i = 1}^m \inf \{ 2^{-n} \;|\; x_i \in B_n \} \;\Big|\; x = \sum_{i = 1}^m x_i \Big\} $$
defines a G-pseudonorm on $G$ such that, given any [[net]] $(x_\nu)$ in $G$, we have that $\rho(x_\nu)$ converges to $0$ if and only if $x_\nu \in N$ holds eventually.

## Families of G-pseudonorms

Any family of G-pseudonorms on an abelian group $G$ makes $G$ into a [[topological abelian group]] (TAG), and every TAG structure on $G$ arises in this way.  However, different collections of G-pseudonorms may determine the same topological structure.

In one direction, let $G$ be an abelian group, and suppose that we equip $G$ with an arbitrary collection $D$ of G-pseudonorms.  Every G-pseudonorm determines a pseudometric, and these pseudometrics generate a [[gauge space]] structure on $G$ and thence a [[topological structure]].  Because the pseudometrics involved are translation-invariant, we have in fact made $G$ into a TAG. In more detail:  A [[subset]] $U$ of $G$ is [[open subset|open]] if and only if, for every $x$ in $U$, for some [[list]] $\rho_1,\ldots,\rho_n$ from $D$ and some [[real number]] $\epsilon \gt 0$, for every $y$ in $G$, if $\rho_i(y - x) \lt \epsilon$ for every $i$, then $y \in U$.  Then you can check that these subsets form a topology relative to which the group operations are continuous.

It may also be nice to look at the [[uniform space]] structure on $G$; the gauge and the TAG structure determine the same uniform structure.  Explicitly, a [[binary relation]] $\sim$ on $G$ is an [[entourage]] if and only if, for some list $\rho_1,\ldots,\rho_n$ from $D$ and some real number $\epsilon \gt 0$, for every $x$ and $y$ in $G$, if $\rho_i(y - x) \lt \epsilon$ for every $i$, then $x \sim y$.  Then these entourages form the uniform structure on $G$ which is compatible with the group structure and whose underlying topological structure is the one above.

Conversely, let $G$ be a TAG.  Then the collection of all [[continuous map|continuous]] G-pseudonorms $\rho\colon G \to \mathbb{R}$ generates the topological structure on $G$.  The proof is complicated, but essentially it amounts to this: applying the final example from the Examples section above to each neighbourhood of $0$ (or at least to each neighbourhood in a neighbourhood base), check that the G-pseudonorms defined are continuous, and check that there are enough of them to generate a topology at least as strong as the actual topology on $G$; the converse is immediate.  In other words, the hard thing that has to be checked is that there are enough continuous G-pseudonorms.

## In weak foundations

The only really tricky part is the proof that there are enough continuous G-pseudonorms to generate the topology on any TAG $G$.  The development seems to be [[predicative mathematics|predicative]] over $\mathbb{R}$; you can\'t speak of the collection of *all* continuous G-pseudonorms if you are being strongly predicative, but you really only need one for each neighbourhood in a neighbourhood base of $G$.  However, it is not [[constructive mathematics|constructive]] or predicative over $\mathbb{N}$, because the infima may not exist.  Also, we use [[dependent choice]].

The same problems arise in proving that every [[topological vector space]] structure is generated by some family of [[F-norm|F-pseudonorms]]; on the other hand, [[locally convex spaces]] (which are generated by [[pseudonormed space|pseudonorms]]) are better behaved.  There may be a similar theory of locally convex TAGs based on homogeneous G-pseudonorms, but I haven\'t looked into this.

It would be natural, in constructive mathematics, to attempt the development with [[locale|localic]] groups.  Even in [[classical mathematics]], however, there may be (and are) [[sober space|sober]] TAGs which cannot be interpreted as localic groups, such as the additive group of [[rational numbers]] with its topology as a [[subspace]] of the [[real line]].  Probably we have to start by requiring a G-pseudonorm to be a [[continuous map]] from the localic group $G$ to the [[locale of real numbers]], rather than starting with a discrete abelian group, but I haven\'t looked into this further.

## References

_[[HAF]]_, Chapters 22 and 26.  See especially Section 26.29 for the last Example.

[[!redirects G-norm]]
[[!redirects G-norms]]
[[!redirects G-pseudonorm]]
[[!redirects G-pseudonorms]]
[[!redirects G-seminorm]]
[[!redirects G-seminorms]]


+-- {: .rightHandSide}
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#### Homotopy theory
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=--
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#Contents#
* table of contents
{:toc}

## Statement

+-- {: .num_prop #RepresentationSpheresAreGCWComplexes}
###### Proposition
**(G-representation spheres are G-CW-complexes)**

For $G$ a [[compact Lie group]] (e.g. a [[finite group]]) and $V \in RO(G)$ a [[finite dimensional vector space|finite-dimensional]] [[orthogonal group|orthogonal]] $G$-[[linear representation]], the [[representation sphere]] $S^V$ admits the structure of a [[G-CW-complex]].

=--

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

Observe that we  have a $G$-equivariant [[homeomorphism]] between the representation sphere of $V$ and the [[unit sphere]] in $\mathbb{R} \oplus V$, where $\mathbb{R}$ is the 1-dimensional [[trivial representation]]  ([this Prop.](representation+sphere#RepresentationSpheresAsUnitSpheres))

\[
  \label{RepresentationSphereAsUnitSphere}
  S^V \;\simeq\; S(\mathbb{R} \oplus V)
  \,.
\]

It is thus sufficient to show that unit spheres in orthogonal representations admit G-CW-complex structure.

This in turn follows as soon as there is a $G$-equivariant [[triangulation]] of $S(\mathbb{R}\oplus V)$, hence a [[triangulation]] with the property that the $G$-[[action]] restricts to a [[bijection]] on its sets of $k$-dimensional cells, for each $k$. Because then if $G/H$ is an [[orbit]] of this $G$-action on the set of $k$-cells, we have a cell $G/H \times D^k$ of an induced [[G-CW-complex]].

Since the unit spheres in (eq:RepresentationSphereAsUnitSphere) are [[smooth manifolds]] with [[smooth function|smooth]] $G$-[[action]], the existence of such $G$-[[equivariant triangulations]] follows for general [[compact Lie groups]] $G$ from the _[[equivariant triangulation theorem]]_ ([Illman 83](#Illman83)).

More explicitly, in the case that $G$ is a [[finite group]] such an equivariant triangulation may be constructed as follows:

Let $\{b_1, b_2, \cdots, b_{n+1}\}$ be an [[orthonormal basis]] of $\mathbb{R} \oplus V$. Take then as [[vertices]] of the triangulation all the distinct points $\pm g(b_i) \in \mathbb{R} \oplus V$, and as [[edges]] the [[geodesics]] (great circle segments) between nearest neighbours of these points, etc.

=--

## Examples

+-- {: .num_example #ZnCWDecompositionOf2SphereWithRotationAction}
###### Example
**($\mathbb{Z}_n$-CW-decomposition of 2-sphere with rotation action)**

For $n \in \mathbb{N}$, $n \geq 2$, let $\mathbb{Z}_n \hookrightarrow SO(2)$ be the [[cyclic group]] [[action|acting]] by [[rotations]] on the [[plane]] $\mathbb{R}^2$. Writing $\mathbb{R}^2_{rot_n}$ for the corresponding [[representation]], its [[representation sphere]] $S^{\mathbb{R}^2_{rot_n}}$ has a [[G-CW-complex]] [[structure]] as follows:

1. The [[vertices]] are the two [[fixed point]] [[poles]] $(G/G) \times \{0\} = \{0\}$ and $(G/G) \times \{\infty\} = \{\infty\}$;

1. the [[edges]] are $n$ great circle arcs obtained from any one such arc from $0$ to $\infty$ together with all its [[images]] under $G$, hence together a free $G$-[[orbit]] $(G/1) \times D^1 = G \times D^1$ of 1-cells;

1. the [[faces]] are the $n$ [[bigons]] between each such arc and the next one, hence together a free [[orbit]] $(G/1) \times D^2 = G \times D^2$ of 2-cells.

The graphics on the right illustrates this cell decomposition for $n = 8$:

> graphics grabbed from [here](https://www.deviantart.com/insanitygonewrong/art/segmented-sphere-103786045)

=--

## References

* {#Illman78} [[Sören Illman]], _Smooth equivariant triangulations ofG-manifolds for $G$ a finite group_, Math. Ann. (1978) 233: 199 ([doi:10.1007/BF01405351](https://doi.org/10.1007/BF01405351))

* {#Illman83} [[Sören Illman]], _The Equivariant Triangulation Theorem for Actions of Compact Lie Groups_, Mathematische Annalen (1983) Volume: 262, page 487-502 ([dml:163720](https://eudml.org/doc/163720))

[[!redirects representation spheres are G-CW-complexes]]


G-set

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[topological group]] $G$, a (continuous) $G$-set is a [[set]] $X$ equipped with a continuous group [[action]] $\mu: G \times X \to X$, where $X$ is given the discrete topology.

In the case where $G$ is a discrete group, the continuity requirement is void, and this is just a [[permutation representation]] of the discrete group $G$.

Note that since $X$ must be given the discrete topology, this behaves rather unlike [[topological G-spaces]]. In particular, a topological group does not act continuously on itself, in general. Thus this notion is not too useful when $G$ is a "usual" topology group like $SU(2)$. Instead, the topology on the group acts as a [[filter]] of subgroups (where the filter contains the open subgroups), and each element of a continuous $G$-set is required to have a "large" stabilizer.

The $G$-sets form a [[category]], where the morphisms are the $G$-invariant maps. See [[category of G sets]].

## Properties

### Relation to $G$-orbits

\begin{remark}\label{GActionsAndGOrbits}
**([[G-set|$G$-sets]] are the [[free coproduct completion]] of [[orbit category|$G$-orbits]])**
\linebreak
  Let $G \,\in\, Grp(Set)$ be a [[discrete group]].
Since every [[G-set]] $X$ decomposes as a [[disjoint union]] of [[transitive actions]], namely of [[orbits]] of [[elements]] of $X$, the defining inclusion of the [[orbit category]] into [[G-set|$G Set$]] exhibits the latter as the [[free coproduct completion]] of the [[orbit category]] (see also [this Prop.](free+coproduct+completion#CategoriesOfCoproductsOfConnectedObjects)).
\end{remark}

### For topological groups

+-- {: .num_prop}
###### Proposition {#ContinuousCharacterization}
Let $G$ be a [[topological group]], and $X$ be a set with a $G$ action $\mu: G \times X \to X$. Then the action is continuous if and only if the stabilizer of each element is open.
=--

+-- {: .proof}
###### Proof
Suppose $\mu$ is continuous. Since $X$ has the discrete topology, $\{x\}$ is an open subset of $X$. So $\mu^{-1}(\{x\})$ is open. So we know the stabilizer
$$
  I_x = \{g \in G: g \cdot x = x\} = \{g \in G: (g, x) \in \mu^{-1}(\{x\})\}
$$
is open.

Conversely, suppose each such set is open. Given any (necessarily open) subset $A \subseteq X$, its inverse image is
$$
  \mu^{-1}(A) = \bigcup_{a \in A} \mu^{-1}(\{a\}).
$$
So it suffices to show that each $\mu^{-1}(\{a\})$ is open. We have
$$
  \mu^{-1}(\{a\}) = \bigcup_{x \in X}\{g \in G: g \cdot x = a\} \times \{x\}.
$$
Thus we only have to show that for each $a, x \in X$, the set $\{g \in G: g \cdot x = a\}$ is open. If there is no such $g$, then this is empty, hence open. Otherwise, let $g_0$ be such that $g_0 \cdot x= a$.  Then we have
$$
  \{g \in G: g \cdot x = a\} = g_0 \cdot I_x.
$$
Since $g_0$ is a homeomorphism, and $I_x$ is open, this is open. So done.
=--

## Examples

### Discrete groups

In the following examples, all groups are discrete.

* A $\mathbb{Z}_2$-set is a set equipped with an [[involution]].

* Any [[permutation]] $\pi : X \to X$ gives $X$ the structure of a $\mathbb{Z}$-set, with the action of $\mathbb{Z}$ on $X$ defined by iterated composition of $\pi$ or $\pi^{-1}$.

* $G$ is itself a $G$-set via the (left or right) [[regular representation]].

* A [[normal subgroup]] $N \lhd G$ defines a $G$-set by the action of [[conjugation]].

* For $G$ a [[finite group]] then [[Mackey functors]] on [[finite set|finite]] $G$-sets are equivalent to genuine [[G-spectra]].

### Continuous groups

* The group $\Sigma_N$ of permutations of the natural numbers can be given the topology generated by the stabilizers of finite subsets of $N$. This acts continuously on $N$. This is used in the construction of the [[basic Fraenkel model]]. See also [[nominal sets]].

## Related concepts

* [[topological G-space]], [[simplicial group action]]

* [[category of G-sets]]

* [[class equation]]

* [[Galois theory]]

* [[action]]: a $G$-set is a set with an _action_ of the given group, $G$.

* [[Borel model structure]]

* [[Burnside ring]], [[Burnside category]]

* [[Burnside marks]]

* [[table of marks]]

## References

An early account (where the term "representation group" is used to refer to a [[finite set]] equipped with a [[permutation]] [[action]]):

* {#Burnside01} [[William Burnside]], _On the Representation of a Group of Finite Order as a Permutation Group, and on the Composition of Permutation Groups_, Proceedings of the American Mathematical Society 1901 ([doi:10.1112/plms/s1-34.1.159](https://doi.org/10.1112/plms/s1-34.1.159))

For a more modern account see

* {#tomDieck79} [[Tammo tom Dieck]], chapter 1 of _[[Transformation Groups and Representation Theory]]_, Springer 1979

Basic exposition of $G$-sets as a [[Grothendieck topos]]:

* [[Jaap van Oosten]], (eg. p. 32 in:) *Topos Theory* (2018) [[pdf](https://webspace.science.uu.nl/~ooste110/syllabi/topostheory.pdf), [[vanOosten-ToposTheory.pdf:file]]]

[[!redirects G-sets]]


G-spectrum

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $G$ a [[compact Lie group]] (or more generally a [[compact topological group]]) the concept of _$G$-spectrum_ (or _$G$-equivariant spectrum_) is the generalization of that of [[spectrum]] as one passes from [[stable homotopy theory]] to [[equivariant stable homotopy theory]], or more generally, as $G$ is allow to vary, to [[global equivariant stable homotopy theory]].

Where the ordinary concept of [[spectrum]] is given in terms of [[looping and delooping]] of ordinary [[topological spaces]] by ordinary [[spheres]], a $G$-spectrum is instead given by looping and delooping of [[topological G-spaces]] with respect to [[representation spheres]] of $G$, namely [[one-point compactifications]] of linear $G$-[[representations]], for all representations appearing in a chosen "[[G-universe]]".

Such a [[G-universe]] is called _complete_ if it contains every [[irreducible representation]] of $G$, and the spectra modeled on such a complete $G$-universe are the _genuine $G$-spectra_. At the other extreme, if the [[G-universe]] contains only the trivial representations, then the resulting spectra are the _[[spectra with G-action]]_, also called _[[naive G-spectra]]_ for emphasis of the distinction to the previous case.

The genuine $G$-spectra are richer than _[[spectra with G-action]]_ and have better homotopy-theoretic properties. In particular the genuine [[equivariant cohomology]] theories which they [[Brown representability theorem|represent]] have [[suspension isomorphisms]] for suspension by all [[representation spheres]] and with respect to [[RO(G)-grading]].

When $G$ is the trivial group, a $G$-spectrum is also known as a [[coordinate-free spectrum]].

## Models

There are various equivalent ways to present genuine $G$-spectra.

### Indexed on all representations
 {#IndexedOnAllRepresentations}

([May 96, chapter XII](#May96), [Greenlees-May, section 2](#GreenleesMay))

Fix a [[G-universe]]. For $V$ any [[orthogonal group|orthogonal]] [[representation]] in the universe, write $S^V$ for its [[representation sphere]]. For $V \hookrightarrow W$ a subrepresentation, write $W-V$ for the [[orthogonal complement]] representation.

A _$G$-prespectrum_ $E$ is an assignment of a [[pointed topological space|pointed]] [[G-space]] $E_V$ to each $G$-[[representation]] $V$ (in the given [[G-universe]]), equipped for each subrepresentation $V \hookrightarrow W$ with a pointed $G$-equivariant [[continuous function]]

$$
  \sigma_{V,W}
  \;\colon\;
  S^{W-V} \wedge E_V \longrightarrow E_W
$$

such that

1. $\sigma_{V,V} = id$;

1. for any $Z \hookrightarrow V \hookrightarrow W$ we have [[commuting diagrams]]

   $$
     \array{
       S^{Z-W}\wedge S^{V-W} \wedge E_V
       &\stackrel{S^{Z-W}\wedge(\sigma_{V,W})}{\longrightarrow}& S^{Z-W}E_W
       \\
       \downarrow && \downarrow^{\mathrlap{\Sigma_{Z,W}}}
       \\
       S^{Z-V}E_V &\stackrel{\sigma_{V,Z}}{\longrightarrow}& E_Z
     }
   $$

Write

$$
  \tilde \sigma_{V,W} \;\colon\; E_V \longrightarrow \Omega^{W-V}E_W
$$

for the [[adjuncts]] of these structure maps.

A $G$-prespectrum is called (at least in ([May 96, chapter XII](#May96)))

* a _$G$-$\Omega$-spectrum_ if all the $\sigma_{V,W}$ are [[weak homotopy equivalences]];

* a _$G$-spectrum_ if all the $\sigma_{V,W}$ are [[homeomorphisms]].

### Via orthogonal spectra with $G$-action

While the definition of [spectra indexed on all representations](#IndexedOnAllRepresentations) manifestly relates to the [[suspension isomorphism]] for smashing with [[representation spheres]] and shifting in [[RO(G)-grading]], the information encoded in the objects in this definition has much redundancy. A "smaller" definition of genuine $G$-spectra is given by [[orthogonal spectra]] equipped with $G$-action ([Mandell-May 04](#Mandellmay04), [Schwede 15](#Schwede15)).

### Via Mackey functors
 {#RelationToMackeyFunctors}

For $G$ a [[finite group]] then genuine $G$-spectra are equivalent to  _[[Mackey functors]]_ on the category of [[finite set|finite]] [[G-sets]].

([Guillou-May 11, theorem 0.1](#GuillouMay11), [Barwick 14, below example B.6](#Barwick14)).

### Via excisive functors

Characterization of $G$-spectra [via excisive functors]() on [[G-spaces]] is in ([Blumberg 05](#Blumberg05)).

## Properties

### Relation to spectra with $G$-action

(e.g. [Carlsson 92, p. 14](#Carlsson92), [GreenleesMay, p.16](#GreenleesMay))

### Relation to Borel-equivariant spectra

In the general context of ([[global equivariant stable homotopy theory|global]]) [[equivariant stable homotopy theory]], [[Borel equivariant cohomology|Borel-equivariant]] spectra are those which are _right induced_ from plain [[spectra]], hence which are in the [[essential image]] of the [[right adjoint]] to the [[forgetful functor]] from [[equivariant spectra]] to plain [[spectra]].

([Schwede 18, Example 4.5.19](#Schwede18))

### Equivariant stable Whitehead theorem
 {#EquivariantWhitehead}

The equivariant version of the stable [[Whitehead theorem]] holds:

a map of $G$-spectra $f \colon E \longrightarrow F$ is a [[weak equivalence]] (e.g. an $RO(G)$-degree-wise [[weak homotopy equivalence]] of [[topological G-spaces]] in the [model via indexing on all representations](#IndexedOnAllRepresentations)) precisely it if induces [[isomorphisms]] $\pi_\bullet(f) \colon \pi_\bullet(E) \longrightarrow \pi_{\bullet}(F)$ on all [[equivariant homotopy group]] [[Mackey functors]].

([Greenlees-May, theorem 2.3](#GreenleesMay))

## Related concepts

* [[equivariant stable homotopy category]]

* [[RO(G)-grading]]

* [[fixed point spectrum]]

* [[geometric fixed points]]

## Examples

* [[equivariant sphere spectrum]]

## References

Good lecture notes are

* {#Blumberg17} [[Andrew Blumberg]], _Equivariant homotopy theory_, 2017 ([pdf](https://www.ma.utexas.edu/users/a.debray/lecture_notes/m392c_EHT_notes.pdf), [GitHub](https://github.com/adebray/equivariant_homotopy_theory))

The concept of genuine $G$-spectra is due to

* [[L. Gaunce Lewis, Jr.]], [[Peter May]], M. Steinberger, and J. E. McClure, _Equivariant stable homotopy theory_, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986, With contributions by J. E. McClure. MR 866482 (88e:55002)

and in terms of [[orthogonal spectra]] due to

* {#MandellMay04} [[Michael Mandell]], [[Peter May]], _Equivariant orthogonal spectra and S-modules_. Preprint, April 29, 2000, ([KTheory:0408](http://www.math.uiuc.edu/K-theory/0408/))

Further accounts:

* {#tomDieck87} [[Tammo tom Dieck]], Section II.6 of: _[[Transformation Groups]]_, de Gruyter 1987  ([doi:10.1515/9783110858372]( https://doi.org/10.1515/9783110858372))

and in the context of the [[Arf-Kervaire invariant problem]]:

* [[Michael Hill]], [[Michael Hopkins]], [[Doug Ravenel]], _The Arf-Kervaire invariant problem in algebraic topology: introduction_, Current developments in mathematics, 2009, Int. Press, Somerville, MA, 2010, pp. 23–57. MR 2757358

  _On the non-existence of elements of Kervaire invariant one_, ([arXiv:0908.3724](http://arxiv.org/abs/0908.3724))

  _The Arf-Kervaire problem in algebraic topology: sketch of the proof_, Current developments in mathematics, 2010, Int. Press, Somerville, MA, 2011, pp. 1–43

Surveys and introductions include

* {#Carlsson92} [[Gunnar Carlsson]], _A survey of equivariant stable homotopy theory_,Topology, Vol 31, No. 1, pp. 1-27, 1992 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/carlsson1.pdf))

* {#GreenleesMay} [[John Greenlees]], [[Peter May]], section 2 of _[[Equivariant stable homotopy theory]]_, in I.M. James (ed.), _[[Handbook of Algebraic Topology]]_ , pp. 279-325. 1995. ([pdf](http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf))

Lecture notes include

* {#May96} [[Peter May]], _Equivariant homotopy and cohomology theory_ CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. ([pdf](http://www.math.uchicago.edu/~may/BOOKS/alaska.pdf))

* {#Schwede15} [[Stefan Schwede]], _[[Lectures on Equivariant Stable Homotopy Theory]]_

See also

* {#Schwede18} [[Stefan Schwede]], _[[Global homotopy theory]]_ 2018  ([arXiv:1802.09382](https://arxiv.org/abs/1802.09382))

Relation to [[Mackey functors]]:

* {#GuillouMay11} [[Bert Guillou]], [[Peter May]],  _Models of $G$-spectra as presheaves of spectra, ([arXiv:1110.3571](http://arxiv.org/abs/1110.3571))

  _Permutative $G$-categories in equivariant infinite loop space theory ([arXiv:1207.3459](http://arxiv.org/abs/1207.3459))

* {#Nardin12} [[Denis Nardin]], section 2.6 of _Stability and distributivity over orbital ∞-categories_, 2012 ([pdf](https://www.math.univ-paris13.fr/~nardin/thesis.pdf))

* {#Barwick14} [[Clark Barwick]], _Spectral Mackey functors and equivariant algebraic K-theory (I)_, Adv. Math., 304:646–727 ([arXiv:1404.0108](http://arxiv.org/abs/1404.0108))

For more references see at _[[equivariant stable homotopy theory]]_ and at _[[Mackey functor]]_

Characterization via [[excisive functors]] is in

* {#Blumberg05} [[Andrew Blumberg]], _Continuous functors as a model for the equivariant stable homotopy category_ ([arXiv:math.AT/0505512](http://arxiv.org/abs/math.AT/0505512))

In the case of a [[cyclic group]] of prime order, genuine $G$-spectra admit a simple model which amounts to specifying a spectrum $E$, a $G$-action on $E$, a genuine fixed point spectrum $E^G$, and a diagram $E_{hG} \to E^G \to E^{hG}$.  See Example 3.29 in:

* [[Saul Glasman]], _Stratified categories, geometric fixed points and a generalized  Arone-Ching theorem_, [arXiv:1507.01976](https://arxiv.org/abs/1507.01976).

A perspective on the category of genuine G-spectra as a lax limit over those of "naive" G-spectra is given in

* {#AyalaMazelGeeRozenblyum17} [[David Ayala]], [[Aaron Mazel-Gee]], [[Nick Rozenblyum]], _A naive approach to genuine $G$-spectra and cyclotomic spectra_ ([arXiv:1710.06416](https://arxiv.org/abs/1710.06416))

Universal properties of the (∞,1)-category of genuine $G$-spectra are considered in

* Bastiaan Cnossen, _Twisted ambidexterity in equivariant homotopy theory_, [arXiv](https://arxiv.org/abs/2303.00736).

* Tim Campion, _Free duals and a new universal property for stable equivariant homotopy theory_, [arXiv](https://arxiv.org/abs/2302.04207).

[[!redirects G-spectra]]

[[!redirects genuine G-spectrum]]
[[!redirects genuine G-spectra]]
[[!redirects G-equivariant spectrum]]
[[!redirects G-equivariant spectra]]
[[!redirects equivariant spectrum]]
[[!redirects equivariant spectra]]

[[!redirects genuine equivariant spectrum]]
[[!redirects genuine equivariant spectra]]


G-structure

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **G-structure** on an $n$-[[manifold]] $M$, for a given structure group $G$, is a $G$-subbundle of the [[frame bundle]] (of the [[tangent bundle]]) of $M$.

Equivalently, this means that a $G$-structure is a choice of [[reduction of structure groups|reduction]] of the canonical structure group [[general linear group|GL(n)]] of the [[principal bundle]] to which the [[tangent bundle]] is [[associated bundle|associated]] along the given inclusion $G \hookrightarrow GL(n)$.

More generally, one can consider the case $G$ is not a [[subgroup]] but equipped with any group homomorphism $G \to GL(n)$. If this is instead an [[epimorphism]] one speaks of a [[lift of structure groups]].

Both cases, in turn, can naturally be understood as special cases of [[twisted differential c-structures]], which is a notion that applies more generally to [[principal infinity-bundles]].

In this language, a $G$-structure is a lift _in [[classifying stacks]]_ ([[differentiable stack|differentiable]] [[delooping]] [[stacks]] $\mathbf{B}G$ of [[Lie groups]] $G$) of the classifying map of the [[tangent bundle]]

$$
  \array{
    && \mathbf{B}G
    \\
    &
    {}^{ \mathllap{G structure} }\nearrow
    & \big\downarrow^{ str }
    \\
    X
      &\underset{
        \vdash T X
      }{\longrightarrow}&
    \mathbf{B} GL(n)
  }
$$

Beware the distinction to _[[tangential structure]]_ on $X$, where such a lift is considered (only) at the level of underlying [[classifying spaces]].

## Definition

### General

Given a [[smooth manifold]] $X$ of [[dimension]] $n$ and given a [[Lie group|Lie]] [[subgroup]] $G \hookrightarrow GL(n)$ of the [[general linear group]], then a _$G$-structure_ on $X$ is a [[reduction of structure groups|reduction of the structure group]] of the [[frame bundle]] of $X$ to $G$.

There are many more explicit (and more abstract) equivalent ways to say this, which we discuss below. There are also many evident variants and generalizations.

Notably one may consider reductions of the frames in the $k$th order [[jet bundle]]. (e. g. [Alekseevskii](#Alekseevskii)) This yields order $k$ $G$-structure and the ordinary $G$-structures above are then first order.

Moreover, the definition makes sense for generalized [[manifolds]] modeled on other base spaces than just [[Cartesian spaces]]. In particular there are evident generalizations to [[supermanifolds]] and to [[complex manifolds]].

### In terms of subbundles of the frame bundle
 {#InTermsOfSubbundlesOfTheFrameBundle}

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

Given a [[smooth manifold]] $X$ of [[dimension]] $n$ with [[frame bundle]] $Fr(X)$, and given a [[Lie group]] [[monomorphism]]

$$
  G \longrightarrow GL(\mathbb{R}^n)
$$

into the [[general linear group]], then a _$G$-structure_ on $X$ is an $G$-[[principal bundle]] $P \to X$ equipped with an inclusion of [[fiber bundles]]

$$
  \array{
    P &&\hookrightarrow&& Fr(X)
    \\
    & \searrow && \swarrow
    \\
    && X
  }
$$

which is $G$-equivariant.

=--

([Sternberg 64, section VII, def. 2.1](#Sternberg64)).

+-- {: .num_remark}
###### Remark

From this perspective, a $G$-structure consists of the collection of all $G$-[[frame field|frames]] on a manifold. For instance for an [[orthogonal structure]] it consists of all those frames which are pointwise an [[orthonormal basis]] of the [[tangent bundle]] (with respect to the [[Riemannian metric]] which is defined by the orthonormal structure).

=--

Accordingly:

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

Given $G \hookrightarrow GL(n)$ and given any one [[frame field]] $\sigma \colon X \to Fr(X)$ over a [[manifold]] $X$, then acting with $G$ on $\sigma$ at each point produces a $G$-subbundle. This is called the $G$-structure _generated_ by the frame field $\sigma$.

=--

### In terms of Cartan connections

A $G$-structure equipped with compatible [[connection on a bundle|connection]] data is equivalently a [[Cartan connection]] for the inclusion $(G \hookrightarrow \mathbb{R}^n \rtimes G)$.

See at _[Cartan connection -- Examples -- G-structures](Cartan+connection#ExampleGStructures)_

### In higher differential geometry

#### $G$-structure on a $K$-principal bundle

We give an equivalent definition of $G$-structures in terms of [[higher differential geometry]] ("from the [[nPOV]]"). This serves to clarify the slightly subtle but important difference between existence and choice of $G$-structure, and seamlessly embeds the notion into the more general context of [[twisted differential c-structures]].

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

Let $G \to K$ be a [[homomorphism]] of [[Lie groups]]. Write

$$
  \mathbf{c} : \mathbf{B}G \to \mathbf{B}K
$$

for the morphism of [[delooping]] [[Lie groupoids]] ( the [[smooth infinity-groupoid|smooth]] [[moduli stacks]] of smooth $K$- and $G$-[[principal bundles]], respectively).

For $X$ a [[smooth manifold]] (or generally an [[orbifold]] or [[Lie groupoid]], etc.) Let $P \to X$ be a $K$-[[principal bundle]] and let

$$
  k \colon X \longrightarrow \mathbf{B}K
$$

be any choice of morphism [[modulating morphism|modulating]] it.

Write $\mathbf{H}(X, \mathbf{B}G)$ etc. for the [[derived hom space|hom-groupoid]] of [[smooth infinity-groupoid|smooth groupoids]] / smooth stacks . This is equivalently the groupoid of $G$-principal bundles over $X$ and smooth [[gauge transformations]] between them.

Then the **groupoid of $G$-structure on $P$** (with respect to the given morphism $G \to K$) is the [[homotopy pullback]]

$$
  \mathbf{c}Struc_{[P]}(X)
  :=
  \mathbf{H}(X, \mathbf{B}G)
   \times_{\mathbf{H}(X, \mathbf{B}K)}
  \{k\}
  \,.
$$

$$
  \array{
    \mathbf{c}Struc_{[P]}(X)
    &\longrightarrow&
    \ast
    \\
    \downarrow & \swArrow_\simeq& \downarrow^{\mathrlap{k}}
    \\
    \mathbf{H}(X, \mathbf{B}G)
     &\stackrel{\mathbf{H}(X, \mathbf{c})}{\longrightarrow}&
    \mathbf{H}(X, \mathbf{B}K)
  }
$$

(the groupoid of _[[twisted differential c-structure|twisted c-structures]]_).

=--

+-- {: .num_remark}
###### Remark

If here $k$ is trivial in that it factors through the point, $k \colon X \to \ast \to \mathbf{B}K$ then this homotopy fiber product is $\mathbf{H}(X,K/G)$, where $K/G$ is the [[coset space]] ([[Klein geometry]]) which itself sits in the [[homotopy fiber sequence]]

$$
   K/G \to \mathbf{B}G \to \mathbf{B}K
  \,.
$$

=--

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

Specifically, when $X$ is a [[smooth manifold]] of [[dimension]] $n$, the [[frame bundle]] $Fr(X)$ is [[modulating moprhism|modulated]] by a morphism $\tau_X \colon X \to \mathbf{B} GL(n)$ into the [[moduli stack]] for the [[general linear group]] $K := GL(n)$. Then for any group homomorphism $G \to GL(n)$, a **$G$-structure** on $X$ is a $G$-structure on $Fr(X)$, as above.

=--

#### $G$-Structure on an etale $\infty$-groupoid

We discuss the concept in the generality of [[higher differential geometry]], formalized in [[differential cohesion]].

See at _[differential cohesion -- G-Structure](differential+cohesive+%28infinity%2C1%29-topos#structures)_

## Properties

### Integrability of $G$-structure

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

A $G$-structure on a [[manifold]] $X$ is called _locally flat_ ([Sternberg 64, section VII, def. 24](#Sternberg64)) or _integrable_ (e.g. [Alekseevskii](#Alekseevskii)) if it is locally equivalent to the _standard flat $G$-structure_, def. \ref{StandardFlatGStructure}.

This means that there is an [[open cover]] $\{U_i \to X\}$ by [[open subsets]] of the [[Cartesian space]] $\mathbb{R}^n$ such that the restriction of the $G$-structure to each of these is equivalent to the standard flat $G$-structure.

=--

See at _[[integrability of G-structures]]_ for more on this

The [[obstruction]] to integrability of $G$-structure is the _[[torsion of a G-structure]]_. See there for more.

### Relation to special holonomy

The existence of $G$-structures on tangent bundles of [[Riemannian manifolds]] is closely related to these having [[special holonomy]].

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

Let $(X,g)$ be a [[connected topological space|connected]] [[Riemannian manifold]]
of [[dimension]] $n$ with [[holonomy group]]  $Hol(g) \subset O(n)$.

For $G \subset O(n)$ some other [[subgroup]], $(X,g)$ admits a torsion-free
G-structure precisely if $Hol(g)$ is [[adjoint action|conjugate]] to
a [[subgroup]] of $G$.

Moreover, the space of such $G$-structures is the [[coset]] $G/L$,
where $L$ is the group of elements suchthat conjugating $Hol(g)$ with them
lands in $G$.

=--

This appears as ([Joyce prop. 3.1.8](#Joyce))

## Examples

### Canonical G-structures

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

For $G \hookrightarrow GL(n)$ a subgroup, the _standard flat $G$-structure_ on the [[Cartesian space]] $\mathbb{R}^n$ is the $G$-structure which is generated, via def. \ref{GStructureGeneratedByFrameField}, from the canonical [[frame field]] on $\mathbb{R}^n$ (the one which is the identity at each point, under the defining identifications).

=--

\begin{example}
  \label{CanonicalHStructureOnFModH}
  For $H \subset G$ any [[Lie group|Lie]] [[subgroup]]-inclusion, the canonical [[quotient space]] [[coprojection]] $G \to G/H$ to the [[coset space]] is an $H$-[[principal bundle]] that exhibits $H$-structure on $G/H$.
\end{example}

(e.g. [Čap-Slovak 09, p. 53](#CapSlovak09))

### Reduction of tangent bundle structure

* For the [[subgroup]] of $GL(n, \mathbb{R})$ of matrices of positive determinant, a $GL(n, \mathbb{R})^+$-structure defines an [[orientation]].

* For the [[orthogonal group]], an $O(n)$-structure defines a [[Riemannian metric]]. (See the discussion at [[vielbein]] and at

* For the [[special linear group]], an $SL(n,R)$-structure defines a [[volume form]].

* For the trivial group, an $\{e\}$-structure consists of an [[absolute parallelism]] of the manifold.

* For $n = 2 m$ even, a $GL(m, \mathbb{C})$-structure defines an [[almost complex structure]] on the manifold. It must satisfy an integrability condition to be a [[complex structure]].

### Lift of tangent bundle structure

An example for a lift of structure groups is

* for the [[spin group]]  $spin(n)$, a $G$-structure is a [[spin structure]].

This continues with lifts to the

* [[string group]] giving [[string structure]];

* [[fivebrane group]] giving [[fivebrane structure]].

### Reduction of more general bundle structure
 {#ExamplesOfReductionsOfNonTangentBundles}

* For general $G \to K$, the corresponding notion of [[Cartan geometry]] involves $G$-structure on $K$-principal bundles (not necessarily underlying a tangent bundle).

* A $U(n,n) \hookrightarrow O(2n,2n)$-structure is a [[generalized complex structure]];

* For $H_n \to E_{n(n)}$ the inclusion of the [[maximal compact subgroup]] into the [[split real form]] of an [[exceptional Lie group]], the corresponding structure is an [[exceptional generalized geometry]].

### Complex geometric examples

* The choice of $SO(n, \mathbb{C})$ as subgroup of $GL(n, \mathbb{C})$, determines a complex Riemannian structure;

* $CO(n, \mathbb{C}) \hookrightarrow GL(n, \mathbb{C})$, a complex conformal structure;

* $Sp(2n, \mathbb{C})\hookrightarrow GL(2n, \mathbb{C})$, an almost symplectic structure;

* $GL(2, \mathbb{C}) GL(n, \mathbb{C}) \hookrightarrow GL(2n, \mathbb{C}), n \geq 3$,  determines an almost quaternionic structure;

* more generally a $GL(m, \mathbb{C}) GL(n, \mathbb{C})$-structure on a $m n$-dimensional manifold is locally identical to a Grassmannian spinor structure.

### Special holonomy examples

[[!include normed division algebra Riemannian geometry -- table]]

### $G$-Structures on 8-manifolds

For discussion of [[G-structures]] on [[closed manifold|closed]] [[8-manifolds]] see [there](8-manifold#GStructuresOn8Manifolds).

### Higher geometric examples

See the list at [[twisted differential c-structure]].

\linebreak

## Related concepts

* [[special holonomy]]

* [[integrability of G-structures]]

* [[homothety]]

* [[tangential structure]]

## References

### Traditional

The concept originates around the work of [[Eli Cartan]] ([[Cartan geometry]]) and

* [[Shiing-Shen Chern]], _The geometry of G-structures_, Bull. Amer. Math. Soc. 72(2): 167–219. 1966 ([pdf](http://www.ams.org/journals/bull/1966-72-02/S0002-9904-1966-11473-8/S0002-9904-1966-11473-8.pdf), [Euclid](http://projecteuclid.org/euclid.bams/1183527777))

Textbook accounts:

* {#Sternberg64} [[Shlomo Sternberg]], chapter VII of: _Lectures on differential geometry_, Prentice-Hall 1964,

  2nd edition AMS 1983 ([ISBN:978-0-8218-1385-0](https://bookstore.ams.org/chel-316))

* [[Shoshichi Kobayashi]], _Transformation Groups in Differential Geometry_ 1972, reprinted as: Classics in Mathematics Vol. 70, Springer 1995 ([doi:10.1007/978-3-642-61981-6](https://link.springer.com/book/10.1007/978-3-642-61981-6))

* [[Pierre Molino]], _Theorie des G-Structures: Le Probleme d'Equivalence_, Lecture Notes in Mathematics, Springer (1977) ([ISBN:978-3-540-37360-5](https://www.springer.com/de/book/9783540082460))

* {#CapSlovak09} [[Andreas Čap]], [[Jan Slovák]], Chapter 1 of: _Parabolic Geometries I -- Background and General Theory_, AMS 2009 ([ISBN:978-1-4704-1381-1](http://bookstore.ams.org/surv-154))

See also:

* [[Pierre Molino]], _Sur quelques propriétés des G-structures_, J. Differential Geom. Volume 7, Number 3-4 (1972), 489-518 ([euclid:jdg/1214431168](https://projecteuclid.org/euclid.jdg/1214431168))

Lecture notes:

* [[Marius Crainic]], Chapters 3 and 4 of: _[Differential geometry course](https://webspace.science.uu.nl/~crain101/DG-2015/)_, 2015 ([pdf](https://webspace.science.uu.nl/~crain101/DG-2015/main10.pdf), [[CrainicDifferentialGeometry15.pdf:file]])

* {#Pasquotto} [[Federica Pasquotto]], _Linear $G$-structures by examples_ ([pdf](http://www.few.vu.nl/~pasquott/course16.pdf), [[PasquottoGStructures.pdf:file]])

Surveys include

* {#Alekseevskii} [[Dmitry Vladimirovich Alekseevsky]], _$G$-structure on a manifold_ in M. Hazewinkel (ed.) _Encyclopedia of Mathematics, Volume 4_ ([eom:G-structure](https://encyclopediaofmath.org/wiki/G-structure))

See also

* Wikipedia, _[G-structure](http://en.wikipedia.org/wiki/G-structure)_

Discussion with an eye towards [[special holonomy]] is in

* {#Joyce} [[Dominic Joyce]], section 2.6 of _Compact manifolds with special holonomy_ , Oxford Mathematical Monogrophs (200)

Discussion with an eye towards [[torsion constraints in supergravity]] is in

* {#Lott90} [[John Lott]], _The Geometry of Supergravity Torsion Constraints_, Comm. Math. Phys. 133 (1990), 563–615, (exposition in [arXiv:0108125](http://arxiv.org/abs/math/0108125))

Discussion of G-structures more generally on [[orbifolds]]:

* A. V. Bagaev, N. I. Zhukova, _The Automorphism Groups of Finite Type $G$-Structures on Orbifolds_, Siberian Mathematical Journal 44, 213–224 (2003) ([doi:10.1023/A:1022920417785](https://doi.org/10.1023/A:1022920417785))

* [[Robert Wolak]], _Orbifolds, geometric structures and foliations. Applications to harmonic maps_, Rendiconti del seminario matematico - Universita politecnico di Torino vol. 73/1 , 3-4 (2016), 173-187 ([arXiv:1605.04190](https://arxiv.org/abs/1605.04190))

On $G$-structure for $G = Br_\infty$ the infinite [[braid group]]:

* [[Frederick R. Cohen]], *Braid orientations and bundles with flat connections*, Inventiones mathematicae **46** (1978) 99–110 [[doi:10.1007/BF01393249](https://doi.org/10.1007/BF01393249)]

* [[Jonathan Beardsley]], *On Braids and Cobordism Theories*, Glasgow (2022) [notes: [pdf](https://www.jonathanbeardsley.com/GlasgowNotes2022.pdf), [[Beardsley-BraidsAndCobordism.pdf:file]]]

### In supergeometry

Discussion of $G$-structures in [[supergeometry]] includes

* [[Dmitri Alekseevsky]], [[Vicente Cortés]], [[Chandrashekar Devchand]], [[Uwe Semmelmann]], _Killing spinors are Killing vector fields in Riemannian Supergeometry_ ([arXiv:dg-ga/9704002](http://arxiv.org/abs/dg-ga/9704002))

  (on [[Killing spinors]] as super-[[Killing vectors]])

### In supergravity
 {#ReferencesInSupergravity}

Discussion of [[G-structures]] in [[supergravity]] and [[superstring theory]]:

In relation to [[torsion constraints in supergravity]]:

* {#Lott90} [[John Lott]], _The Geometry of Supergravity Torsion Constraints_, Comm. Math. Phys. 133 (1990), 563–615, (exposition in [arXiv:0108125](http://arxiv.org/abs/math/0108125))

As a way of speaking about [[Calabi-Yau structure]] and [[generalized Calabi-Yau structure]]:

* {#Prins16} [[Daniël Prins]], _On flux vacua, $SU(n)$-structures and generalised complex geometry_, Université Claude Bernard -- Lyon I, 2015.  ([arXiv:1602.05415](https://arxiv.org/abs/1602.05415), [tel:01280717](https://tel.archives-ouvertes.fr/tel-01280717))

In relation to [[BPS states]]/partial reduction of [[number of supersymmetries]] under [[KK-compactification]]:

* [[Paul Koerber]], _Lectures on Generalized Complex Geometry for Physicists_, Fortsch. Phys. 59: 169-242, 2011 ([arXiv:1006.1536](https://arxiv.org/abs/1006.1536))

  (with application to [[generalized complex geometry]])

* [[Jerome Gauntlett]], Dario Martelli, Stathis Pakis, [[Daniel Waldram]], _G-Structures and Wrapped NS5-branes_, Commun.Math.Phys. 247 (2004) 421-445 ([arxiv:hep-th/0205050](https://arxiv.org/abs/hep-th/0205050))

  (application to [[flux compactifications]])

* [[Jérôme Gaillard]], _On $G$-structures in gauge/string duality_, 2011 ([cronfa:42569](https://cronfa.swan.ac.uk/Record/cronfa42569) [spire:1340775](http://inspirehep.net/record/1340775), [[GaillardGStructure.pdf:file]])

  (with application to [[holographic QCD]])

* [[Ulf Danielsson]], [[Giuseppe Dibitetto]], Adolfo Guarino, _KK-monopoles and $G$-structures in M-theory/type IIA reductions_, JHEP 1502 (2015) 096 ([arXiv:1411.0575](https://arxiv.org/abs/1411.0575))

  (with application to [[D6-branes]]/[[KK-monopoles]] in [[M-theory]])

* [[Ruben Minasian]], [[Daniël Prins]], [[Hagen Triendl]], _Supersymmetric branes and instantons on curved spaces_, JHEP 10 (2017) 159 ([arXiv:1707.07002](https://arxiv.org/abs/1707.07002))

and specifically so for [[M-theory on 8-manifolds]]:

* {#IshamPope88} [[Chris Isham]], [[Christopher Pope]], _Nowhere Vanishing Spinors and Topological Obstructions to the Equivalence of the NSR and GS Superstrings_, Class. Quant. Grav. 5 (1988) 257 ([spire:251240](http://inspirehep.net/record/251240), [doi:10.1088/0264-9381/5/2/006](https://doi.org/10.1088/0264-9381/5/2/006))

  (focus on [[Spin(7)-structure]])

* {#IshamPopeWarner88} [[Chris Isham]], [[Christopher Pope]], [[Nicholas Warner]], _Nowhere-vanishing spinors and triality rotations in 8-manifolds_,  Classical and Quantum Gravity, Volume 5, Number 10, 1988 ([cds:185144](http://cds.cern.ch/record/185144), [doi:10.1088/0264-9381/5/10/009](https://iopscience.iop.org/article/10.1088/0264-9381/5/10/009))

  (focus on [[Spin(7)-structure]])

* Cezar Condeescu, [[Andrei Micu]], [[Eran Palti]], _M-theory Compactifications to Three Dimensions with M2-brane Potentials_, JHEP 04 (2014) 026 ([arxiv:1311.5901](https://arxiv.org/abs/1311.5901))

* [[Daniël Prins]], [[Dimitrios Tsimpis]], _IIA supergravity and M-theory on manifolds with $SU(4)$ structure_, Phys. Rev. D 89.064030 ([arXiv:1312.1692](https://arxiv.org/abs/1312.1692))

* [[Elena Babalic]], [[Calin Lazaroiu]], _Singular foliations for M-theory compactification_, JHEP 03 (2015) 116 ([arXiv:1411.3497](https://arxiv.org/abs/1411.3497))

* [[Elena Babalic]], [[Calin Lazaroiu]], _Foliated eight-manifolds for M-theory compactification_, JHEP 01 (2015) 140 ([arXiv:1411.3148](https://arxiv.org/abs/1411.3148))

* [[Carlos S. Shahbazi]], _M-theory on non-Kähler manifolds_, JHEP 09 (2015) 178 [[arXiv:1503.00733](https://arxiv.org/abs/1503.00733)]

* [[Elena Babalic]], [[Calin Lazaroiu]], _The landscape of $G$-structures in eight-manifold compactifications of M-theory_, JHEP 11 (2015) 007 ([arXiv:1505.02270](https://arxiv.org/abs/1505.02270))

* [[Elena Babalic]], [[Calin Lazaroiu]], _Internal circle uplifts, transversality and stratified $G$-structures_, JHEP 11 (2015) 174 ([arXiv:1505.05238](https://arxiv.org/abs/1505.05238))

* [[Robin Terrisse]], [[Dimitrios Tsimpis]], _$SU(3)$ structures on $S^2$ bundles over four-manifolds_, JHEP09 (2017) 133 ([arXiv:1707.04636](https://arxiv.org/abs/1707.04636))

* Vikas Yadav, Aalok Misra, *On M-Theory Dual of Large-$N$ Thermal QCD-Like Theories up to $\mathcal{O}(R^4)$ and $G$-Structure Classification of Underlying Non-Supersymmetric Geometries* ([arXiv:2004.07259](https://arxiv.org/abs/2004.07259))

  > (relation to [[holographic QCD]])

* Mateo Galdeano, *The Geometry and Superconformal Algebras of String Compactifications with a $G$-structure* [[arXiv:2212.13781](https://arxiv.org/abs/2212.13781)]

  > (relation to [[superconformal algebras]])

See also

* Johannes Held, Section 3 of: _Non-SupersymmetricFlux Compactifications of Heterotic String- and M-theory_ ([pdf](https://edoc.ub.uni-muenchen.de/14341/1/Held_Johannes.pdf))

* {#Papadopoulos18} [[George Papadopoulos]], _Geometry and symmetries of null $G$-structures_ ([arXiv:1811.03500](https://arxiv.org/abs/1811.03500))

* [[Daniel Waldram]], *Fluxes, holography and uses of Exceptional Generalized Geometry*, talk at *[[Strings 2022]]*  [[indico:4940880](https://indico.cern.ch/event/1085701/contributions/4940880), [slides](https://indico.cern.ch/event/1085701/contributions/4940880/attachments/2482886/4262576/Daniel%20Waldram%20Strings2022-waldram.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=e222a3a9-ebf4-445c-ada0-f7b1d592d908) ]

  > on generalized [[G-structures]] and [[exceptional generalized geometry]] of [[flux compactifications]] and in [[AdS/CFT]]

### In complex geometry

* Sergey Merkulov, _On group theoretic aspects of the non-linear twistor transform_, ([pdf](http://people.maths.ox.ac.uk/lmason/Tn/40/TN40-07.pdf))

and his chapter A in

* [[Yuri Manin]], _Gauge Field Theory and Complex Geometry_, Springer.

* Norman Wildberger, _On the complexication of the classical geometries and exceptional numbers_, ([pdf](http://web.maths.unsw.edu.au/~norman/papers/L.pdf))

* Jun-Muk Hwang, _Rational curves and prolongations of G-structures_,[arXiv:1703.03160](https://arxiv.org/abs/1703.03160)

### In higher geometry

Some discussion in [[higher differential geometry]] is in section 4.4.2 of

* _[[schreiber:differential cohomology in a cohesive topos]]_

Formalization in [[modal type theory|modal]] [[homotopy type theory]] is in

* {#Wellen17} [[Felix Wellen]], _[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]_, 2017

* {#Wellen18} [[Felix Wellen]], _[[schreiber:Cartan Geometry in Modal Homotopy Type Theory]]_ ([arXiv:1806.05966](https://arxiv.org/abs/1806.05966))

[[!redirects G-structures]]


G-topology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Analytic geometry
+--{: .hide}
[[!include analytic geometry -- contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

While [[p-adic numbers]] and other [[non-archimedean fields]] may be [[complete topological space|complete]] with respect to the [[norm]] [[topology]] induced from their [[non-archimedean valuation]], this [[topology]] is generally not a useful topology on the [[affinoid domains]] over these spaces. For instance [[analytic continuation]] does not in general exist for these topologies.

To remedy this ([Tate 71](#Tate71)) introduced a
[[Grothendieck topology]] for such [[analytic spaces]] with respect to which there is a good [[analytic geometry]], namely what is called _[[rigid analytic geometry]]_.

This is simply the Grothendieck topology whose [[coverings]] are the covers by [[affinoid domain embeddings]]. It should probably be called the _Tate topology_. But in ([Bosch-Güntzer-Remmert 84,section 9.1.4](#BoschGuntzerRemmert84)) it is called the _G-topology_ (where actually "G" is short for "Grothendieck"...) and since then that name stuck.

## References

The concept was introduced in

* {#Tate71} [[John Tate]], _Rigid analytic spaces_, Invent. Math. __12__:257–289, 1971

A standard textbook account is in section 9 of

* {#BoschGuntzerRemmert84} S. Bosch, U. Güntzer, [[Reinhold Remmert]], section 9.1 of _[[Non-Archimedean Analysis]] -- A systematic approach to rigid analytic geometry_, 1984 ([pdf](http://math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf))

Lectures notes in the context of [[Berkovich analytic spaces]] include

* {#Berkovich09} [[Vladimir Berkovich]], section 3.2 of _Non-archimedean analytic spaces_, lectures at the _Advanced School on $p$-adic Analysis and Applications_, ICTP, Trieste, 31 August - 11 September 2009 ([pdf](http://www.wisdom.weizmann.ac.il/~vova/Trieste_2009.pdf))

In theorem 8.14 of

* {#BassatKremnitzer13} [[Oren Ben-Bassat]], [[Kobi Kremnizer]],  _Non-Archimedean analytic geometry as relative algebraic geometry_ ([arXiv:1312.0338](http://arxiv.org/abs/1312.0338))

the G-topology is re-derived from more general abstract grounds.


G-universe

#Contents#
* table of contents
{:toc}

## Idea

In [[equivariant stable homotopy theory]] over a [[compact Lie group]] $G$, a _$G$-universe_ is a $G$-[[representation]] that contains "all" representations of $G$ of sorts.

This is used in one definition of [[G-spectra]] via [[looping and delooping]] by [[representation spheres]].

## Definition

A **[[G-universe]]** in this context is (e.g. [Greenlees-May, p. 10](#GreenleesMay)) an infinite dimensional real [[inner product space]] equipped with a linear $G$-[[action]] that is the [[direct sum]] of countably many copies of a given [[set]] of (finite dimensional) [[representations]] of $G$, at least containing the trivial representation on $\mathbb{R}$ (so that $U$ contains at least a copy of $\mathbb{R}^\infty$).

## Applications

### Infinite complex projective $G$-space

+-- {: .num_defn #InfiniteProjectiveGSpace}
###### Definition
**([[infinite complex projective G-space]])**

For $G$ an [[abelian group|abelian]] [[compact Lie group]], let

\[
  \label{GUniverseForInfiniteComplexProjectiveSpace}
  \mathcal{U}_G
  \;\coloneqq\;
  \underset{k \in \mathbb{N}}{\bigoplus}
  \underset{\mathbf{1}_V \in R(G)}{\bigoplus}
  \mathbf{1}_V
\]

be the [[G-universe]] being the infinite [[direct sum]] of all [[complex numbers|complex]] 1-[[dimension|dimensional]] [[linear representations]] of $G$, regarded as a [[topological G-space]] with [[topological space|toplogy]] the [[colimit]] of its [[finite dimensional vector space|finite-dimensional]] [[linear subspaces]].

Then the _[[infinite complex projective G-space]]_ is the [[colimit]]

$$
  P\big(
    \mathcal{U}_G
  \big)
  \;\coloneqq\;
  \underset{
    \underset{
      {
        V \subset \mathcal{U}_G
      }
      \atop
      {
         dim(V) \lt \infty
      }
    }{\longrightarrow}
  }{\lim}
  P\big( V \big)
$$

of the [[projective G-spaces]] for all the [[finite dimensional vector space|finite-dimensional]] $G$-[[linear representations]] inside the [[G-universe]] (eq:GUniverseForInfiniteComplexProjectiveSpace).

=--

(e.g. [Greenlees 01, Sec. 9.2](#Greenlees01))

This concept is at the heart of _[[equivariant complex oriented cohomology theory]]_.

## Related concepts

* [[RO(G)-grading]]

* [[infinite projective G-space]]

## References

* {#GreenleesMay} [[John Greenlees]], [[Peter May]], _Equivariant stable homotopy theory_ ([pdf](http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf))

[[!redirects G-universes]]


G. E. Murphy

* [GoogleScholar page](https://scholar.google.com/citations?user=B_gnjUsAAAAJ&hl=en)

## Selected writings

Introducing the [[Jucys-Murphy elements]] in the context of [[Alfred Young]]'s [[seminormal representation]] in the [[representation theory of the symmetric group]]:

* [[G. E. Murphy]], *A new construction of Young's seminormal representation of the symmetric groups*, Journal of Algebra Volume 69, Issue 2, April 1981, Pages 287-297 (doi:10.1016/0021-8693(81)90205-2)

* {#Murphy83} [[G. E. Murphy]], *The idempotents of the symmetric group and Nakayama's conjecture*, Journal of Algebra Volume 81, Issue 1, March 1983, Pages 258-265 ([pdf](https://core.ac.uk/download/pdf/82692868.pdf))

* {#Murphy92} [[G. E. Murphy]], *On the representation theory of the symmetric groups and associated Hecke algebras*, J. Algebra 152 (1992) 492–513 ([pdf](https://core.ac.uk/download/pdf/82422274.pdf), doi:10.1016/0021-8693(92)90045-N))

category: people


G. H. Hardy

__G. H. Hardy__ (Godfrey Harold Hardy) was a remarkable 20th British mathematical analysist and analytic number theorist in first half of 20th century. In addition to important works of his own, he is well known as proponent of rigorous analysis and pure mathematics in Great Britain which was at the time dominated by so-called [[applied mathematics]], for his fruitful collaborations with Littlewood and [[Ramanujan]] and for some influential books including the one on divergent series ([[resummation]] methods) and the popular book "A mathematician's apology" about the mathematical profession and the reasons of mathematicians.

* wikipedia: [G. H. Hardy](https://en.wikipedia.org/wiki/G._H._Hardy)

## Selected writings

On the [[asymptotic expansion]] of the [[partition function (number theory)|partition function]] in [[number theory]]:

* [[Godfrey Hardy]], [[Srinivasa Ramanujan]], *Asymptotic formulaæ in combinatory analysis*, Proceedings of the London Mathematical Society, 2(1):75–115, 1918 ([doi:10.1112/plms/s2-17.1.75](https://doi.org/10.1112/plms/s2-17.1.75), [[HardyRamanujanAsymptoticFormulae.pdf:file]])

[[!redirects Godfrey Harold Hardy]]
[[!redirects Godfrey Hardy]]


G. Olshanski

* [personal page](http://iitp.ru/en/users/88.htm)

## Selected writings

On [[asymptotic representation theory]]:

* [[G. Olshanski]], *Asymptotic representation theory*, Lecture notes 2009-2010 ([webpage](https://lpetrov.cc/art), [pdf 1](https://storage.lpetrov.cc/Olshanski_ART_course_1.pdf), [pdf 2](https://storage.lpetrov.cc/Olshanski_ART_course_2.pdf))

category: people


G. Peter Lepage

* [institute page](https://physics.cornell.edu/peter-lepage)

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

## Selected writings

Applying [[light-cone gauge quantization]] to [[QCD]]:

* [[G. Peter Lepage]], [[Stanley J. Brodsky]], *Exclusive processes in perturbative quantum chromodynamics*, Phys. Rev. D **22** (1980) 2157 [[doi:10.1103/PhysRevD.22.2157](https://doi.org/10.1103/PhysRevD.22.2157)]

category: people

[[!redirects Peter Lepage]]


G. Winskel

Professor Glynn Winskel is a professor in the Computer Laboratory,
University of Cambridge.  He was formerly Professor at the Computer Science Department, University of Aarhus and
Director of [[BRICS]] (centre for Basic Research in Computer Science).

* [home page](http://www.cl.cam.ac.uk/~gw104/)