alasdairforsythe/text-english-code-fiction-nonfiction

text stringlengths 0 634k
$X^n g^j\in L\ \forall j$.
I.e.\ elements of the form $X^n g^j$ lie separately in $L$ and it is
sufficient to consider such elements. From the coaction we learn that
if $X^n g^j\in L$ we have $X^m g^j\in L\ \forall m\le n$.
The action
by $X$ leads to $X^n g^j\in L \Rightarrow X^{n+1} g^j\in
L$ except if
$n+j=0$. The classification is given by the possible choices we have
for each power in $g$. For every positive integer $j$ we can
choose wether or not to include the span of
$\{ X^n g^j\|\forall n\}$ in $L$ and for
every non-positive
integer we can choose to include either the span of $\{ X^n
g^j\|\forall n\}$
or just
$\{ X^n g^j\|\forall n\le -j\}$ or neither. I.e.\ for positive
integers ($\mathbb{N}$) we have two choices while for non-positive (identified
with $\mathbb{N}_0$) ones we have three choices.

Clearly, the finite dimensional $L$ are those where we choose only to
include finitely many powers of $g$ and also only finitely many powers
of $X$. The latter is only possible for the non-positive powers
of $g$.
By identifying positive integers $n$ with powers $1-n$ of $g$, we
obtain a classification by finite subsets of $\mathbb{N}$.

(b) Irreducibility clearly corresponds to just including one power of $g$
in the finite dimensional case.

(c) The decomposition property is obvious from the discussion.
\end{proof}


\begin{cor}
\label{cor:uqbp_eclass}
(a) Left crossed \ensuremath{U_q(\lalg{b_+})}-submodules $L\subseteq\ker\cou\subset\ensuremath{U_q(\lalg{b_+})}$ via
the left adjoint
action and left regular coaction (with subsequent projection to
$\ker\cou$ via $x\mapsto x-\cou(x)1$) are in one-to-one correspondence to
the set $3^{\mathbb{N}}\times2^{\mathbb{N}_0}$.
Finite dimensional $L$ are in one-to-one correspondence to
finite sets
$I\subset\mathbb{N}\setminus\{1\}$ and $\dim L=\sum_{n\in I}n$.

(b) Finite dimensional irreducible $L$ correspond to $\{n\}$
with $n\ge 2$ the dimension.

(c) Finite dimensional $L$ are direct sums of irreducible ones. In
particular $L=\oplus_{n\in I} L^n$ with $L^n$ corresponding to $\{n\}$.
\end{cor}
\begin{proof}
Only a small modification of lemma \ref{lem:uqbp_class} is
necessary. Elements of
the form $P(g)$ are replaced by elements of the form
$P(g)-P(1)$. Monomials with non-vanishing degree in $X$ are unchanged.
The choices for elements of degree $0$ in $g$ are reduced to either
including the span of
$\{ X^k \|\forall k>0 \}$ in the crossed submodule or not. In
particular, the crossed submodule characterised by \{1\} in lemma
\ref{lem:uqbp_class} is projected out.
\end{proof}

Differential calculi in the original sense of Woronowicz are
classified by corollary \ref{cor:cqbp_eclass} while from the quantum
tangent space
point of view the
classification is given by corollary \ref{cor:uqbp_eclass}.
In the finite dimensional case the duality is strict in the sense of a
one-to-one correspondence.
The infinite dimensional case on the other hand depends strongly on
the algebraic models we use for the function or enveloping
algebras. It is therefore not surprising that in the present purely
algebraic context the classifications are quite different in this
case. We will restrict ourselves to the finite dimensional
case in the following description of the differential calculi.


\begin{thm}
\label{thm:q_calc}
(a) Finite dimensional differential calculi $\Gamma$ on \ensuremath{C_q(B_+)}{} and
corresponding quantum tangent spaces $L$ on \ensuremath{U_q(\lalg{b_+})}{} are
in one-to-one correspondence to
finite sets $I\subset\mathbb{N}\setminus\{1\}$. In particular
$\dim\Gamma=\dim L=\sum_{n\in I}n$.

(b) Coirreducible $\Gamma$ and irreducible $L$ correspond to
$\{n\}$ with $n\ge 2$ the dimension.
Such a $\Gamma$ has a
right invariant basis $\eta_0,\dots,\eta_{n-1}$ so that the relations
\begin{gather*}
\diff X=\eta_1+(q^{n-1}-1)\eta_0 X \qquad
\diff g=(q^{n-1}-1)\eta_0 g\\
[a,\eta_0]=\diff a\quad \forall a\in\ensuremath{C_q(B_+)}\\
[g,\eta_i]_{q^{n-1-i}}=0\quad \forall i\qquad
[X,\eta_i]_{q^{n-1-i}}=\begin{cases}
\eta_{i+1} & \text{if}\ i<n-1 \\
0 & \text{if}\ i=n-1
\end{cases}
\end{gather*}
hold, where $[a,b]_p := a b - p b a$. By choosing the dual basis on

$X^n g^j\in L\ \forall j$.

I.e.\ elements of the form $X^n g^j$ lie separately in $L$ and it is

sufficient to consider such elements. From the coaction we learn that

if $X^n g^j\in L$ we have $X^m g^j\in L\ \forall m\le n$.

The action

by $X$ leads to $X^n g^j\in L \Rightarrow X^{n+1} g^j\in

L$ except if

$n+j=0$. The classification is given by the possible choices we have

for each power in $g$. For every positive integer $j$ we can

choose wether or not to include the span of

$\{ X^n g^j|\forall n\}$ in $L$ and for

every non-positive

integer we can choose to include either the span of $\{ X^n

g^j|\forall n\}$

or just

$\{ X^n g^j|\forall n\le -j\}$ or neither. I.e.\ for positive

integers ($\mathbb{N}$) we have two choices while for non-positive (identified

with $\mathbb{N}_0$) ones we have three choices.

Clearly, the finite dimensional $L$ are those where we choose only to

include finitely many powers of $g$ and also only finitely many powers

of $X$. The latter is only possible for the non-positive powers

of $g$.

By identifying positive integers $n$ with powers $1-n$ of $g$, we

obtain a classification by finite subsets of $\mathbb{N}$.

(b) Irreducibility clearly corresponds to just including one power of $g$

in the finite dimensional case.

(c) The decomposition property is obvious from the discussion.

\end{proof}

\begin{cor}

\label{cor:uqbp_eclass}

(a) Left crossed \ensuremath{U_q(\lalg{b_+})}-submodules $L\subseteq\ker\cou\subset\ensuremath{U_q(\lalg{b_+})}$ via

the left adjoint

action and left regular coaction (with subsequent projection to

$\ker\cou$ via $x\mapsto x-\cou(x)1$) are in one-to-one correspondence to

the set $3^{\mathbb{N}}\times2^{\mathbb{N}_0}$.

Finite dimensional $L$ are in one-to-one correspondence to

finite sets

$I\subset\mathbb{N}\setminus\{1\}$ and $\dim L=\sum_{n\in I}n$.

(b) Finite dimensional irreducible $L$ correspond to $\{n\}$

with $n\ge 2$ the dimension.

(c) Finite dimensional $L$ are direct sums of irreducible ones. In

particular $L=\oplus_{n\in I} L^n$ with $L^n$ corresponding to $\{n\}$.

\end{cor}

\begin{proof}

Only a small modification of lemma \ref{lem:uqbp_class} is

necessary. Elements of

the form $P(g)$ are replaced by elements of the form

$P(g)-P(1)$. Monomials with non-vanishing degree in $X$ are unchanged.

The choices for elements of degree $0$ in $g$ are reduced to either

including the span of

$\{ X^k |\forall k>0 \}$ in the crossed submodule or not. In

particular, the crossed submodule characterised by \{1\} in lemma

\ref{lem:uqbp_class} is projected out.

\end{proof}

Differential calculi in the original sense of Woronowicz are

classified by corollary \ref{cor:cqbp_eclass} while from the quantum

tangent space

point of view the

classification is given by corollary \ref{cor:uqbp_eclass}.

In the finite dimensional case the duality is strict in the sense of a

one-to-one correspondence.

The infinite dimensional case on the other hand depends strongly on

the algebraic models we use for the function or enveloping

algebras. It is therefore not surprising that in the present purely

algebraic context the classifications are quite different in this

case. We will restrict ourselves to the finite dimensional

case in the following description of the differential calculi.

\begin{thm}

\label{thm:q_calc}

(a) Finite dimensional differential calculi $\Gamma$ on \ensuremath{C_q(B_+)}{} and

corresponding quantum tangent spaces $L$ on \ensuremath{U_q(\lalg{b_+})}{} are

in one-to-one correspondence to

finite sets $I\subset\mathbb{N}\setminus\{1\}$. In particular

$\dim\Gamma=\dim L=\sum_{n\in I}n$.

(b) Coirreducible $\Gamma$ and irreducible $L$ correspond to

$\{n\}$ with $n\ge 2$ the dimension.

Such a $\Gamma$ has a

right invariant basis $\eta_0,\dots,\eta_{n-1}$ so that the relations

\begin{gather*}

\diff X=\eta_1+(q^{n-1}-1)\eta_0 X \qquad

\diff g=(q^{n-1}-1)\eta_0 g\\

[a,\eta_0]=\diff a\quad \forall a\in\ensuremath{C_q(B_+)}\\

[g,\eta_i]_{q^{n-1-i}}=0\quad \forall i\qquad

[X,\eta_i]_{q^{n-1-i}}=\begin{cases}

\eta_{i+1} & \text{if}\ i<n-1 \\

0 & \text{if}\ i=n-1

\end{cases}

\end{gather*}

hold, where $[a,b]_p := a b - p b a$. By choosing the dual basis on