alasdairforsythe/text-english-code-fiction-nonfiction

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the corresponding irreducible $L$ we obtain
the braided derivations
\begin{gather*}
\partial_i\no{f}=
\no{Q_{n-1-i,g} Q_{n-1-i,X} \frac{1}{[i]_q!} (\partial_{q,X})^i f}
\qquad\forall i\ge 1\\
\partial_0\no{f}=
\no{Q_{n-1,g} Q_{n-1,X} f - f}
\end{gather*}
for $f\in \k(q)[X,g,g^{-1}]$ with normal ordering
$\k(q)[X,g,g^{-1}]\to \ensuremath{C_q(B_+)}$ given by \mbox{$g^n X^m\mapsto g^n X^m$}.

(c) Finite dimensional $\Gamma$ and $L$ decompose into direct sums of
coirreducible respectively irreducible ones.
In particular $\Gamma=\oplus_{n\in I}\Gamma^n$ and
$L=\oplus_{n\in I}L^n$ with $\Gamma^n$ and $L^n$ corresponding to $\{n\}$.
\end{thm}
\begin{proof}
(a) We observe that the classifications of lemma
\ref{lem:cqbp_class} and lemma \ref{lem:uqbp_class} or
corollary \ref{cor:cqbp_eclass} and corollary \ref{cor:uqbp_eclass}
are dual to each other in the finite (co){}dimensional case. More
precisely, for $I\subset\mathbb{N}$ finite the crossed submodule $M$
corresponding to $(1,I)$ in lemma \ref{lem:cqbp_class} is the
annihilator of the crossed
submodule $L$ corresponding to $I$ in lemma \ref{lem:uqbp_class}
and vice versa.
$\ensuremath{C_q(B_+)}/M$ and $L$ are dual spaces with the induced pairing.
For $I\subset\mathbb{N}\setminus\{1\}$ finite this descends to
$M$ corresponding to $(1,I\cup\{1\})$ in corollary
\ref{cor:cqbp_eclass} and $L$ corresponding to $I$ in corollary
\ref{cor:uqbp_eclass}.
For the dimension of $\Gamma$ observe
$\dim\Gamma=\dim{\ker\cou/M}=\codim M$.

(b) Coirreducibility (having no proper quotient) of $\Gamma$
clearly corresponds to maximality of $M$. The statement then follows
from parts (b) of corollaries
\ref{cor:cqbp_eclass} and \ref{cor:uqbp_eclass}. The formulas are
obtained by choosing the basis $\eta_0,\dots,\eta_{n-1}$ of
$\ker\cou/M$ as the equivalence classes of
\[(g-1)/(q^{n-1}-1),X,\dots,X^{n-1}\]
The dual basis of $L$ is then given by
\[g^{1-n}-1, X g^{1-n},\dots, q^{k(k-1)} \frac{1}{[k]_q!} X^k g^{1-n},
\dots,q^{(n-1)(n-2)} \frac{1}{[n-1]_q!} X^{n-1} g^{1-n}\]

(c) The statement follows from corollaries \ref{cor:cqbp_eclass} and
\ref{cor:uqbp_eclass} parts (c) with the observation
\[\ker\cou/M=\ker\cou/{\bigcap_{n\in I}}M^n
=\oplus_{n\in I}\ker\cou/M^n\]
\end{proof}

\begin{cor}
There is precisely one differential calculus on \ensuremath{C_q(B_+)}{} which is
natural in the sense that it
has dimension $2$.
It is coirreducible and obeys the relations
\begin{gather*}
[g,\diff X]=0\qquad [g,\diff g]_q=0\qquad
[X,\diff X]_q=0\qquad [X,\diff g]_q=(q-1)({\diff X}) g
\end{gather*}
with $[a,b]_q:=ab-qba$. In particular we have
\begin{gather*}
\diff\no{f} = {\diff g} \no{\partial_{q,g} f} + {\diff X}
\no{\partial_{q,X} f}\qquad\forall f\in \k(q)[X,g,g^{-1}]
\end{gather*}
\end{cor}
\begin{proof}
This is a special case of theorem \ref{thm:q_calc}.
The formulas follow from (b) with $n=2$.
\end{proof}



\section{Classification in the Classical Limit}
\label{sec:class}

In this section we give the complete classification of differential
calculi and quantum tangent spaces in the classical case of \ensuremath{C(B_+)}{}
along the lines of the previous section.
We pay particular
attention to the relation to the $q$-deformed setting.


The classical limit \ensuremath{C(B_+)}{} of the quantum group \ensuremath{C_q(B_+)}{} is
simply obtained by substituting the parameter $q$ with $1$.
The
classification of left crossed submodules in part (a) of lemma
\ref{lem:cqbp_class} remains
unchanged, as one may check by going through the proof.
In particular, we get a correspondence of crossed modules in the
$q$-deformed setting with crossed modules in the
classical setting
as a map of
pairs $(P,I)\mapsto (P,I)$
that converts polynomials $\k(q)[g]$ to polynomials $\k[g]$ (if
defined) and leaves
sets $I$ unchanged. This is one-to-one in the finite
dimensional case.
However, we did use the distinctness of powers of $q$ in part (b) and

the corresponding irreducible $L$ we obtain

the braided derivations

\begin{gather*}

\partial_i\no{f}=

\no{Q_{n-1-i,g} Q_{n-1-i,X} \frac{1}{[i]_q!} (\partial_{q,X})^i f}

\qquad\forall i\ge 1\\

\partial_0\no{f}=

\no{Q_{n-1,g} Q_{n-1,X} f - f}

\end{gather*}

for $f\in \k(q)[X,g,g^{-1}]$ with normal ordering

$\k(q)[X,g,g^{-1}]\to \ensuremath{C_q(B_+)}$ given by \mbox{$g^n X^m\mapsto g^n X^m$}.

(c) Finite dimensional $\Gamma$ and $L$ decompose into direct sums of

coirreducible respectively irreducible ones.

In particular $\Gamma=\oplus_{n\in I}\Gamma^n$ and

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

\end{thm}

\begin{proof}

(a) We observe that the classifications of lemma

\ref{lem:cqbp_class} and lemma \ref{lem:uqbp_class} or

corollary \ref{cor:cqbp_eclass} and corollary \ref{cor:uqbp_eclass}

are dual to each other in the finite (co){}dimensional case. More

precisely, for $I\subset\mathbb{N}$ finite the crossed submodule $M$

corresponding to $(1,I)$ in lemma \ref{lem:cqbp_class} is the

annihilator of the crossed

submodule $L$ corresponding to $I$ in lemma \ref{lem:uqbp_class}

and vice versa.

$\ensuremath{C_q(B_+)}/M$ and $L$ are dual spaces with the induced pairing.

For $I\subset\mathbb{N}\setminus\{1\}$ finite this descends to

$M$ corresponding to $(1,I\cup\{1\})$ in corollary

\ref{cor:cqbp_eclass} and $L$ corresponding to $I$ in corollary

\ref{cor:uqbp_eclass}.

For the dimension of $\Gamma$ observe

$\dim\Gamma=\dim{\ker\cou/M}=\codim M$.

(b) Coirreducibility (having no proper quotient) of $\Gamma$

clearly corresponds to maximality of $M$. The statement then follows

from parts (b) of corollaries

\ref{cor:cqbp_eclass} and \ref{cor:uqbp_eclass}. The formulas are

obtained by choosing the basis $\eta_0,\dots,\eta_{n-1}$ of

$\ker\cou/M$ as the equivalence classes of

\[(g-1)/(q^{n-1}-1),X,\dots,X^{n-1}\]

The dual basis of $L$ is then given by

\[g^{1-n}-1, X g^{1-n},\dots, q^{k(k-1)} \frac{1}{[k]_q!} X^k g^{1-n},

\dots,q^{(n-1)(n-2)} \frac{1}{[n-1]_q!} X^{n-1} g^{1-n}\]

(c) The statement follows from corollaries \ref{cor:cqbp_eclass} and

\ref{cor:uqbp_eclass} parts (c) with the observation

\[\ker\cou/M=\ker\cou/{\bigcap_{n\in I}}M^n

=\oplus_{n\in I}\ker\cou/M^n\]

\end{proof}

\begin{cor}

There is precisely one differential calculus on \ensuremath{C_q(B_+)}{} which is

natural in the sense that it

has dimension $2$.

It is coirreducible and obeys the relations

\begin{gather*}

[g,\diff X]=0\qquad [g,\diff g]_q=0\qquad

[X,\diff X]_q=0\qquad [X,\diff g]_q=(q-1)({\diff X}) g

\end{gather*}

with $[a,b]_q:=ab-qba$. In particular we have

\begin{gather*}

\diff\no{f} = {\diff g} \no{\partial_{q,g} f} + {\diff X}

\no{\partial_{q,X} f}\qquad\forall f\in \k(q)[X,g,g^{-1}]

\end{gather*}

\end{cor}

\begin{proof}

This is a special case of theorem \ref{thm:q_calc}.

The formulas follow from (b) with $n=2$.

\end{proof}

\section{Classification in the Classical Limit}

\label{sec:class}

In this section we give the complete classification of differential

calculi and quantum tangent spaces in the classical case of \ensuremath{C(B_+)}{}

along the lines of the previous section.

We pay particular

attention to the relation to the $q$-deformed setting.

The classical limit \ensuremath{C(B_+)}{} of the quantum group \ensuremath{C_q(B_+)}{} is

simply obtained by substituting the parameter $q$ with $1$.

The

classification of left crossed submodules in part (a) of lemma

\ref{lem:cqbp_class} remains

unchanged, as one may check by going through the proof.

In particular, we get a correspondence of crossed modules in the

$q$-deformed setting with crossed modules in the

classical setting

as a map of

pairs $(P,I)\mapsto (P,I)$

that converts polynomials $\k(q)[g]$ to polynomials $\k[g]$ (if

defined) and leaves

sets $I$ unchanged. This is one-to-one in the finite

dimensional case.

However, we did use the distinctness of powers of $q$ in part (b) and