alasdairforsythe/text-english-code-fiction-nonfiction

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\end{gather*}
as a crossed module.
\end{proof}

For the transition from the $q$-deformed (lemma
\ref{lem:uqbp_class}) to the classical case we
observe that the space spanned by $g^{s_1},\dots,g^{s_m}$ with $m$
different integers $s_i\in\mathbb{Z}$ maps to the space spanned by
$1, H, \dots, H^{m-1}$ in the
prescription of the classical limit (as described in section
\ref{sec:intro_limits}). I.e.\ the classical crossed submodule
characterised by an integer $l$ and a finite set $I\subset\mathbb{N}$ comes
from a crossed submodule characterised by this same $I$ and additionally $l$
other integers $j\in\mathbb{Z}$ for which $X^k g^{1-j}$ is included. In
particular, we have a one-to-one correspondence in the finite
dimensional case.

To formulate the analogue of corollary \ref{cor:uqbp_eclass} for the
classical case is essentially straightforward now. However, as for
\ensuremath{C(B_+)}{}, we obtain more crossed submodules than those from the $q$-deformed
setting. This is due to the degeneracy introduced by forgetting the
powers of $g$ and just retaining the number of different powers.

\begin{cor}
\label{cor:ubp_eclass}
(a) Proper left crossed \ensuremath{U(\lalg{b_+})}-submodules
$L\subset\ker\cou\subset\ensuremath{U(\lalg{b_+})}$ 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
pairs $(l,I)$ with $l\in\mathbb{N}_0$ and $I\subset\mathbb{N}$ finite where $l\neq 0$
or $I\neq\emptyset$.
$\dim L<\infty$ iff $l=0$. In particular $\dim
L=(\sum_{n\in I}n)-1$ if $l=0$.
\end{cor}


As in the $q$-deformed setting, we give a description of the finite
dimensional differential calculi where we have a strict duality to
quantum tangent spaces.

\begin{prop}
(a) Finite dimensional differential calculi $\Gamma$ on \ensuremath{C(B_+)}{} and
finite dimensional quantum tangent spaces $L$ on \ensuremath{U(\lalg{b_+})}{} are
in one-to-one correspondence to non-empty finite sets $I\subset\mathbb{N}$.
In particular $\dim\Gamma=\dim L=(\sum_{n\in I}) n)-1$.

The $\Gamma$ with $1\in\mathbb{N}$ are in
one-to-one correspondence to the finite dimensional
calculi and quantum tangent spaces of the $q$-deformed setting
(theorem \ref{thm:q_calc}(a)).

(b) The differential calculus $\Gamma$ of dimension $n\ge 2$
corresponding to the
coirreducible one of \ensuremath{C_q(B_+)}{} (theorem \ref{thm:q_calc}(b)) has a right
invariant
basis $\eta_0,\dots,\eta_{n-1}$ so that
\begin{gather*}
\diff X=\eta_1+\eta_0 X \qquad
\diff g=\eta_0 g\\
[g, \eta_i]=0\ \forall i \qquad
[X, \eta_i]=\begin{cases}
0 & \text{if}\ i=0\ \text{or}\ i=n-1\\
\eta_{i+1} & \text{if}\ 0<i<n-1
\end{cases}
\end{gather*}
hold. The braided derivations obtained from the dual basis of the
corresponding $L$ are
given by
\begin{gather*}
\partial_i f=\frac{1}{i!}
\left(\frac{\partial}{\partial X}\right)^i f\qquad
\forall i\ge 1\\
\partial_0 f=\left(X \frac{\partial}{X}+
g \frac{\partial}{g}\right) f
\end{gather*}
for $f\in\ensuremath{C(B_+)}$.

(c) The differential calculus of dimension $n-1$
corresponding to the
one in (b) with $1$ removed from the characterising set is
the same as the one above, except that we set $\eta_0=0$ and
$\partial_0=0$.
\end{prop}
\begin{proof}
(a) We observe that the classifications of corollary
\ref{cor:cbp_class} and lemma \ref{lem:ubp_class} or
corollary \ref{cor:cbp_eclass} and corollary \ref{cor:ubp_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 corollary \ref{cor:cbp_class} is the
annihilator of the crossed
submodule $L$ corresponding to $(0,I)$ in lemma \ref{lem:ubp_class}
and vice versa.
$\ensuremath{C(B_+)}/M$ and $L$ are dual spaces with the induced pairing.
For non-empty $I$ this descends to
$M$ corresponding to $(1,I)$ in corollary
\ref{cor:cbp_eclass} and $L$ corresponding to $(0,I)$ in corollary
\ref{cor:ubp_eclass}.

\end{gather*}

as a crossed module.

\end{proof}

For the transition from the $q$-deformed (lemma

\ref{lem:uqbp_class}) to the classical case we

observe that the space spanned by $g^{s_1},\dots,g^{s_m}$ with $m$

different integers $s_i\in\mathbb{Z}$ maps to the space spanned by

$1, H, \dots, H^{m-1}$ in the

prescription of the classical limit (as described in section

\ref{sec:intro_limits}). I.e.\ the classical crossed submodule

characterised by an integer $l$ and a finite set $I\subset\mathbb{N}$ comes

from a crossed submodule characterised by this same $I$ and additionally $l$

other integers $j\in\mathbb{Z}$ for which $X^k g^{1-j}$ is included. In

particular, we have a one-to-one correspondence in the finite

dimensional case.

To formulate the analogue of corollary \ref{cor:uqbp_eclass} for the

classical case is essentially straightforward now. However, as for

\ensuremath{C(B_+)}{}, we obtain more crossed submodules than those from the $q$-deformed

setting. This is due to the degeneracy introduced by forgetting the

powers of $g$ and just retaining the number of different powers.

\begin{cor}

\label{cor:ubp_eclass}

(a) Proper left crossed \ensuremath{U(\lalg{b_+})}-submodules

$L\subset\ker\cou\subset\ensuremath{U(\lalg{b_+})}$ 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

pairs $(l,I)$ with $l\in\mathbb{N}_0$ and $I\subset\mathbb{N}$ finite where $l\neq 0$

or $I\neq\emptyset$.

$\dim L<\infty$ iff $l=0$. In particular $\dim

L=(\sum_{n\in I}n)-1$ if $l=0$.

\end{cor}

As in the $q$-deformed setting, we give a description of the finite

dimensional differential calculi where we have a strict duality to

quantum tangent spaces.

\begin{prop}

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

finite dimensional quantum tangent spaces $L$ on \ensuremath{U(\lalg{b_+})}{} are

in one-to-one correspondence to non-empty finite sets $I\subset\mathbb{N}$.

In particular $\dim\Gamma=\dim L=(\sum_{n\in I}) n)-1$.

The $\Gamma$ with $1\in\mathbb{N}$ are in

one-to-one correspondence to the finite dimensional

calculi and quantum tangent spaces of the $q$-deformed setting

(theorem \ref{thm:q_calc}(a)).

(b) The differential calculus $\Gamma$ of dimension $n\ge 2$

corresponding to the

coirreducible one of \ensuremath{C_q(B_+)}{} (theorem \ref{thm:q_calc}(b)) has a right

invariant

basis $\eta_0,\dots,\eta_{n-1}$ so that

\begin{gather*}

\diff X=\eta_1+\eta_0 X \qquad

\diff g=\eta_0 g\\

[g, \eta_i]=0\ \forall i \qquad

[X, \eta_i]=\begin{cases}

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

\eta_{i+1} & \text{if}\ 0<i<n-1

\end{cases}

\end{gather*}

hold. The braided derivations obtained from the dual basis of the

corresponding $L$ are

given by

\begin{gather*}

\partial_i f=\frac{1}{i!}

\left(\frac{\partial}{\partial X}\right)^i f\qquad

\forall i\ge 1\\

\partial_0 f=\left(X \frac{\partial}{X}+

g \frac{\partial}{g}\right) f

\end{gather*}

for $f\in\ensuremath{C(B_+)}$.

(c) The differential calculus of dimension $n-1$

corresponding to the

one in (b) with $1$ removed from the characterising set is

the same as the one above, except that we set $\eta_0=0$ and

$\partial_0=0$.

\end{prop}

\begin{proof}

(a) We observe that the classifications of corollary

\ref{cor:cbp_class} and lemma \ref{lem:ubp_class} or

corollary \ref{cor:cbp_eclass} and corollary \ref{cor:ubp_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 corollary \ref{cor:cbp_class} is the

annihilator of the crossed

submodule $L$ corresponding to $(0,I)$ in lemma \ref{lem:ubp_class}

and vice versa.

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

For non-empty $I$ this descends to

$M$ corresponding to $(1,I)$ in corollary

\ref{cor:cbp_eclass} and $L$ corresponding to $(0,I)$ in corollary

\ref{cor:ubp_eclass}.