alasdairforsythe/text-english-code-fiction-nonfiction

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regular action and adjoint coaction to \ensuremath{U(\lalg{b_+})}-submodules $\subset\ensuremath{U(\lalg{b_+})}$
by regular
action. This map is injective in the finite codimensional
case. Intersections and codimensions are preserved in this case.

This descends to $\ker\cou$.
\end{lem}
\begin{proof}
To obtain the classical limit of a left ideal it is enough to
apply the limiting process (as described in section
\ref{sec:intro_limits}) to the
module generators (We can forget the additional comodule
structure). On the one hand,
any element generated by left multiplication with polynomials in
$g$ corresponds to some element generated by left multiplication with a
polynomial in $H$, that is, there will be no more generators in the
classical setting. On the other hand, left multiplication by a
polynomial in $H$ comes
from left multiplication by the same polynomial in $g-1$, that is,
there will be no fewer generators.

The maximal left crossed \ensuremath{U_q(\lalg{b_+})}-submodule $\subseteq\ensuremath{U_q(\lalg{b_+})}$
by left multiplication and adjoint coaction of
codimension $n$ ($n\ge 1$) is generated as a left ideal by
$\{1-q^{1-n}g,X^n\}$ (see lemma
\ref{lem:cqbp_class}). Applying the limiting process to this
leads to the
left ideal of \ensuremath{U(\lalg{b_+})}{} (which is not maximal for $n\neq 1$) generated by
$\{H+n-1,X^n\}$ having also codimension $n$.

More generally, the picture given for arbitrary finite codimensional left
crossed modules of \ensuremath{U_q(\lalg{b_+})}{} in terms of generators with respect to
polynomials in $g,g^{-1}$ in lemma \ref{lem:cqbp_class} carries over
by replacing factors
$1-q^{1-n}g$ with factors $H+n-1$ leading to generators with
respect to polynomials in $H$. In particular,
intersections go to intersections since the distinctness of
the factors for different $n$ is conserved.

The restriction to $\ker\cou$ is straightforward.
\end{proof}


We are now in a position to give a detailed description of the
differential calculi induced from the $q$-deformed setting by the
limiting process.

\begin{prop}
(a) Certain finite dimensional
differential calculi $\Gamma$ on \ensuremath{U(\lalg{b_+})}{} and quantum tangent spaces $L$
on \ensuremath{C(B_+)}{}
are in one-to-one correspondence to finite dimensional differential
calculi on \ensuremath{U_q(\lalg{b_+})}{} and quantum
tangent spaces on \ensuremath{C_q(B_+)}{}. Intersections correspond to intersections.

(b) In particular,
$\Gamma$ and $L$ corresponding to coirreducible differential calculi
on \ensuremath{U_q(\lalg{b_+})}{} and
irreducible quantum tangent spaces on \ensuremath{C_q(B_+)}{} via the limiting process
are given as follows:
$\Gamma$ has a right invariant basis
$\eta_0,\dots,\eta_{n-1}$ so that
\begin{gather*}
\diff X=\eta_1 \qquad \diff H=(1-n)\eta_0 \\
[H, \eta_i]=(1-n+i)\eta_i\quad\forall i\qquad
[X, \eta_i]=\begin{cases}
\eta_{i+1} & \text{if}\ \ i<n-1\\
0 & \text{if}\ \ i=n-1
\end{cases}
\end{gather*}
holds. The braided derivations corresponding to the dual basis of
$L$ are given by
\begin{gather*}
\partial_i\no{f}=\no{T_{1-n+i,H}
\frac{1}{i!}\left(\frac{\partial}{\partial X}\right)^i f}
\qquad\forall i\ge 1\\
\partial_0\no{f}=\no{T_{1-n,H} f - f}
\end{gather*}
for $f\in\k[X,H]$
with the normal ordering $\k[X,H]\to \ensuremath{U(\lalg{b_+})}$ via $H^n X^m\mapsto H^n X^m$.
\end{prop}
\begin{proof}
(a) The strict duality between \ensuremath{C(B_+)}-subcomodules $L\subseteq\ker\cou$
given by lemma \ref{lem:cbp_dual} and corollary \ref{cor:uqbp_eclass}
and \ensuremath{U(\lalg{b_+})}-modules $\ensuremath{U(\lalg{b_+})}/(\k 1+M)$ with $M$ given by lemma
\ref{lem:ubp_dual} and
corollary \ref{cor:cqbp_eclass} can be checked explicitly.
It is essentially due to mutual annihilation of factors $H+k$ in
\ensuremath{U(\lalg{b_+})}{} with elements $g^k$ in \ensuremath{C(B_+)}{}.

(b) $L$ is generated by
$\{g^{1-n}-1,Xg^{1-n},\dots,
X^{n-1}g^{1-n}\}$ and
$M$ is generated by $\{H(H+n-1),X(H+n-1),X^n \}$.
The formulas are obtained by denoting with
$\eta_0,\dots,\eta_{n-1}$ the equivalence classes of
$H/(1-n),X,\dots,X^{n-1}$ in $\ensuremath{U(\lalg{b_+})}/(\k 1+M)$.
The dual basis of $L$ is then
\[g^{1-n}-1,X g^{1-n},
\dots,\frac{1}{(n-1)!}X^{n-1}

regular action and adjoint coaction to \ensuremath{U(\lalg{b_+})}-submodules $\subset\ensuremath{U(\lalg{b_+})}$

by regular

action. This map is injective in the finite codimensional

case. Intersections and codimensions are preserved in this case.

This descends to $\ker\cou$.

\end{lem}

\begin{proof}

To obtain the classical limit of a left ideal it is enough to

apply the limiting process (as described in section

\ref{sec:intro_limits}) to the

module generators (We can forget the additional comodule

structure). On the one hand,

any element generated by left multiplication with polynomials in

$g$ corresponds to some element generated by left multiplication with a

polynomial in $H$, that is, there will be no more generators in the

classical setting. On the other hand, left multiplication by a

polynomial in $H$ comes

from left multiplication by the same polynomial in $g-1$, that is,

there will be no fewer generators.

The maximal left crossed \ensuremath{U_q(\lalg{b_+})}-submodule $\subseteq\ensuremath{U_q(\lalg{b_+})}$

by left multiplication and adjoint coaction of

codimension $n$ ($n\ge 1$) is generated as a left ideal by

$\{1-q^{1-n}g,X^n\}$ (see lemma

\ref{lem:cqbp_class}). Applying the limiting process to this

leads to the

left ideal of \ensuremath{U(\lalg{b_+})}{} (which is not maximal for $n\neq 1$) generated by

$\{H+n-1,X^n\}$ having also codimension $n$.

More generally, the picture given for arbitrary finite codimensional left

crossed modules of \ensuremath{U_q(\lalg{b_+})}{} in terms of generators with respect to

polynomials in $g,g^{-1}$ in lemma \ref{lem:cqbp_class} carries over

by replacing factors

$1-q^{1-n}g$ with factors $H+n-1$ leading to generators with

respect to polynomials in $H$. In particular,

intersections go to intersections since the distinctness of

the factors for different $n$ is conserved.

The restriction to $\ker\cou$ is straightforward.

\end{proof}

We are now in a position to give a detailed description of the

differential calculi induced from the $q$-deformed setting by the

limiting process.

\begin{prop}

(a) Certain finite dimensional

differential calculi $\Gamma$ on \ensuremath{U(\lalg{b_+})}{} and quantum tangent spaces $L$

on \ensuremath{C(B_+)}{}

are in one-to-one correspondence to finite dimensional differential

calculi on \ensuremath{U_q(\lalg{b_+})}{} and quantum

tangent spaces on \ensuremath{C_q(B_+)}{}. Intersections correspond to intersections.

(b) In particular,

$\Gamma$ and $L$ corresponding to coirreducible differential calculi

on \ensuremath{U_q(\lalg{b_+})}{} and

irreducible quantum tangent spaces on \ensuremath{C_q(B_+)}{} via the limiting process

are given as follows:

$\Gamma$ has a right invariant basis

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

\begin{gather*}

\diff X=\eta_1 \qquad \diff H=(1-n)\eta_0 \\

[H, \eta_i]=(1-n+i)\eta_i\quad\forall i\qquad

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

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

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

\end{cases}

\end{gather*}

holds. The braided derivations corresponding to the dual basis of

$L$ are given by

\begin{gather*}

\partial_i\no{f}=\no{T_{1-n+i,H}

\frac{1}{i!}\left(\frac{\partial}{\partial X}\right)^i f}

\qquad\forall i\ge 1\\

\partial_0\no{f}=\no{T_{1-n,H} f - f}

\end{gather*}

for $f\in\k[X,H]$

with the normal ordering $\k[X,H]\to \ensuremath{U(\lalg{b_+})}$ via $H^n X^m\mapsto H^n X^m$.

\end{prop}

\begin{proof}

(a) The strict duality between \ensuremath{C(B_+)}-subcomodules $L\subseteq\ker\cou$

given by lemma \ref{lem:cbp_dual} and corollary \ref{cor:uqbp_eclass}

and \ensuremath{U(\lalg{b_+})}-modules $\ensuremath{U(\lalg{b_+})}/(\k 1+M)$ with $M$ given by lemma

\ref{lem:ubp_dual} and

corollary \ref{cor:cqbp_eclass} can be checked explicitly.

It is essentially due to mutual annihilation of factors $H+k$ in

\ensuremath{U(\lalg{b_+})}{} with elements $g^k$ in \ensuremath{C(B_+)}{}.

(b) $L$ is generated by

$\{g^{1-n}-1,Xg^{1-n},\dots,

X^{n-1}g^{1-n}\}$ and

$M$ is generated by $\{H(H+n-1),X(H+n-1),X^n \}$.

The formulas are obtained by denoting with

$\eta_0,\dots,\eta_{n-1}$ the equivalence classes of

$H/(1-n),X,\dots,X^{n-1}$ in $\ensuremath{U(\lalg{b_+})}/(\k 1+M)$.

The dual basis of $L$ is then

\[g^{1-n}-1,X g^{1-n},

\dots,\frac{1}{(n-1)!}X^{n-1}