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\[\partial_x(a b)=(\partial_x a) b |
+ a_{(2)} \partial_{a_{(1)}\triangleright x}b\qquad\forall x\in L, a\in A\] |
Given a right invariant basis $\{\eta_i\}_{i\in I}$ of $\Gamma$ with a |
dual basis $\{\phi_i\}_{i\in I}$ of $L$ we have |
\[{\diff a}=\sum_{i\in I} \eta_i\cdot \partial_i(a)\qquad\forall a\in A\] |
where we denote $\partial_i=\partial_{\phi_i}$. (This can be easily |
seen to hold by evaluation against $\phi_i\ \forall i$.) |
\section{Classification on \ensuremath{C_q(B_+)}{} and \ensuremath{U_q(\lalg{b_+})}{}} |
\label{sec:q} |
In this section we completely classify differential calculi on \ensuremath{C_q(B_+)}{} |
and, dually, quantum tangent spaces on \ensuremath{U_q(\lalg{b_+})}{}. We start by |
classifying the relevant crossed modules and then proceed to a |
detailed description of the calculi. |
\begin{lem} |
\label{lem:cqbp_class} |
(a) Left crossed \ensuremath{C_q(B_+)}-submodules $M\subseteq\ensuremath{C_q(B_+)}$ by left |
multiplication and left |
adjoint coaction are in one-to-one correspondence to |
pairs $(P,I)$ |
where $P\in\k(q)[g]$ is a polynomial with $P(0)=1$ and $I\subset\mathbb{N}$ is |
finite. |
$\codim M<\infty$ iff $P=1$. In particular $\codim M=\sum_{n\in I}n$ |
if $P=1$. |
(b) The finite codimensional maximal $M$ |
correspond to the pairs $(1,\{n\})$ with $n$ the |
codimension. The infinite codimensional maximal $M$ are characterised by |
$(P,\emptyset)$ with $P$ irreducible and $P(g)\neq 1-q^{-k}g$ for any |
$k\in\mathbb{N}_0$. |
(c) Crossed submodules $M$ of finite |
codimension are intersections of maximal ones. |
In particular $M=\bigcap_{n\in I} M^n$, with $M^n$ corresponding to |
$(1,\{n\})$. |
\end{lem} |
\begin{proof} |
(a) Let $M\subseteq\ensuremath{C_q(B_+)}$ be a crossed \ensuremath{C_q(B_+)}-submodule by left |
multiplication and left adjoint coaction and let |
$\sum_n X^n P_n(g) \in M$, where $P_n$ are polynomials in $g,g^{-1}$ |
(every element of \ensuremath{C_q(B_+)}{} can be expressed in |
this form). From the formula for the coaction ((\ref{eq:adl}), see appendix) |
we observe that for all $n$ and for all $t\le n$ the element |
\[X^t P_n(g) \prod_{s=1}^{n-t} (1-q^{s-n}g)\] |
lies in $M$. |
In particular |
this is true for $t=n$, meaning that elements of constant degree in $X$ |
lie separately in $M$. It is therefore enough to consider such |
elements. |
Let now $X^n P(g) \in M$. |
By left multiplication $X^n P(g)$ generates any element of the form |
$X^k P(g) Q(g)$, where $k\ge n$ and $Q$ is any polynomial in |
$g,g^{-1}$. (Note that $Q(q^kg) X^k=X^k Q(g)$.) |
We see that $M$ contains the following elements: |
\[\begin{array}{ll} |
\vdots & \\ |
X^{n+2} & P(g) \\ |
X^{n+1} & P(g) \\ |
X^n & P(g) \\ |
X^{n-1} & P(g) (1-q^{1-n}g) \\ |
X^{n-2} & P(g) (1-q^{1-n}g) (1-q^{2-n}g) \\ |
\vdots & \\ |
X & P(g) (1-q^{1-n}g) (1-q^{2-n}g) \ldots (1-q^{-1}g) \\ |
& P(g) (1-q^{1-n}g) (1-q^{2-n}g) \ldots (1-q^{-1}g)(1-g) |
\end{array} |
\] |
Moreover, if $M$ is generated by $X^n P(g)$ as a module |
then these elements generate a basis for $M$ as a vector |
space by left |
multiplication with polynomials in $g,g^{-1}$. (Observe that the |
application of the coaction to any of the elements shown does not |
generate elements of new type.) |
Now, let $M$ be a given crossed submodule. We pick, among the |
elements in $M$ of the form $X^n P(g)$ with $P$ of minimal |
length, |
one |
with lowest degree in $X$. Then certainly the elements listed above are |
in $M$. Furthermore for any element of the form $X^k Q(g)$, $Q$ must |
contain $P$ as a factor and for $k<n$, $Q$ must contain $P(g) (1-q^{1-n}g)$ |
as a factor. We continue by picking the smallest $n_2$, so that |
$X^{n_2} P(g) (1-q^{1-n}g) \in M$. Certainly $n_2<n$. Again, for any |
element of $X^l Q(g)$ in $M$ with $l<n_2$, we have that |
$P(g) (1-q^{1-n}g) (1-q^{1-n_2}g)$ divides Q(g). We proceed by |
induction, until we arrive at degree zero in $X$. |
We obtain the following elements generating a basis for $M$ by left |
multiplication with polynomials in $g,g^{-1}$ (rename $n_1=n$): |
\[ \begin{array}{ll} |
\vdots & \\ |
X^{n_1+1} & P(g) \\ |
X^{n_1} & P(g) \\ |
X^{n_1-1} & P(g) (1-q^{1-{n_1}}g) \\ |
\vdots & \\ |