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{\large \bf Math 8300 -- Quantum Groups -- Problem Set 2} 

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{\large \bf Due: Friday, February 21}
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\noindent 1. A Hopf algebra $H$ is a bialgebra with additional structure. That is, it has an antipode map $S: H \longrightarrow H$ such that
$$ m (S \otimes \textrm{id}) \circ \Delta = m (\textrm{id} \otimes S) \circ \Delta = \eta \circ \varepsilon, $$
where $m$ and $\Delta$ are multiplication and comultiplication, $\eta$ and $\varepsilon$ are unit and counit in $H$. 
\begin{enumerate}[a)]
\item Draw commutative diagrams representing the above displayed axioms. 
\item Explain, using diagrams, why every finite dimensional Hopf algebra $H$ has a dual Hopf algebra structure on the underlying
dual vector space.
\item Show that $S$ is an antialgebra map. That is, $S(m(a,b)) = m(S(b),S(a))$ and $S(1)=1$. Again, $m$ is multiplication here. (Note: We
do not assume that $S$ as a linear map has an inverse.)
\end{enumerate}

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\noindent 2. Show that the tensor algebra $T(V) = \displaystyle\bigoplus_{i=0}^\infty V^{\otimes i}$ can be given a Hopf algebra structure. (Lectures 8 and 9 included definitions of the necessary comultiplication, counit, and antipode maps.)
 
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\noindent 3. Show that the Hopf algebra structure on $k G$, the group algebra of a finite group $G$, has dual Hopf algebra $k(G)$, the algebra of functions on $G$ valued in $k$, with multiplication defined pointwise: $(f \cdot g)(x) := f(x) g(x)$. That is, write down the five pairing axioms from Lecture 10, p. 3, and show that they are satisfied for an appropriate choice of non-degenerate bilinear pairing.

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\noindent 4. In lecture 12, we weakened cocommutativity in a bialgebra, asking for a family of isomorphisms $c_{A,B} : A \otimes B \longrightarrow B \otimes A$ for all modules $A, B$ (as opposed to requiring it to be the simple ``flip'' map $\tau: a \otimes b \mapsto b \otimes a$). These must be compatible with associativity (whose diagrams are a pair of a commutative hexagons - see p. 2A in the lecture 12 notes). The existence of the map $c_{A,B}$ is equivalent to the existence of an element $R$ in $H \otimes H$ with certain properties. Show that the compatibility of $c_{A,B}$ with associativity is equivalent to the element $R$ satisfying the pair of properties:
$$ (\Delta \otimes \textrm{id}) R = R_{13} R_{23}, \qquad  (\textrm{id} \otimes \Delta) R = R_{13} R_{12} $$

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\noindent 5. Let $E, F, K, K^{-1}$ denote the generators of $U_q(\mathfrak{sl}_2)$ with relations as given in Lecture 16. Show the following relations for any integer $m > 0$:
$$ [E, F^m] = [m]_q F^{m-1} \frac{q^{-(m-1)} K - q^{(m-1)} K^{-1}}{q-q^{-1}},$$  
$$ [E^m, F] = [m]_q E^{m-1} \frac{q^{(m-1)} K - q^{-(m-1)} K^{-1}}{q-q^{-1}} $$
where
$$ [m]_q := \frac{q^m - q^{-m}}{q-q^{-1}}$$

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