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\begin{document}
\begin{center}
Math 261B---Fall 2020\\
Problem Set 1
\end{center}
\medskip
1. (a) Show that the polynomial ring $A = \ZZ [x_{1},\ldots,x_{n}]$
has a unique structure of Hopf algebra over $\ZZ $ such that the
variables $x_{i}$ are primitive, meaning that $\Delta x_{i} =
x_{1}\otimes 1+1\otimes x_{i}$, the antipode is $S(x_{i})=-x_{i}$, and
the co-unit is $\varepsilon (f) = f(0)$.
\smallskip
(b) If $K$ is an algebraically closed field, $K\otimes _{\ZZ }A$
becomes the coordinate ring of the additive group $\GG _{a}^{n}(K)$.
Show that this is a special case of the more general property that for
any commutative ring $R$, the Hopf algebra structure on $A$ gives the
set of ring homomorphisms $A\rightarrow R$ the structure of a group
isomorphic to $(R^{n},+)$.
\smallskip
(c) Verify that the group structure in part (b) is functorial with
respect to ring homomorphisms $R\rightarrow R'$.
In the language of algebraic geometry, there is an affine scheme
$\Spec (A)$ over $\ZZ $ associated to $A$. Any scheme $X$ over a
commutative ring $k$ induces a functor $X(-)$ from $k$-algebras $R$ to
sets, where $X(R)$ is the set of $k$-scheme morphisms $\Spec
(R)\rightarrow X$, also called the set of $R$-valued points of $X$.
Morphisms of affine $k$-schemes $\Spec (R)\rightarrow \Spec (A)$ are
in functorial one-to-one correspondence with $k$-algebra homomorphisms
$A\rightarrow R$. Note that every commutative ring is a $\ZZ
$-algebra in a unique way.
What this problem shows is that for $A$ as in part (a), $\Spec
(A)$ is a group scheme $\GG _{a}^{n}$ over $\ZZ $ whose functor
of points send $R$ to the additive group $R^{n}$.
\medskip
2. Work out the analog of Problem 1 for the Laurent polynomial ring
$A = \ZZ [x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$, with each variable
grouplike, meaning that $\Delta x_{i} = x_{i}\otimes x_{i}$, and with
antipode $S(x_{i})=x_{i}^{-1}$ and co-unit $\varepsilon (f) = f(1)$.
\medskip
3. The axioms of a non-commutative Hopf algebra (over a commutative
ground ring $k$) are essentially the same as in the commutative case,
except that the antipode $S$ is required to be an a $k$-algebra
antihomomorphism (i.e., it reverses multiplication). The coproduct
$\Delta \colon A\rightarrow A\otimes A$ and co-unit $\varepsilon
\colon A\rightarrow k$ are ordinary algebra homomorphisms.
\smallskip
(a) Write down the axioms explicitly, using Sweedler notation $\Delta
f = \sum f_{(1)}\otimes f_{(2)}$ for the coproduct, recalling that
they should formally make $A$ a group object in the opposite of the
category of $k$-algebras, with $\otimes $ playing the role of a
product.
\smallskip
(b) Show that, even in the commutative case, the antipode is a
homomorphism from the coproduct to the opposite coproduct.
\smallskip
(c) Show that the axioms are self-dual. Hence, in particular, if $k$
is a field and $A$ is a finite-dimensional Hopf algebra, then $A^{*}$
has the structure of a Hopf algebra.
\smallskip
(d) Prove that if the antipode $S$ in a Hopf algebra $A$ is invertible
(this holds automatically in some cases, including commutative and
co-commutative Hopf algebras), then $S^{-1}$ is the antipode for the
same algebra with opposite coproduct, and also the antipode for the
algebra with the same coproduct and the opposite product. In
particular, if $A$ is either commutative or co-commutative, then
$S^{2}$ is the identity.
\medskip
4. Show that the group algebra $kG$ of a finite group $G$ has a
co-commutative Hopf algebra structure in which the elements of $g$ are
grouplike, and that this makes $kG$ dual to the Hopf algebra of
$k$-valued functions on $G$. This works for any commutative ring $k$.
\medskip
5. If $A$ is an infinite-dimensional Hopf algebra over a field $k$,
its dual space $A^{*}$ becomes an algebra, but not a co-algebra,
because $\Delta ^{*}\colon A ^{*}\rightarrow (A\otimes A)^{*}$ need
not map $A^{*}$ into $A^{*}\otimes A^{*}$, which is a proper subspace
of $(A\otimes A)^{*}$.
Let $A^{\circ }\subseteq A^{*}$ be the preimage $(\Delta ^{*})^{-1}
(A^{*}\otimes A^{*})$. Show that $A^{\circ }$ is a subalgebra of
$A^{*}$ and that the Hopf algebra structure on $A$ induces a dual Hopf
algebra structure on $A^{\circ }$. This alebra is called the Hopf
dual of $A$.
\medskip
6. Show that the free abelian group generated by the elements
$x^{n}/n!$ in $\QQ [x]$ is a $\ZZ $-subalgebra $D\subseteq \QQ [x]$,
called the divided power algebra in one variable. Show that $D$ has a
natural co-commutative Hopf algebra structure, dual to the Hopf
algebra structure on $\ZZ [x]$ in which $x$ is primitive. Duality in
this case means a perfect pairing $D\otimes _{\ZZ }\ZZ [x]\rightarrow
\ZZ $ such that the coproduct, co-unit and antipode in each algebra
are dual to the product, unit and antipode in the other.
\medskip
7. Consider the case of the pair of dual Hopf algebras $kG$ and
$\Ocal (G)$ in Problem 4, where $\Ocal (G)$ is the algebra of
functions $G\rightarrow k$, when $k$ is a field of characteristic $p$
and $G = \ZZ /p\ZZ $ is a cyclic group of order $p$.
\smallskip
(a) Show that $k(G) = k[x]/(x^{p}-1)$, with $x$ group-like. We can
think of it as $\Ocal (\mu _{p})$ for the non-reduced group scheme of
`$p$-th roots of unity' over $k$.
\smallskip
(b) Show that linear representations of $G$ over $k$, or $kG$ modules,
are the same as $\Ocal (G)$ comodules, and that except for the fact
that $G$ is not connected group, it behaves just like a unipotent
linear algebraic group, in the equivalent senses that (i) the only
irreducibe $kG$ module is the trivial representation; (ii) in every
finite-dimensional representation, $G$ acts by upper unitriangular
matrices in some basis; (iii) $G$ acts unipotently on $\Ocal (G)$.
\smallskip
(c) Show that $\Ocal (G)$ modules, or $kG$ comodules, are the same as
($\ZZ /p\ZZ $)-graded vector spaces. In particular, $\Ocal (G)$ has
$p$ distinct non-isomorphic one-dimensional irreducible modules, and
every module is a direct sum of these. In this sense the group scheme
$\mu _{p}$ is `reductive' and its representation theory in any
characteristic resembles the characteristic zero representation theory
of a cyclic group of order $p$.
\medskip
8. (a) Write out explicitly the axioms of a right coaction
$W\rightarrow W\otimes A$, where $A$ is a Hopf algebra over $k$ and
$W$ is a $k$ module, dual to the axioms of a group action. You can
assume $k$ is a field if you like, but the axioms are the same over
any commutative ring.
\smallskip
(b) Verify in detail that if $G\times V\rightarrow V$ is a linear
algebraic action of an affine algebraic group on a finite dimensional
vector space $V$ (considered as an algebraic variety), and $\rho
\colon \Ocal (V)\rightarrow \Ocal (V)\otimes \Ocal (G)$ is the
corresponding homomorphism of algebras of functions, then $\rho $ maps
$V^{*}$ into $V^{*}\otimes \Ocal (G)$ and makes $V^{*}$ a
$\Ocal (G)$ comodule. Note that $V^{*}$ is the subspace of $\Ocal
(V)$ consisting of linear functions.
\smallskip
(c) Verify in detail that every finite-dimensional right $\Ocal (G)$
comodule $W$ arises from a unique linear algebraic action of $G$ on $V
= W^{*}$ as in part (b), and more explicitly, that the right action of
$g\in G$ on $W$ dual to the left action on $V$ is given by composing
$\rho \colon W\rightarrow W\otimes \Ocal (G)$ with the evaluation map
$\ev _{g}\colon \Ocal (G)\rightarrow k$.
\medskip
9. (a) Show that if $G$ is an algebraic group, the grouplike elements of
the Hopf algebra $\Ocal (G)$ are the 1-dimensional characters of $G$,
that is, the group homomorphisms $G\rightarrow \GG _{m}(K)= K^{\times
}$, considered as functions on $G$.
\smallskip
(b) More generally, show that if $A$ is a Hopf algebra over a
commutative ring $k$, then grouplike elements of $A$ correspond
naturally to $A$ comodules isomorphic to $k$ as a $k$ module.
\medskip
10. Let $A$ be a Hopf algebra over $k$. Show that a $k$-linear map
$\lambda \colon A\rightarrow k$ is a grouplike element of the Hopf
dual of $A$ if and only if is an algebra homomorphism.
In the case $A = \Ocal (G)$ for an affine algebraic group $G$, this
means that the grouplike elements of the Hopf dual of $A$ correspond
to the group elements $g\in G$.
\medskip
11. Let $G$ be an affine algebraic group and let $A = \Ocal (G)$.
Recall that $G$ embeds in the the algebra $A^{*}$ by $g\rightarrow \ev
_{g}$ and that its Lie algebra $\gfrak = T_{e}G$ embeds in $A^{*}$ as
the space of linear functionals that kill the ideal $\mfrak _{e}^{2}$
and the constant functions. Verify in detail that the adjoint action
of $G$ on $\gfrak $, given for $g\in G$ by the diffential at $e$ of
conjugation by $g$ on $G$, corresponds to conjugation by $G$ on
$\gfrak $ in $A^{*}$.
\medskip
12. Show that in characteristic zero, the finite dimensional
representations of $GL_{2}(K)$ are completely reducible, and the
irreducible representations are the standard representations on
homogeneous polyonomials of each degree $d$ in $K[x,y]$, tensored with
integer powers of the 1-dimensional representation whose character is
the determinant. More precisely, with $T\subset B\subset GL_{2}$ the
diagonal and upper triangular matrices, the weight lattice is $X = \ZZ
^{2}$, the dominant weights are $(\lambda _{1},\lambda _{2})$ such
that $\lambda _{1}\geq \lambda _{2}$, and $V_{\lambda } = (\det
)^{\otimes \lambda _{2}}\otimes K[x,y]_{\lambda _{1}-\lambda _{2}}$ is
the irreducible representation with highest weight $\lambda $.
More explicitly, show that the coordinate ring $\Ocal (GL_{2})$
decomposes into a direct sum of subspaces spanned by the matrix
coefficients of the representations described above. This and the
irreducibility of these represenations implies the other conclusions.
\end{document}