# 2021-05-03

## Representations of Hopf Algebras

Tags: #lie-theory

See Hopf algebra

• Algebras: $$m: A^{\otimes 2} \to A$$ and $$u:k\to A$$ the unit with associativity:

• coalgebra : $$\Delta: A\to A^{\otimes 2}$$, $${\varepsilon}: A\to k$$ the counit. Reverse the arrows in the diagram for coassociativity. This yields a bialgebra, for Hopf structure need an antipode $$s:M\to M$$:

Why Hopf algebra? Some natural examples:

• $$kG$$ the group algebra.

• $$\Delta(g) \coloneqq g^{\otimes 2}$$
• $${\varepsilon}(g) = 1_G$$
• $$s(g) = g^{-1}$$
• $$k^G = \mathop{\mathrm{Hom}}_k(kG, k)$$ an algebra of functions, forcing distinct group elements to be orthogonal idempotents, take $$\left\{{ P_x {~\mathrel{\Big\vert}~}x\in G }\right\}$$ with $$P_x P_y = \delta_{xy} P_y$$ ??

• Consider category $${}_{H}{\mathsf{Mod}}^{\mathrm{fd}}$$ of finite-dimensional Unsorted/Representation Theory (Subject MOC) of $$H$$.

• Issue: tensor product of $$R{\hbox{-}}$$modules may not again be an $$R{\hbox{-}}$$module.
• Antipode will be invertible when $$H$$ is finite dimensional

• A lot of structures here: closed under tensors, duals, contains $$k$$.

• Finite tensor category : looks like $${}_{H}{\mathsf{Mod}}$$, Unsorted/enriched category over vector spaces, Monoidal category, coherent associativity via pentagon axiom, triangle axiom.

• Evaluation $$M {}^{ \vee }\otimes M \to \one$$ and coevaulation $$\one \to X\otimes X {}^{ \vee }$$.
• For finite dimensional vector spaces, $$k\mapsto \sum k e_i \otimes e_i {}^{ \vee }$$?
• Finite rank: finitely many simples up to isomorphism. Can still have infinitely many indecomposables.
• Define $$\operatorname{Ext} ^n_{\mathsf{C}}(X, Y)$$ to be equivalence classes of $$n{\hbox{-}}$$fold extensions, i.e. exact sequences $$0 \to Y \to E_n \to \cdots \to E_1 \to X \to 0$$, and $$H^*(\mathsf{C}) \coloneqq H^*_{\mathsf{C}}(\one, \one) = \bigoplus _{n\geq 0} \operatorname{Ext} ^n_{\mathsf{C}} (\one, \one )$$. Can similarly replace $$\one$$ with $$X$$ to define $$H^*(X)$$, which will be a module over $$H^*(\mathsf{C})$$.

• support variety : $$V_{\mathsf{C}}(\one) = \operatorname{mSpec}H^*(\mathsf{C})$$, $$V_{\mathsf{C}}(X)$$ is a more complicated quotient.

• Representation theory of categories: module categories over a category!

• Big question: tensor product property. Is there an equality \begin{align*} V_{\mathsf{C}}(X\otimes Y) \overset{?}{=} V_{\mathsf{C}}(X) \cap V_{\mathsf{C}}(Y) .\end{align*}

• True for cocommutative Hopf algebra, some quantum groups.

• Some counterexamples in non-braided monoidal categories. Uses a smash product of modules

• See thick ideals.

## Clausen, the K-theory of adic spaces.

Reference: Clausen, the K-theory of adic spaces. https://www.youtube.com/watch?v=e_0PTVzViRQ

• adic spaces : formalism for non-Archimedean geometry.

• formal scheme : e.g. formal thickening of a subvariety. Sometimes want to delete a special fiber.

• Definition of (adic ring, complete with respect to a finitely-generated ideal, so $$R = \varprojlim R/I^n$$

• Yields scheme as a subcategory?
• Some nice features:

• Topological bases of quasicompact open subsets
• Has a nice ring attached to each subspace.
• Subtleties:

• Structure sheaf is only a presheaf and not necessarily a sheaf
• Not even great when it is a sheaf: can’t work locally
• $$\mathsf{Solid}_{\mathbf{Z}}$$: abelian bicomplete category of solid sets Full subcategory of condensed sets. Has compact projective generators $$\prod_I {\mathbf{Z}}$$

• compact generators : mapping out to filtered colimits..?
• projective generators : lift along surjections
• generators of a category : everything is a cokernel of direct sums of these
• For morphisms, note $$\mathop{\mathrm{Hom}}( \prod_I {\mathbf{Z}}, {\mathbf{Z}}) = \bigoplus_I {\mathbf{Z}}$$

• Let $$\mathsf{Perf}$$ be perfect complexes, why not consider $$K({ {\mathsf{Bun}}\qty{\operatorname{GL}_r} }(X))$$ or $$K(\mathsf{Perf}(X))$$?

• Doesn’t satisfy descent
• Need a good category of quasicoherent sheaves

• What is a presentable category?

• What is a Tate algebra?

• What is Arakelov theory?

• Something to do with arithmetic surfaces.
• Some apparent contributions by Faltings:
• A Riemann-Roch theorem
• A Noether formula
• A Hodge index theorem
• Non-negativity of the self-intersection of the dualizing sheaf.
• Vojta 1991: new proof of the Mordell conjecture
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