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:

Link to Diagram

  • 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 \(s:M\to M\):

Link to Diagram

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.

Tags: #higher-algebra/K-theory #projects/notes/seminars

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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