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:
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\(kG\) the group algebra.
- \(\Delta(g) \coloneqq g^{\otimes 2}\)
- \({\varepsilon}(g) = 1_G\)
- \(s(g) = g^{-1}\)
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\(k^G = \mathop{\mathrm{Hom}}_k(kG, k)\) an algebra of functions, forcing distinct group elements to be orthogonal idempotents, take \(\left\{{ P_x {~\mathrel{\Big\vert}~}x\in G }\right\}\) with \(P_x P_y = \delta_{xy} P_y\) ??
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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.
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Antipode will be invertible when \(H\) is finite dimensional
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A lot of structures here: closed under tensors, duals, contains \(k\).
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Finite tensor category : looks like \({}_{H}{\mathsf{Mod}}\), Unsorted/enriched category over vector spaces, Monoidal category, coherent associativity via pentagon axiom, triangle axiom.
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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.
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Evaluation \(M {}^{ \vee }\otimes M \to \one\) and coevaulation \(\one \to X\otimes X {}^{ \vee }\).
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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 \(H^*(\mathsf{C})\).
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support variety : \(V_{\mathsf{C}}(\one) = \operatorname{mSpec}H^*(\mathsf{C})\), \(V_{\mathsf{C}}(X)\) is a more complicated quotient.
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Representation theory of categories: module categories over a category!
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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*}
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True for cocommutative Hopf algebra, some quantum groups.
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Some counterexamples in non-braided monoidal categories. Uses a smash product of modules
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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
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adic spaces : formalism for non-Archimedean geometry.
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formal scheme : e.g. formal thickening of a subvariety. Sometimes want to delete a special fiber.
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Definition of (adic ring, complete with respect to a finitely-generated ideal, so \(R = \varprojlim R/I^n\)
- Yields scheme as a subcategory?
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Some nice features:
- Topological bases of quasicompact open subsets
- Has a nice ring attached to each subspace.
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Subtleties:
- Structure sheaf is only a presheaf and not necessarily a sheaf
- Not even great when it is a sheaf: can’t work locally
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\(\mathsf{Solid}_{\mathbf{Z}}\): 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
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For morphisms, note \(\mathop{\mathrm{Hom}}( \prod_I {\mathbf{Z}}, {\mathbf{Z}}) = \bigoplus_I {\mathbf{Z}}\)
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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
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Need a good category of quasicoherent sheaves
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What is a presentable category?
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What is a Tate algebra?
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What is Arakelov theory?
- Something to do with arithmetic surfaces.
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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