--- date: 2021-05-03 tags: [ "web/quick-notes" ] --- # 2021-05-03 ## Representations of Hopf Algebras Tags: #lie-theory See [Hopf algebra](Hopf%20algebra.md) - Algebras: $m: A^{\tensor 2} \to A$ and $u:k\to A$ the unit with associativity: \begin{tikzcd} {M^{\tensor 3}} && {M^{\tensor 2}} \\ \\ {M^{\tensor 2}} && M \arrow["m", from=3-1, to=3-3] \arrow["m", from=1-3, to=3-3] \arrow["{m\tensor \id_M}", from=1-1, to=1-3] \arrow["{\id_M \tensor m}"', from=1-1, to=3-1] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJNXntcXHRlbnNvciAzfSJdLFsyLDAsIk1ee1xcdGVuc29yIDJ9Il0sWzAsMiwiTV57XFx0ZW5zb3IgMn0iXSxbMiwyLCJNIl0sWzIsMywibSJdLFsxLDMsIm0iXSxbMCwxLCJtXFx0ZW5zb3IgXFxvbmUiXSxbMCwyLCJcXG9uZVxcdGVuc29yIG0iLDJdXQ==) - [coalgebra](coalgebra.md) : $\Delta: A\to A^{\tensor 2}$, $\eps: A\to k$ the counit. Reverse the arrows in the diagram for coassociativity. This yields a bialgebra, for Hopf structure need an antipode $s:M\to M$: \begin{tikzcd} &&& M \\ \\ k &&& {M^{\tensor 2}} \\ \\ &&& {M^{\tensor 2}} \arrow["{s\tensor \id_M}", from=3-4, to=5-4] \arrow["u"', from=3-1, to=5-4] \arrow["\eps"', from=1-4, to=3-1] \arrow["{?}", from=1-4, to=3-4] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMywwLCJNIl0sWzMsMiwiTV57XFx0ZW5zb3IgMn0iXSxbMyw0LCJNXntcXHRlbnNvciAyfSJdLFswLDIsImsiXSxbMSwyLCJzXFx0ZW5zb3IgXFxvbmUiXSxbMywyLCJ1IiwyXSxbMCwzLCJcXGVwcyIsMl0sWzAsMSwiPyJdXQ==) Why [Hopf algebra](Hopf%20algebra.md)? Some natural examples: - $kG$ the [group algebra](group%20algebra.md). - $\Delta(g) \da g^{\tensor 2}$ - $\eps(g) = 1_G$ - $s(g) = g\inv$ - $k^G = \Hom_k(kG, k)$ an algebra of functions, forcing distinct group elements to be orthogonal idempotents, take \( \ts{ P_x \st x\in G } \) with \( P_x P_y = \delta_{xy} P_y \) ?? - Consider category $\modsleft{H}^\fd$ of finite-dimensional [Unsorted/Representation Theory (Subject MOC)](Unsorted/Representation%20Theory%20(Subject%20MOC).md) of $H$. - Issue: tensor product of $R\dash$modules may not again be an $R\dash$module. - Antipode will be invertible when $H$ is finite dimensional - A lot of structures here: closed under tensors, duals, contains $k$. - Finite [tensor category](tensor%20category.md) : looks like $\modsleft{H}$, [Unsorted/enriched category](Unsorted/enriched%20category.md) over vector spaces, [Monoidal category](Monoidal%20category.md), coherent associativity via [pentagon axiom](pentagon%20axiom), [triangle axiom](triangle%20axiom). - Evaluation $M\dual \tensor M \to \one$ and coevaulation $\one \to X\tensor X\dual$. - For finite dimensional vector spaces, $k\mapsto \sum k e_i \tensor e_i\dual$? - Finite rank: finitely many simples up to isomorphism. Can still have infinitely many indecomposables. - Define $\Ext^n_{\cat C}(X, Y)$ to be equivalence classes of $n\dash$fold extensions, i.e. exact sequences $0 \to Y \to E_n \to \cdots \to E_1 \to X \to 0$, and $H^*(\cat{C}) \da H^*_{\cat C}(\one, \one) = \bigoplus _{n\geq 0} \Ext^n_{\cat C} (\one, \one )$. Can similarly replace $\one$ with $X$ to define $H^*(X)$, which will be a module over $H^*(\cat C)$. - [support variety](support%20variety) : $V_{\cat C}(\one) = \mspec H^*(\cat C)$, $V_{\cat C}(X)$ is a more complicated quotient. - Representation theory *of categories*: module categories over a category! - Big question: **tensor product property**. Is there an equality \[ V_{\cat C}(X\tensor Y) \equalsbecause{?} V_{\cat C}(X) \intersect V_{\cat C}(Y) .\] - True for cocommutative [Hopf algebra](Hopf%20algebra.md), some [quantum groups](quantum%20groups). - Some counterexamples in non-braided monoidal categories. Uses a *smash product* of modules - See [thick ideals](thick%20ideals). ## Clausen, the K-theory of adic spaces. Tags: #higher-algebra/K-theory #projects/notes/seminars > Reference: Clausen, the K-theory of adic spaces. - [adic spaces](adic%20spaces) : formalism for non-Archimedean geometry. - [formal scheme](formal%20scheme) : e.g. formal thickening of a subvariety. Sometimes want to delete a [special fiber](special%20fiber.md). - Definition of ([adic ring](adic%20ring), complete with respect to a finitely-generated ideal, so $R = \inverselim R/I^n$ - Yields [scheme](scheme.md) as a subcategory? - Some nice features: - Topological bases of [quasicompact](quasicompact.md) 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 - $\Solid_\ZZ$: abelian bicomplete category of [solid sets](solid%20mathematics.md) Full subcategory of [condensed sets](condensed%20sets). Has compact projective generators $\prod_I \ZZ$ - [compact generators](compact%20generators) : mapping out to filtered colimits..? - [projective generators](projective%20generators) : lift along surjections - [generators of a category](generators%20of%20a%20category) : everything is a cokernel of direct sums of these - For morphisms, note $\Hom( \prod_I \ZZ, \ZZ) = \bigoplus_I \ZZ$ - Let $\Perf$ be [perfect complexes](perfect%20complexes.md), why not consider $K(\VectBundle(X))$ or $K(\Perf(X))$? - Doesn't satisfy [descent](descent.md) - Need a good category of [quasicoherent sheaves](quasicoherent%20sheaves) - What is a [presentable category](presentable%20category.md)? - What is a [Tate algebra](Tate%20algebra)? - What is [Arakelov theory](Arakelov%20theory)? - Something to do with [arithmetic surfaces](arithmetic%20surfaces). - Some apparent contributions by Faltings: - A [Riemann-Roch](Riemann-Roch.md) theorem - A [Noether formula](Noether%20formula) - A [Hodge index theorem](Hodge%20index%20theorem.md) - Non-negativity of the self-intersection of the [dualizing sheaf](dualizing%20sheaf.md). - Vojta 1991: new proof of the [Mordell conjecture](Mordell%20conjecture)