# 2022-01-21

Tags: #web/quick-notes

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## Chelsea Walton, Stanford AG Seminar (15:07)

• Representations of algebras: $$\rho: A\to { \operatorname{End} }_k(V)$$ for some $V\in { \mathsf{Vect} }{/ {k}} $$, possibly infinite dimensional, i.e.$$V\in {}{A}{\mathsf{Mod}} _{/ {k}}$.
• Irreps of commutative algebras are all 1-dimensional, so representation theory is really the study of noncommutative algebras.
• PI (polynomial identity) algebras: almost commutative. Requires a uniform monic multilinear polynomial $$f$$ such that $$f(a_1, \cdots, a_n)$$ for all $$\mathbf{a} \in A{ {}^{ \scriptscriptstyle\times^{n} } }$$.
• Prime algebras: $$A$$ such that $$aAb \neq 0$$ when $$a, b\neq 0$$, slightly weaker than being an integral domain.
• PI degree of $$A$$: half of the minimal degree of any polynomial identity for $$A$$. Roughly the rank of $$A$$ over its center $$Z(A)$$. More precisely, $$\operatorname{rank}_{Z(A)} A = \qty{\mathrm{PIdeg}(A)}^2$$.
• Examples of how this measures noncommutativity:
• $$A$$ commutative implies degree 1.
• $$\operatorname{Mat}_{n\times n}({\mathbf{C}})$$ has degree $$d$$.
• $$A\coloneqq{\mathbf{C}}\left\langle{x, y}\right\rangle/\left\langle{xy+yx}\right\rangle$$ has degree 2. Note $$Z(A) = {\mathbf{C}}[x^2, y^2]$$.
• $$U_q({\mathfrak{g}})$$, an enveloping algebra quantized at a root of unity, has degree depending on $$q$$.
• $${\mathcal{O}}_q(G)$$ quantized function algebras, also has degree depending on $$q$$.
• In an Azumaya algebra all of the irreps have the same dimension.
• Theorem (many, many people): for $$A$$ prime, PI, finitely generated as an algebra, Noetherian, the dimensions of finite-dimensional irreps are bounded above by the PI degree.
• Central characters: $$\ker \rho \cap Z(A) \in \operatorname{mSpec}Z(A)$$.
• There is a well-defined surjection $${\mathsf{Irr}}{\mathsf{Rep}}(A) \to \operatorname{mSpec}Z(A)$$ where $$[\rho] \mapsto {\mathfrak{m}}_\rho \coloneqq\ker \rho \cap Z(A)$$.
• For $$A$$ commutative, $${\mathsf{Irr}}{\mathsf{Rep}}(A) \cong \operatorname{mSpec}A$$.
• For $$A=\operatorname{Mat}_{n\times n}({\mathbf{C}}), \ker \rho = 0$$.
• For $$A\coloneqq{\mathbf{C}}\left\langle{x, y}\right\rangle/\left\langle{xy+yx}\right\rangle$$, $$\operatorname{mSpec}Z(A) = \operatorname{mSpec}{\mathbf{C}}[x^2, y^2]\cong {\mathbf{C}}^2$$. The 1-dim reps are on the axes $$u=0$$ and $$v=0$$, and the 2-dim reps are in the complement.
• Irreps of highest dimensions: Azumaya locus $${\mathcal{A}}_A$$, open and dense in the smooth locus of $$Y\coloneqq\operatorname{mSpec}Z(A)$$. For lower dimensions: ramification locus $${\mathcal{R}}_A$$
• Very convenient when $${\mathcal{A}}_A = Y^\setminus$$ is the entire smooth locus! Yields $${\mathcal{R}}_A = Y^{\mathrm{sing}}$$, the singular locus.
• In this case, smooth points become maximal dimension irreps, and singular points are lower dimension irreps.
• Some questions: what varieties are in the image of the correspondence? What does the algebra tell you about the singular locus?
• Some obstructions in the forward direction: computing centers of noncommutative algebras is hard, as is computing $$\operatorname{mSpec}$$ and what their singular locus is.
• See global dimension of an algebra, Gorenstein conditions.
• 3-dim Sklkyanin algebras: $$S(a,b,c) = {\mathbf{C}}\left\langle{x,y,z}\right\rangle/R$$ where $$R$$ consists of 3 quadratic relations with some genericity properties (e.g. not all parameters simultaneously vanish).
• Recover $$S(1,-1,0) = {\mathbf{C}}[x,y,z]$$, interpret as deformations of a polynomial algebra. These show up in string theory.
• Hard to compute Groebner bases, so instead associate AG data: $$E \subseteq {\mathbf{P}}^2$$ an elliptic curve, $$\sigma\in \mathop{\mathrm{Aut}}(E)$$.
• Theorem (Artin, Tate, Vandenbergh, W.): $$S(a,b,c)$$ is always prime, and is PI iff $$\sigma$$ is a finite order automorphism.
• Theorem (W, Wang, Yakimov): For $$A \coloneqq S(a,b,c)$$, $${\mathcal{A}}_A = Y^\setminus$$ and $${\mathcal{R}}_A = Y^{\mathrm{sing}}$$.
• How Poisson geometry is used: slice variety into 2d symplectic slices (symplectic core).
• 3-dim Sklyanin algebras show up in string theory, 4-dim in some kind of quantum scattering phenomenon?