2022-01-21

Tags: #web/quick-notes

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Chelsea Walton, Stanford AG Seminar (15:07)

  • Representations of algebras: \(\rho: A\to \mathop{\mathrm{End}}_k(V)\) for some $V\in { \mathsf{Vect} }{/ {k}} \(, possibly infinite dimensional, i.e. \)V\in {\mathsf{A}{\hbox{-}}\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.
    • \(\operatorname{Mat}_{n\times n}({\mathbb{C}})\) has degree \(d\).
    • \(A\coloneqq{\mathbb{C}}\left\langle{x, y}\right\rangle/\left\langle{xy+yx}\right\rangle\) has degree 2. Note \(Z(A) = {\mathbb{C}}[x^2, y^2]\).
    • \(U_q({\mathfrak{g}})\), 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 \(A=\operatorname{Mat}_{n\times n}({\mathbb{C}}), \ker \rho = 0\).
    • For \(A\coloneqq{\mathbb{C}}\left\langle{x, y}\right\rangle/\left\langle{xy+yx}\right\rangle\), \(\operatorname{mSpec}Z(A) = \operatorname{mSpec}{\mathbb{C}}[x^2, y^2]\cong {\mathbb{C}}^2\). 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: \(S(a,b,c) = {\mathbb{C}}\left\langle{x,y,z}\right\rangle/R\) where \(R\) consists of 3 quadratic relations with some genericity properties (e.g. not all parameters simultaneously vanish).
    • Recover \(S(1,-1,0) = {\mathbb{C}}[x,y,z]\), interpret as deformations of a polynomial algebra. These show up in string theory.
    • Hard to compute Groebner bases, so instead associate AG data: \(E \subseteq {\mathbb{P}}^2\) 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?
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