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date: 2022-01-21
tags: [ web/quick-notes ]
---

# 2022-01-21

Tags:
#web/quick-notes

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

- Representations of algebras: $\rho: A\to \Endo_k(V)$ for some $V\in \Vect\slice k$, possibly infinite dimensional, i.e. $V\in \mods{A}\slice k$.
- 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 $\vector a \in A\cartpower{n}$.
  - 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, $\rank_{Z(A)} A = \qty{\mathrm{PIdeg}(A)}^2$.
- Examples of how this measures noncommutativity:
  - $A$ commutative implies degree 1.
  - $\Mat_{n\times n}(\CC)$ has degree $d$.
  - $A\da\CC\gens{x, y}/\gens{xy+yx}$ has degree 2.
  Note $Z(A) = \CC[x^2, y^2]$.
  - $U_q(\lieg)$, an enveloping algebra quantized at a root of unity, has degree depending on $q$.
  - $\OO_q(G)$ quantized function algebras, also has degree depending on $q$.

- In an [Azumaya algebra](Azumaya%20algebra.md) 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: $\ker \rho \intersect Z(A) \in \mspec Z(A)$.
- There is a well-defined surjection $\Irr\Rep(A) \to \mspec Z(A)$ where $[\rho] \mapsto \mfm_\rho \da \ker \rho \intersect Z(A)$.
  - For $A$ commutative, $\Irr\Rep(A) \cong \mspec A$.
  - For $A=\Mat_{n\times n}(\CC), \ker \rho = 0$.
  - For $A\da \CC\gens{x, y}/\gens{xy+yx}$, $\mspec Z(A) = \mspec \CC[x^2, y^2]\cong \CC^2$.
    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 $\mca_A$, open and dense in the smooth locus of $Y\da \mspec Z(A)$.
  For lower dimensions: ramification locus $\mcr_A$
  - Very convenient when $\mca_A = Y^\sm$ is the entire smooth locus!
    Yields $\mcr_A = Y^\sing$, 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 $\mspec$ and what their singular locus is.
- See [global dimension](global%20dimension) of an algebra, [Gorenstein](Gorenstein.md) conditions.
- 3-dim Sklkyanin algebras: $S(a,b,c) = \CC\gens{x,y,z}/R$ where $R$ consists of 3 quadratic relations with some genericity properties (e.g. not all parameters simultaneously vanish).
  - Recover $S(1,-1,0) = \CC[x,y,z]$, interpret as deformations of a polynomial algebra. These show up in string theory.
  - Hard to compute Groebner bases, so instead associate AG data: $E \subseteq \PP^2$ an elliptic curve, $\sigma\in \Aut(E)$.

- 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 $A \da S(a,b,c)$, $\mca_A = Y^\sm$ and $\mcr_A = Y^\sing$.
- 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?