2022-01-21 – Notes

Tags: #web/quick-notes

Refs: ?

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?