2022-01-21

Tags: #web/quick-notes

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Chelsea Walton, Stanford AG Seminar (15:07)

  • Representations of algebras: ρ:AEndk(V) for some $V\in { \mathsf{Vect} }{/ {k}} ,possiblyinfinitedimensional,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(a1,,an) for all aA×n.
    • Prime algebras: A such that aAb0 when a,b0, 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, rankZ(A)A=(PIdeg(A))2.
  • Examples of how this measures noncommutativity:
    • A commutative implies degree 1.
    • Matn×n(C) has degree d.
    • A:=Cx,y/xy+yx has degree 2. Note Z(A)=C[x2,y2].
    • Uq(g), an enveloping algebra quantized at a root of unity, has degree depending on q.
    • Oq(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: kerρZ(A)mSpecZ(A).
  • There is a well-defined surjection IrrRep(A)mSpecZ(A) where [ρ]mρ:=kerρZ(A).
    • For A commutative, IrrRep(A)mSpecA.
    • For A=Matn×n(C),kerρ=0.
    • For A:=Cx,y/xy+yx, mSpecZ(A)=mSpecC[x2,y2]C2. 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 AA, open and dense in the smooth locus of Y:=mSpecZ(A). For lower dimensions: ramification locus RA
    • Very convenient when AA=Y is the entire smooth locus! Yields RA=Ysing, 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 of an algebra, Gorenstein conditions.
  • 3-dim Sklkyanin algebras: S(a,b,c)=Cx,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)=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: EP2 an elliptic curve, σAut(E).
  • Theorem (Artin, Tate, Vandenbergh, W.): S(a,b,c) is always prime, and is PI iff σ is a finite order automorphism.
  • Theorem (W, Wang, Yakimov): For A:=S(a,b,c), AA=Y and RA=Ysing.
  • 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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