Tags: #web/quick-notes
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Chelsea Walton, Stanford AG Seminar (15:07)
- Representations of algebras: ρ:A→Endk(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.
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PI (polynomial identity) algebras: almost commutative. Requires a uniform monic multilinear polynomial f such that f(a1,⋯,an) for all a∈A×n.
- Prime algebras: A such that aAb≠0 when a,b≠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, rankZ(A)A=(PIdeg(A))2.
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Examples of how this measures noncommutativity:
- A commutative implies degree 1.
- Matn×n(C) has degree d.
- A:=C⟨x,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: kerρ∩Z(A)∈mSpecZ(A).
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There is a well-defined surjection IrrRep(A)→mSpecZ(A) where [ρ]↦mρ:=kerρ∩Z(A).
- For A commutative, IrrRep(A)≅mSpecA.
- For A=Matn×n(C),kerρ=0.
- For A:=C⟨x,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.
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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.
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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.
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3-dim Sklkyanin algebras: S(a,b,c)=C⟨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)=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⊆P2 an elliptic curve, σ∈Aut(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?