Invariant Theory and Hochschild Cohomology of Skew Group Algebras
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1 Invariant Theory and Hochschild Cohomology of Skew Group Algebras Briana Foster-Greenwood UNT April 23, 2011 Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
2 Outline 1 Hochschild Cohomology and Associative Deformations 2 Hochschild Cohomology of Skew Group Algebras 3 Classical invariant theory tools Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
3 Hochschild Cohomology and Associative Deformations Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
4 Deformations of algebras Start with an associative F-algebra: A Adjoin a central parameter: A[t] (this will be underlying vector space structure) Define a new multiplication: a b = ab + µ 1 (a, b)t + µ 2 (a, b)t 2 + (first define for pairs of elements in A; then extend F[t]-bilinearly) Specialize to t F to get many algebras t = 0 original algebra t = 1??? Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
5 Hochschild cohomology and associative deformations A deform A[t] with multiplication a b = ab + µ 1 (a, b)t + µ 2 (a, b)t 2 + picking any old µ k s will not usually yield an associative algebra to even get started, µ 1 must be a Hochschild 2-cocycle: aµ 1 (b, c) µ 1 (ab, c) + µ 1 (a, bc) µ 1 (a, b)c = 0 Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
6 What is Hochschild cohomology? Hochschild cohomology of an algebra A 1-maps: A A 2-maps: A A A 3-maps: A A A A. HH k (A) =equivalence classes of k-maps satisfying cocycle conditions Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
7 What is Hochschild cohomology? Hochschild cohomology of an algebra A 1-maps: A A 2-maps: A A A 3-maps: A A A A. HH k (A) =equivalence classes of k-maps satisfying cocycle conditions HH 0 (A) = center of A HH 1 (A) = derivations inner derivations (can use to construct some 2-cocycles!) HH 2 (A) first multiplication maps of deformations HH 3 (A) obstructions to obtaining associative deformations Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
8 Hochschild Cohomology of Skew Group Algebras Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
9 Skew group algebras group G acts on vectorspace V group algebra CG acts on polynomial algebra S(V ) = C[v 1,..., v n ] Skew group algebra S(V )#G Elements: C-linear combos of monomials v e 1 v eng 1 n Relations: vw wv = 0 group elements multiply as in group gv = g(v)g Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
10 HH 0 (S(V )#G) - Invariant polynomials The center of a skew group algebra is the set of G-invariant polynomials: S(V ) G = {f S(V ) : g(f ) = f for all g G} Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
11 HH 0 (S(V )#G) - Invariant polynomials The center of a skew group algebra is the set of G-invariant polynomials: S(V ) G = {f S(V ) : g(f ) = f for all g G} Example - Sym 3 acts on C[x, y, z] by permuting the variables elementary symmetric polynomials f 1 = x + y + z f 2 = yz + xz + xy f 3 = xyz Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
12 HH 0 (S(V )#G) - Invariant polynomials The center of a skew group algebra is the set of G-invariant polynomials: S(V ) G = {f S(V ) : g(f ) = f for all g G} Example - Sym 3 acts on C[x, y, z] by permuting the variables elementary symmetric polynomials f 1 = x + y + z f 2 = yz + xz + xy f 3 = xyz use these to build more: e.g. 3f 5 1 7f 2f 3 is also invariant Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
13 HH 0 (S(V )#G) - Invariant polynomials The center of a skew group algebra is the set of G-invariant polynomials: S(V ) G = {f S(V ) : g(f ) = f for all g G} Example - Sym 3 acts on C[x, y, z] by permuting the variables elementary symmetric polynomials f 1 = x + y + z f 2 = yz + xz + xy f 3 = xyz use these to build more: e.g. 3f 5 1 7f 2f 3 is also invariant How many invariant polynomials are there? Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
14 HH 1 (S(V )#G) - Invariant derivations HH 1 (S(V )#G) = {invariant derivations on S(V )} Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
15 HH 1 (S(V )#G) - Invariant derivations HH 1 (S(V )#G) = {invariant derivations on S(V )} Example - Sym 3 also permutes the dual vectors x, y, z a few invariant derivations θ 0 = x + y + z θ 1 = x x + y y + z z θ 2 = yz x + xz y + xy z Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
16 HH 1 (S(V )#G) - Invariant derivations HH 1 (S(V )#G) = {invariant derivations on S(V )} Example - Sym 3 also permutes the dual vectors x, y, z a few invariant derivations θ 0 = x + y + z θ 1 = x x + y y + z z θ 2 = yz x + xz y + xy z build more by multiplying by invariant polynomials: f 3 θ 2 = xy 2 z 2 x + x 2 yz 2 y + x 2 y 2 z z Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
17 HH 1 (S(V )#G) - Invariant derivations HH 1 (S(V )#G) = {invariant derivations on S(V )} Example - Sym 3 also permutes the dual vectors x, y, z a few invariant derivations θ 0 = x + y + z θ 1 = x x + y y + z z θ 2 = yz x + xz y + xy z build more by multiplying by invariant polynomials: f 3 θ 2 = xy 2 z 2 x + x 2 yz 2 y + x 2 y 2 z z How many invariant derivations are there? Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
18 Big invariant theory description of HH(S(V )#G) Theorem (Ginzburg-Kaledin, Farinati) HH k (S(V )#G) = (S(V ) k V ) G more! Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
19 Big invariant theory description of HH(S(V )#G) Theorem (Ginzburg-Kaledin, Farinati) HH k (S(V )#G) = (S(V ) k V ) G more! The remaining summands are spaces of semi-invariants under centralizer subgroups: ( ) k c Z(g) S(V g ) (V g ) C χ (one summand per conjugacy class) Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
20 Classical invariant theory tools Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
21 Graded vector spaces and Poincaré series Start with a graded vector space: S = d 0 S d (with S d S e S d+e ) The Poincaré series for S is a power series with the coefficient of t d recording the dimension of S d : P t (S) = d 0 (dim C S d )t d Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
22 Poincaré series for C[f 1, f 2, f 3 ] C[x, y, z] Sym 3 f 1 = x + y + z, f 2 = yz + xz + xy, f 3 = xyz Build more polynomials Note: f 1, f 2, f 3 are algebraically independent, which forces all these new polynomials to be linearly independent. degree 0 1 polynomials 1 f 1 2 f 2 1, f 2 3 f 3 1, f 1f 2, f 3.. Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
23 Poincaré series for C[f 1, f 2, f 3 ] C[x, y, z] Sym 3 f 1 = x + y + z, f 2 = yz + xz + xy, f 3 = xyz Build more polynomials Note: f 1, f 2, f 3 are algebraically independent, which forces all these new polynomials to be linearly independent. degree 0 1 polynomials 1 f 1 2 f 2 1, f 2 3 f 3 1, f 1f 2, f 3.. P t (C[f 1, f 2, f 3 ]) = 1 + t + 2t 2 + 3t 3 + Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
24 Poincaré series for C[f 1, f 2, f 3 ] C[x, y, z] Sym 3 f 1 = x + y + z, f 2 = yz + xz + xy, f 3 = xyz Build more polynomials Note: f 1, f 2, f 3 are algebraically independent, which forces all these new polynomials to be linearly independent. degree 0 1 polynomials 1 f 1 2 f 2 1, f 2 3 f 3 1, f 1f 2, f 3.. P t (C[f 1, f 2, f 3 ]) = 1 + t + 2t 2 + 3t 3 + = (1 + t + t 2 + )(1 + t 2 + t 4 + )(1 + t 3 + t 6 + ) Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
25 Poincaré series for C[f 1, f 2, f 3 ] C[x, y, z] Sym 3 f 1 = x + y + z, f 2 = yz + xz + xy, f 3 = xyz Build more polynomials Note: f 1, f 2, f 3 are algebraically independent, which forces all these new polynomials to be linearly independent. degree 0 1 polynomials 1 f 1 2 f 2 1, f 2 3 f 3 1, f 1f 2, f 3.. P t (C[f 1, f 2, f 3 ]) = 1 + t + 2t 2 + 3t 3 + = (1 + t + t 2 + )(1 + t 2 + t 4 + )(1 + t 3 + t 6 + ) 1 = (1 t)(1 t 2 )(1 t 3 ) Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
26 Molien s theorem (1897) Poincaré series for ring of invariant polynomials Poincaré series for S G P t (S G ) = 1 1 G det(1 gt) g G det(1 gt) is the characteristic polynomial for the matrix of g. Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
27 Poincaré series for C[x, y, z] Sym 3 elements eigenvalues characteristic polynomial 1 1, 1, 1 (1 t) 3 (12), (13), (23) 1, 1, 1 (1 t) 2 (1 + t) (123), (321) 1, ω, ω 2 (1 t)(1 ωt)(1 ω 2 t) Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
28 Poincaré series for C[x, y, z] Sym 3 elements eigenvalues characteristic polynomial 1 1, 1, 1 (1 t) 3 (12), (13), (23) 1, 1, 1 (1 t) 2 (1 + t) (123), (321) 1, ω, ω 2 (1 t)(1 ωt)(1 ω 2 t) P t (S G ) = 1 [ ] 1 6 (1 t) (1 t) 2 (1 + t) + 2 (1 t 3 ) = 1 (1 t)(1 t 2 )(1 t 3 ) Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
29 Poincaré series for C[x, y, z] Sym 3 elements eigenvalues characteristic polynomial 1 1, 1, 1 (1 t) 3 (12), (13), (23) 1, 1, 1 (1 t) 2 (1 + t) (123), (321) 1, ω, ω 2 (1 t)(1 ωt)(1 ω 2 t) P t (S G ) = 1 [ ] 1 6 (1 t) (1 t) 2 (1 + t) + 2 (1 t 3 ) = 1 (1 t)(1 t 2 )(1 t 3 ) By comparing Poincaré series: C[f 1, f 2, f 3 ] = C[x, y, z] Sym 3 Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
30 Big invariant theory description of HH(S(V )#G) Theorem (Ginzburg-Kaledin, Farinati) HH k (S(V )#G) = (S(V ) k V ) G more! The remaining summands are spaces of semi-invariants under centralizer subgroups: ( ) k c Z(g) S(V g ) (V g ) C χ (one summand per conjugacy class) Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
31 Generalizations of Molien s theorem Tensoring with an exterior algebra Poincaré series for (S(V ) V ) G P X,Y = 1 det(1 + g Y ) G det(1 gx ) g G The coefficient of X i Y j tells you the dimension of (S i (V ) j V ) G. Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
32 Generalizations of Molien s theorem Tensoring with an exterior algebra Poincaré series for (S(V ) V ) G P X,Y = 1 det(1 + g Y ) G det(1 gx ) g G The coefficient of X i Y j tells you the dimension of (S i (V ) j V ) G. More generally, the Poincaré series for (S(V ) V C χ ) G is P X,Y = 1 χ (g) det(1 + g Y ) G det(1 gx ) g G Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
33 Predictive power of Poincaré Module structure for identity component of HH(S(V )#Alt 4) Let G = Alt 4 act irreducibly on V = C 3. Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
34 Predictive power of Poincaré Module structure for identity component of HH(S(V )#Alt 4) Let G = Alt 4 act irreducibly on V = C 3. Poincaré series for identity component of HH(S(V )#G): 1 + x 6 (1 x 2 )(1 x 3 )(1 x 4 ) + x + x 2 + 2x 3 + x 4 + x 5 (1 x 2 )(1 x 3 )(1 x 4 ) y + x + x 2 + 2x 3 + x 4 + x 5 (1 x 2 )(1 x 3 )(1 x 4 ) y x 6 + (1 x 2 )(1 x 3 )(1 x 4 ) y 3 invariant polynomials invariant derivations invariant 2-forms invariant 3-forms Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
35 Predictive power of Poincaré Module structure for identity component of HH(S(V )#Alt 4) Let G = Alt 4 act irreducibly on V = C 3. Poincaré series for identity component of HH(S(V )#G): 1 + x 6 (1 x 2 )(1 x 3 )(1 x 4 ) + x + x 2 + 2x 3 + x 4 + x 5 (1 x 2 )(1 x 3 )(1 x 4 ) y + x + x 2 + 2x 3 + x 4 + x 5 (1 x 2 )(1 x 3 )(1 x 4 ) y x 6 + (1 x 2 )(1 x 3 )(1 x 4 ) y 3 invariant polynomials invariant derivations invariant 2-forms invariant 3-forms Read off finitely-generated free module structure: Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
36 Predictive power of Poincaré Module structure for identity component of HH(S(V )#Alt 4) Let G = Alt 4 act irreducibly on V = C 3. Poincaré series for identity component of HH(S(V )#G): 1 + x 6 (1 x 2 )(1 x 3 )(1 x 4 ) + x + x 2 + 2x 3 + x 4 + x 5 (1 x 2 )(1 x 3 )(1 x 4 ) y + x + x 2 + 2x 3 + x 4 + x 5 (1 x 2 )(1 x 3 )(1 x 4 ) y x 6 + (1 x 2 )(1 x 3 )(1 x 4 ) y 3 invariant polynomials invariant derivations invariant 2-forms invariant 3-forms Read off finitely-generated free module structure: denominators free module over R = C[f 2, f 3, f 4 ] where f 2, f 3, f 4 are algebraically independent invariant polynomials of degrees 2, 3, 4 Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
37 Predictive power of Poincaré Module structure for identity component of HH(S(V )#Alt 4) Let G = Alt 4 act irreducibly on V = C 3. Poincaré series for identity component of HH(S(V )#G): 1 + x 6 (1 x 2 )(1 x 3 )(1 x 4 ) + x + x 2 + 2x 3 + x 4 + x 5 (1 x 2 )(1 x 3 )(1 x 4 ) y + x + x 2 + 2x 3 + x 4 + x 5 (1 x 2 )(1 x 3 )(1 x 4 ) y x 6 + (1 x 2 )(1 x 3 )(1 x 4 ) y 3 invariant polynomials invariant derivations invariant 2-forms invariant 3-forms Read off finitely-generated free module structure: denominators free module over R = C[f 2, f 3, f 4 ] where f 2, f 3, f 4 are algebraically independent invariant polynomials of degrees 2, 3, 4 numerators how many generators? polynomial degree? Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
38 Predictive power of Poincaré Module structure for identity component of HH(S(V )#Alt 4) Let G = Alt 4 act irreducibly on V = C 3. Poincaré series for identity component of HH(S(V )#G): 1 + x 6 (1 x 2 )(1 x 3 )(1 x 4 ) + x + x 2 + 2x 3 + x 4 + x 5 (1 x 2 )(1 x 3 )(1 x 4 ) y + x + x 2 + 2x 3 + x 4 + x 5 (1 x 2 )(1 x 3 )(1 x 4 ) y x 6 + (1 x 2 )(1 x 3 )(1 x 4 ) y 3 invariant polynomials invariant derivations invariant 2-forms invariant 3-forms Invariant derivations: the invariant derivations are the free R-span of six derivations of polynomial degrees 1, 2, 3, 3, 4, 5 it is now a finite linear algebra problem to find all invariant derivations Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
39 Thanks! Briana Foster-Greenwood (UNT) Invariant Theory and HH(S(V )#G) April 23, / 21
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