Quantum exterior algebras extended by groups

Transcription

1 Quantum exterior algebras extended by groups Lauren Grimley 1 Christine Uhl 2 1 Spring Hill College lgrimley@shc.edu 2 St. Bonaventure University cuhl@sbu.edu June 4, 2017 Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

2 Group extensions of quantum exterior algebras Let K be a field and V be a K-vector space with basis {v 1, v 2,..., v n }. Let q = {q ij q ij K {0} and q ij = q 1 ji } be the set of quantum scalars. Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

3 Group extensions of quantum exterior algebras Let K be a field and V be a K-vector space with basis {v 1, v 2,..., v n }. Let q = {q ij q ij K {0} and q ij = q 1 ji } be the set of quantum scalars. Let G GL(V ). Restrict to K with char(k) G and char(k) 2. Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

4 Group extensions of quantum exterior algebras Let K be a field and V be a K-vector space with basis {v 1, v 2,..., v n }. Let q = {q ij q ij K {0} and q ij = q 1 ji } be the set of quantum scalars. Let G GL(V ). Restrict to K with char(k) G and char(k) 2. Definition The group extension, A G, is A KG as a vector space with multiplication (vg)(wh) = v( g w)gh. Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

5 Quantum Drinfeld Hecke algebras Let H q,κ := T (V ) G/(v j v i + q ji v i v j g G κ g (v i, v j )g). Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

6 Quantum Drinfeld Hecke algebras Let H q,κ := T (V ) G/(v j v i + q ji v i v j g G κ g (v i, v j )g). Define the H q,κ for which the associated graded algebra is isomorphic to S q (V ) G to be a quantum Drinfeld Hecke algebra over K. Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

7 Quantum Drinfeld Hecke algebras Let H q,κ := T (V ) G/(v j v i + q ji v i v j g G κ g (v i, v j )g). Define the H q,κ for which the associated graded algebra is isomorphic to S q (V ) G to be a quantum Drinfeld Hecke algebra over K. Theorem (Levandovskyy, Shepler) The factor algebra H q,κ is a quantum Drinfeld Hecke algebra if and only if for all g, h G and 1 i < j < k n Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

8 Quantum Drinfeld Hecke algebras Let H q,κ := T (V ) G/(v j v i + q ji v i v j g G κ g (v i, v j )g). Define the H q,κ for which the associated graded algebra is isomorphic to S q (V ) G to be a quantum Drinfeld Hecke algebra over K. Theorem (Levandovskyy, Shepler) The factor algebra H q,κ is a quantum Drinfeld Hecke algebra if and only if for all g, h G and 1 i < j < k n (i) G acts on S q (V ) by automorphisms and q ij = q 1 ji, q ii = 1, (ii) κ(v j, v i ) = q 1 ij κ(v i, v j ), (iii) 0 = (q ik q jk h v k v k )κ h (v i, v j ) + (q jk v j q ij h v j )κ h (v i, v k ) + ( h v i q ij q ik v i )κ h (v j, v k ), (iv) κ h 1 gh(v r, v s ) = i<j det rsij(h)κ g (v i, v j ), Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

9 Deformations Let H q,κ := T (V ) G/(v j v i + q ji v i v j g G κ g (v i, v j )g). Note: Quantum Drinfeld Hecke algebras are deformations of S q (V ) G. Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

10 Deformations Let H q,κ := T (V ) G/(v j v i + q ji v i v j g G κ g (v i, v j )g). Note: Quantum Drinfeld Hecke algebras are deformations of S q (V ) G. Definition Let t be an indeterminant. A deformation of A over K[t] is an associative algebra A[t] with multiplication given by a b = ab + µ 1 (a b)t + µ 2 (a b)t for linear maps µ i : A A A. Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

11 Deformations Let H q,κ := T (V ) G/(v j v i + q ji v i v j g G κ g (v i, v j )g). Note: Quantum Drinfeld Hecke algebras are deformations of S q (V ) G. Definition Let t be an indeterminant. A deformation of A over K[t] is an associative algebra A[t] with multiplication given by a b = ab + µ 1 (a b)t + µ 2 (a b)t for linear maps µ i : A A A. In order for the deformation to be associative, µ 1 must be a Hochschild 2-cocycle with [µ 1, µ 1 ] a coboundary. Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

12 Translating to HH Theorem (Naidu, Witherspoon) The quantum Drinfeld Hecke algebras over C[t] are precisely the deformations of S q (V ) G over C[t] with deg µ i = 2i for all i 1. Theorem (Naidu, Witherspoon) Assume that the action of G on V extends to an action on Λ q (V ) and S q (V ) by algebra automorphisms. Then each constant Hochschild 2-cocycle on S q (V ) G gives rise to a quantum Drinfeld Hecke algebra. Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

13 What happens with Λ q (V ) G? Λ q (V ) := T (V )/(v i v j q ji v j v i, v 2 i ). Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

14 What happens with Λ q (V ) G? Λ q (V ) := T (V )/(v i v j q ji v j v i, v 2 i ). Let H q,κ,2 := T (V ) G[t]/(v j v i q ji v i v j g G κ g (v i, v j )g, v 2 i ). Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

15 What happens with Λ q (V ) G? Λ q (V ) := T (V )/(v i v j q ji v j v i, v 2 i ). Let H q,κ,2 := T (V ) G[t]/(v j v i q ji v i v j g G κ g (v i, v j )g, v 2 i ). Define the H q,κ,2 for which the associated graded algebra is isomorphic to Λ q (V ) G to be a truncated quantum Drinfeld Hecke algebra over K. Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

16 What happens with Λ q (V ) G? Theorem (G, Uhl) The factor algebra Hˆ q,κ,2 is a truncated quantum Drinfeld Hecke algebra if and only if for all g, h G and 1 i < j < k n (i) G acts on Λ q (V )by automorphisms and q ij = q 1 ji, (ii) κ(v j, v i ) = q 1 ij κ(v i, v j ) and κ(v i, v i ) = 0, (iii) 0 = (q ik q jk h v k v k )κ h (v i, v j ) + (q jk v j q ij h v j )κ h (v i, v k ) + ( h v i q ij q ik v i )κ h (v j, v k ), (iv) κ h 1 gh(v r, v s ) = i<j det rsij(h)κ g (v i, v j ), (v) 0 = q ij v i κ h (v i, v j )+ h v i κ h (v i, v j ) and 0 = q ij h v j κ h (v i, v j )+v j κ h (v i, v j ), (vi) i<j (g r i g r j )κ h(v i, v j ) = 0. Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

17 Example ( ) Let q 12 = 1, q 23 = ω = q 31 where ω = e 2πi ω 3 and G = ω Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

18 Example ( ) Let q 12 = 1, q 23 = ω = q 31 where ω = e 2πi ω 3 and G = ω HH 2 (Λ 3 q G) = span k {(v 1 v 2 I )ɛ 0,0,2, (v 1v 2 I )ɛ 1,1,0, (v 1v 3 I )ɛ 1,0,1, (v 2 v 3 I )ɛ 0,1,1, (I )ɛ 1,1,0, (v 1v 2 g 2 )ɛ 0,0,2, (g 2 )ɛ 0,0,2, (v 1 v 2 g 3 )ɛ 0,0,2, (v 2g 3 )ɛ 2,0,0, (v 1g 3 )ɛ 0,2,0, (v 1v 2 g 4 )ɛ 0,0,2, (v 1 v 2 g 5 )ɛ 0,0,2 }. Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

19 Example ( ) Let q 12 = 1, q 23 = ω = q 31 where ω = e 2πi ω 3 and G = ω HH 2 (Λ 3 q G) = span k {(v 1 v 2 I )ɛ 0,0,2, (v 1v 2 I )ɛ 1,1,0, (v 1v 3 I )ɛ 1,0,1, (v 2 v 3 I )ɛ 0,1,1, (I )ɛ 1,1,0, (v 1v 2 g 2 )ɛ 0,0,2, (g 2 )ɛ 0,0,2, (v 1 v 2 g 3 )ɛ 0,0,2, (v 2g 3 )ɛ 2,0,0, (v 1g 3 )ɛ 0,2,0, (v 1v 2 g 4 )ɛ 0,0,2, (v 1 v 2 g 5 )ɛ 0,0,2 }. All µ HH 2 (Λ 3 q G) have [µ, µ] = 0. Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

20 From Hochschild cohomology Theorem (G, Uhl) The truncated quantum Drinfeld Hecke algebras over K[t] are the deformations of Λ q (V ) G with polynomial deg µ i = 2i for all i > 0 and µ i (v j, v j ) = 0. Theorem (G, Uhl) If the G action on V extends to an action on Λ q (V ), then each constant Hochschild 2-cocyle of Λ q (V ) G that sends v i v i 0 for all i {1, 2,..., n}produces a truncated quantum Drinfeld Hecke algebra. Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

21 Example (revisited) Let q 12 = 1, q 23 = ω = q 31 where ω = e 2πi 3 and G = ( ) ω ω Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

22 Example (revisited) Let q 12 = 1, q 23 = ω = q 31 where ω = e 2πi 3 and G = ( HH 2 (Λ 3 q G) = span k {(v 1 v 2 I )ɛ 0,0,2, (v 1v 2 I )ɛ 1,1,0, (v 1v 3 I )ɛ 1,0,1, (v 2 v 3 I )ɛ 0,1,1, (I )ɛ 1,1,0, (v 1v 2 g 2 )ɛ 0,0,2, (g 2 )ɛ 0,0,2, (v 1 v 2 g 3 )ɛ 0,0,2, (v 2g 3 )ɛ 2,0,0, (v 1g 3 )ɛ 0,2,0, (v 1v 2 g 4 )ɛ 0,0,2, (v 1 v 2 g 5 )ɛ 0,0,2 }. ) ω ω Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

23 Example (revisited) Let q 12 = 1, q 23 = ω = q 31 where ω = e 2πi 3 and G = ( HH 2 (Λ 3 q G) = span k {(v 1 v 2 I )ɛ 0,0,2, (v 1v 2 I )ɛ 1,1,0, (v 1v 3 I )ɛ 1,0,1, (v 2 v 3 I )ɛ 0,1,1, (I )ɛ 1,1,0, (v 1v 2 g 2 )ɛ 0,0,2, (g 2 )ɛ 0,0,2, (v 1 v 2 g 3 )ɛ 0,0,2, (v 2g 3 )ɛ 2,0,0, (v 1g 3 )ɛ 0,2,0, (v 1v 2 g 4 )ɛ 0,0,2, (v 1 v 2 g 5 )ɛ 0,0,2 }. ) ω ω The only PBW deformation H q,κ,2 is a quotient of T (V ) G[t] given by the relations for i {1, 2, 3} and m k. v 2 v 1 = v 1 v 2 + mi, v 3 v 2 = ωv 2 v 3, v 3 v 1 = ω 2 v 1 v 3, and v i v i = 0 Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

24 Example Let G = {( ) ( )} 0 1 0, = {g, I } and q 12 = 1, q 13 = q 23 = HH 2 (Λ 3 q G) = span K {(I )ɛ 1,1,0, (g)ɛ 1,0,1, (g)ɛ 0,1,1 }. The PBW deformation H q,κ,2 is a quotient of T (V ) G[t] given by the relations where m 1, m 2, m 3 K. v 2 v 1 = v 1 v 2 + m 1 I, v 3 v 1 = v 1 v 3 + m 2 g, v 3 v 2 = v 2 v 3 + m 3 g, v 2 i = 0 for i = 1, 2, 3 Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

25 Thank you! Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

26 Parameter space Definition The parameter κ is called admissible if it defines a truncated quantum Drinfeld Hecke algebra. Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

27 Parameter space Definition The parameter κ is called admissible if it defines a truncated quantum Drinfeld Hecke algebra. The subspace of admissible parameters, P G = {κ Hom K (V V, KG) κ is admissible}, we call the parameter space. Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12

28 Parameter space Definition The parameter κ is called admissible if it defines a truncated quantum Drinfeld Hecke algebra. The subspace of admissible parameters, P G = {κ Hom K (V V, KG) κ is admissible}, we call the parameter space. Proposition (G, Uhl) In a truncated quantum Drinfeld Hecke algebra, only group elements that act diagonally on the vector space can support the parameter space. Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, / 12