Quantum exterior algebras extended by groups Lauren Grimley 1 - PowerPoint PPT Presentation

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, 2017 1 / 12

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, 2017 2 / 12

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, 2017 2 / 12

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 ⊗ K G as a vector space with multiplication ( vg )( wh ) = v ( g w ) gh . Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, 2017 2 / 12

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, 2017 3 / 12

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, 2017 3 / 12

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, 2017 3 / 12

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 jkh 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, 2017 3 / 12

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, 2017 4 / 12

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 2 + ... for linear maps µ i : A ⊗ A → A . Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, 2017 4 / 12

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 2 + ... 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, 2017 4 / 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 = − 2 i 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, 2017 5 / 12

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, 2017 6 / 12

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 ) . g ∈ G κ g ( v i , v j ) g , v 2 Let H q ,κ, 2 := T ( V ) ⋊ G [ t ] / ( v j v i − q ji v i v j − � i ). Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, 2017 6 / 12

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 ) . g ∈ G κ g ( v i , v j ) g , v 2 Let H q ,κ, 2 := T ( V ) ⋊ G [ t ] / ( v j v i − q ji v i v j − � 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, 2017 6 / 12

What happens with Λ q ( V ) ⋊ G ? Theorem (G, Uhl) ˆ The factor algebra H 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 jkh 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 ) , i < j ( g r i g r (vi) � j ) κ h ( v i , v j ) = 0 . Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, 2017 7 / 12

Example �� �� − ω 2 0 0 2 π i 3 and G = Let q 12 = − 1 , q 23 = ω = q 31 where ω = e . 0 − ω 0 0 0 1 Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, 2017 8 / 12

Example �� �� − ω 2 0 0 2 π i 3 and G = Let q 12 = − 1 , q 23 = ω = q 31 where ω = e . 0 − ω 0 0 0 1 HH 2 (Λ 3 q ⋊ G ) = span k { ( v 1 v 2 I ) ǫ ∗ 0 , 0 , 2 , ( v 1 v 2 I ) ǫ ∗ 1 , 1 , 0 , ( v 1 v 3 I ) ǫ ∗ 1 , 0 , 1 , 1 , 1 , 0 , ( v 1 v 2 g 2 ) ǫ ∗ 0 , 0 , 2 , ( g 2 ) ǫ ∗ ( v 2 v 3 I ) ǫ ∗ 0 , 1 , 1 , ( I ) ǫ ∗ 0 , 0 , 2 , ( v 1 v 2 g 3 ) ǫ ∗ 0 , 0 , 2 , ( v 2 g 3 ) ǫ ∗ 2 , 0 , 0 , ( v 1 g 3 ) ǫ ∗ 0 , 2 , 0 , ( v 1 v 2 g 4 ) ǫ ∗ 0 , 0 , 2 , ( v 1 v 2 g 5 ) ǫ ∗ 0 , 0 , 2 } . Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, 2017 8 / 12

Example �� �� − ω 2 0 0 2 π i 3 and G = Let q 12 = − 1 , q 23 = ω = q 31 where ω = e . 0 − ω 0 0 0 1 HH 2 (Λ 3 q ⋊ G ) = span k { ( v 1 v 2 I ) ǫ ∗ 0 , 0 , 2 , ( v 1 v 2 I ) ǫ ∗ 1 , 1 , 0 , ( v 1 v 3 I ) ǫ ∗ 1 , 0 , 1 , 1 , 1 , 0 , ( v 1 v 2 g 2 ) ǫ ∗ 0 , 0 , 2 , ( g 2 ) ǫ ∗ ( v 2 v 3 I ) ǫ ∗ 0 , 1 , 1 , ( I ) ǫ ∗ 0 , 0 , 2 , ( v 1 v 2 g 3 ) ǫ ∗ 0 , 0 , 2 , ( v 2 g 3 ) ǫ ∗ 2 , 0 , 0 , ( v 1 g 3 ) ǫ ∗ 0 , 2 , 0 , ( v 1 v 2 g 4 ) ǫ ∗ 0 , 0 , 2 , ( v 1 v 2 g 5 ) ǫ ∗ 0 , 0 , 2 } . All µ ∈ HH 2 (Λ 3 q ⋊ G ) have [ µ, µ ] = 0. Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, 2017 8 / 12

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 = − 2 i 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, 2017 9 / 12

Example (revisited) �� �� − ω 2 0 0 2 π i 3 and G = Let q 12 = − 1 , q 23 = ω = q 31 where ω = e . 0 − ω 0 0 0 1 Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, 2017 10 / 12