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

Quantum exterior algebras extended by groups

Lauren Grimley 1 Christine Uhl 2

1Spring Hill College

lgrimley@shc.edu

• 2St. Bonaventure University

cuhl@sbu.edu

June 4, 2017

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Group extensions of quantum exterior algebras

Let K be a field and V be a K-vector space with basis {v1, v2, ..., vn}. Let q = {qij|qij ∈ K − {0} and qij = q−1

ji } be the set of quantum scalars.

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Group extensions of quantum exterior algebras

Let K be a field and V be a K-vector space with basis {v1, v2, ..., vn}. Let q = {qij|qij ∈ K − {0} and qij = q−1

ji } be the set of quantum scalars.

Let G ⊂ GL(V ). Restrict to K with char(K) ∤ |G| and char(K) = 2.

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Group extensions of quantum exterior algebras

Let K be a field and V be a K-vector space with basis {v1, v2, ..., vn}. Let q = {qij|qij ∈ K − {0} and qij = 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(gw)gh.

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Quantum Drinfeld Hecke algebras

Let Hq,κ := T(V ) ⋊ G/(vjvi + qjivivj −

g∈G κg(vi, vj)g).

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Quantum Drinfeld Hecke algebras

Let Hq,κ := T(V ) ⋊ G/(vjvi + qjivivj −

g∈G κg(vi, vj)g).

Define the Hq,κ for which the associated graded algebra is isomorphic to Sq(V ) ⋊ G to be a quantum Drinfeld Hecke algebra over K.

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Quantum Drinfeld Hecke algebras

Let Hq,κ := T(V ) ⋊ G/(vjvi + qjivivj −

g∈G κg(vi, vj)g).

Define the Hq,κ for which the associated graded algebra is isomorphic to Sq(V ) ⋊ G to be a quantum Drinfeld Hecke algebra over K.

Theorem (Levandovskyy, Shepler)

The factor algebra Hq,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 Hq,κ := T(V ) ⋊ G/(vjvi + qjivivj −

g∈G κg(vi, vj)g).

Define the Hq,κ for which the associated graded algebra is isomorphic to Sq(V ) ⋊ G to be a quantum Drinfeld Hecke algebra over K.

Theorem (Levandovskyy, Shepler)

The factor algebra Hq,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 Sq(V ) by automorphisms and qij = q−1

ji , qii = 1,

(ii) κ(vj, vi) = −q−1

ij κ(vi, vj),

(iii) 0 = (qikqjkhvk − vk)κh(vi, vj) + (qjkvj − qij hvj)κh(vi, vk) + (hvi − qijqikvi)κh(vj, vk), (iv) κh−1gh(vr, vs) =

i<j detrsij(h)κg(vi, vj),

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Deformations

Let Hq,κ := T(V ) ⋊ G/(vjvi + qjivivj −

g∈G κg(vi, vj)g).

Note: Quantum Drinfeld Hecke algebras are deformations of Sq(V ) ⋊ G.

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Deformations

Let Hq,κ := T(V ) ⋊ G/(vjvi + qjivivj −

g∈G κg(vi, vj)g).

Note: Quantum Drinfeld Hecke algebras are deformations of Sq(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 + µ1(a ⊗ b)t + µ2(a ⊗ b)t2 + ... for linear maps µi : A ⊗ A → A.

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Deformations

Let Hq,κ := T(V ) ⋊ G/(vjvi + qjivivj −

g∈G κg(vi, vj)g).

Note: Quantum Drinfeld Hecke algebras are deformations of Sq(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 + µ1(a ⊗ b)t + µ2(a ⊗ b)t2 + ... for linear maps µi : A ⊗ A → A. In order for the deformation to be associative, µ1 must be a Hochschild 2-cocycle with [µ1, µ1] a coboundary.

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Translating to HH

Theorem (Naidu, Witherspoon)

The quantum Drinfeld Hecke algebras over C[t] are precisely the deformations of Sq(V ) ⋊ G over C[t] with deg µi = −2i for all i ≥ 1.

Theorem (Naidu, Witherspoon)

Assume that the action of G on V extends to an action on Λq(V ) and Sq(V ) by algebra automorphisms. Then each constant Hochschild 2-cocycle on Sq(V ) ⋊ G gives rise to a quantum Drinfeld Hecke algebra.

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What happens with Λq(V ) ⋊ G?

Λq(V ) := T(V )/(vivj − qjivjvi, v2

i ).

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What happens with Λq(V ) ⋊ G?

Λq(V ) := T(V )/(vivj − qjivjvi, v2

i ).

Let Hq,κ,2 := T(V ) ⋊ G[t]/(vjvi − qjivivj −

g∈G κg(vi, vj)g, v2 i ).

Grimley (SHC) and Uhl (SBU) Truncated QDHA June 4, 2017 6 / 12

What happens with Λq(V ) ⋊ G?

Λq(V ) := T(V )/(vivj − qjivjvi, v2

i ).

Let Hq,κ,2 := T(V ) ⋊ G[t]/(vjvi − qjivivj −

g∈G κg(vi, vj)g, v2 i ).

Define the Hq,κ,2 for which the associated graded algebra is isomorphic to Λq(V ) ⋊ G to be a truncated quantum Drinfeld Hecke algebra over K.

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What happens with Λq(V ) ⋊ G?

Theorem (G, Uhl)

The factor algebra ˆ Hq,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 Λq(V )by automorphisms and qij = q−1

ji ,

(ii) κ(vj, vi) = −q−1

ij κ(vi, vj) and κ(vi, vi) = 0,

(iii) 0 = (qikqjkhvk − vk)κh(vi, vj) + (qjkvj − qij hvj)κh(vi, vk) + (hvi − qijqikvi)κh(vj, vk), (iv) κh−1gh(vr, vs) =

i<j detrsij(h)κg(vi, vj),

(v) 0 = qijviκh(vi, vj)+hviκh(vi, vj) and 0 = qij hvjκh(vi, vj)+vjκh(vi, vj), (vi)

i<j(gr i gr j )κh(vi, vj) = 0.

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Example

Let q12 = −1, q23 = ω = q31 where ω = e

2πi 3 and G =

• −ω2

−ω 0 1

• .

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Example

Let q12 = −1, q23 = ω = q31 where ω = e

2πi 3 and G =

• −ω2

−ω 0 1

• .

HH2(Λ3

q ⋊ G) = spank{(v1v2I)ǫ∗ 0,0,2, (v1v2I)ǫ∗ 1,1,0, (v1v3I)ǫ∗ 1,0,1,

(v2v3I)ǫ∗

0,1,1, (I)ǫ∗ 1,1,0, (v1v2g2)ǫ∗ 0,0,2, (g2)ǫ∗ 0,0,2,

(v1v2g3)ǫ∗

0,0,2, (v2g3)ǫ∗ 2,0,0, (v1g3)ǫ∗ 0,2,0, (v1v2g4)ǫ∗ 0,0,2,

(v1v2g5)ǫ∗

0,0,2}.

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Example

Let q12 = −1, q23 = ω = q31 where ω = e

2πi 3 and G =

• −ω2

−ω 0 1

• .

HH2(Λ3

q ⋊ G) = spank{(v1v2I)ǫ∗ 0,0,2, (v1v2I)ǫ∗ 1,1,0, (v1v3I)ǫ∗ 1,0,1,

(v2v3I)ǫ∗

0,1,1, (I)ǫ∗ 1,1,0, (v1v2g2)ǫ∗ 0,0,2, (g2)ǫ∗ 0,0,2,

(v1v2g3)ǫ∗

0,0,2, (v2g3)ǫ∗ 2,0,0, (v1g3)ǫ∗ 0,2,0, (v1v2g4)ǫ∗ 0,0,2,

(v1v2g5)ǫ∗

0,0,2}.

All µ ∈ HH2(Λ3

q ⋊ G) have [µ, µ] = 0.

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From Hochschild cohomology

Theorem (G, Uhl)

The truncated quantum Drinfeld Hecke algebras over K[t] are the deformations of Λq(V )⋊G with polynomial deg µi = −2i for all i > 0 and µi(vj, vj) = 0.

Theorem (G, Uhl)

If the G action on V extends to an action on Λq(V ), then each constant Hochschild 2-cocyle of Λq(V ) ⋊ G that sends vi ⊗ vi → 0 for all i ∈ {1, 2, ..., n}produces a truncated quantum Drinfeld Hecke algebra.

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Example (revisited)

Let q12 = −1, q23 = ω = q31 where ω = e

2πi 3 and G =

• −ω2

−ω 0 1

• .

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Example (revisited)

Let q12 = −1, q23 = ω = q31 where ω = e

2πi 3 and G =

• −ω2

−ω 0 1

• .

HH2(Λ3

q ⋊ G) = spank{(v1v2I)ǫ∗ 0,0,2, (v1v2I)ǫ∗ 1,1,0, (v1v3I)ǫ∗ 1,0,1,

(v2v3I)ǫ∗

0,1,1, (I)ǫ∗ 1,1,0, (v1v2g2)ǫ∗ 0,0,2, (g2)ǫ∗ 0,0,2,

(v1v2g3)ǫ∗

0,0,2, (v2g3)ǫ∗ 2,0,0, (v1g3)ǫ∗ 0,2,0, (v1v2g4)ǫ∗ 0,0,2,

(v1v2g5)ǫ∗

0,0,2}.

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Example (revisited)

Let q12 = −1, q23 = ω = q31 where ω = e

2πi 3 and G =

• −ω2

−ω 0 1

• .

HH2(Λ3

q ⋊ G) = spank{(v1v2I)ǫ∗ 0,0,2, (v1v2I)ǫ∗ 1,1,0, (v1v3I)ǫ∗ 1,0,1,

(v2v3I)ǫ∗

0,1,1, (I)ǫ∗ 1,1,0, (v1v2g2)ǫ∗ 0,0,2, (g2)ǫ∗ 0,0,2,

(v1v2g3)ǫ∗

0,0,2, (v2g3)ǫ∗ 2,0,0, (v1g3)ǫ∗ 0,2,0, (v1v2g4)ǫ∗ 0,0,2,

(v1v2g5)ǫ∗

0,0,2}.

The only PBW deformation Hq,κ,2 is a quotient of T(V ) ⋊ G[t] given by the relations v2v1 = −v1v2 + mI, v3v2 = ωv2v3, v3v1 = ω2v1v3, and vivi = 0 for i ∈ {1, 2, 3} and m ∈ k.

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Example

Let G = −1 0

0 −1 0 0 −1

• ,

1 0 0

0 1 0 0 0 1

• = {g, I} and q12 = −1, q13 = q23 = 1.

HH2(Λ3

q ⋊ G) = spanK{(I)ǫ∗ 1,1,0, (g)ǫ∗ 1,0,1, (g)ǫ∗ 0,1,1}.

The PBW deformation Hq,κ,2 is a quotient of T(V ) ⋊ G[t] given by the relations v2v1 = −v1v2 + m1I, v3v1 = v1v3 + m2g, v3v2 = v2v3 + m3g, v2

i = 0 for i = 1, 2, 3

where m1, m2, m3 ∈ K.

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Thank you!

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Parameter space

Definition

The parameter κ is called admissible if it defines a truncated quantum Drinfeld Hecke algebra.

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Parameter space

Definition

The parameter κ is called admissible if it defines a truncated quantum Drinfeld Hecke algebra. The subspace of admissible parameters, PG = {κ ∈ HomK(V ⊗ V , KG)|κ is admissible}, we call the parameter space.

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Parameter space

Definition

The parameter κ is called admissible if it defines a truncated quantum Drinfeld Hecke algebra. The subspace of admissible parameters, PG = {κ ∈ HomK(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.

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