Quantum polynomial functors

Hong, Jiuzu; Yacobi, Oded

doi:10.1016/j.jalgebra.2017.01.030

Cited by 4 publications

(22 citation statements)

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“…In this section we introduce several objects and prove some properties which will be of use in defining quantum polynomial functors. We note that several of these definitions and some of the properties are taken directly from [HY17].…”

Section: Preliminariesmentioning

confidence: 99%

“…We now define the quantum Hom-space algebra as it is defined by Hong and Yacobi [HY17]. Given two Yang-Baxter spaces V and W with basis {v i } and {w j }, respectively, let T (V, W ) be the tensor algebra of Hom(V, W ), that is…”

Section: Preliminariesmentioning

confidence: 99%

“…In this section we propose a different definition of quantum polynomial functors that generalizes the definition in [HY17]. Our category of quantum polynomial functors enjoys many of the properties presented in [HY17], and additionally, it has a composition. Let d, e be non-negative integers.…”

Section: Quantum Polynomial Functorsmentioning

confidence: 99%

“…Hong and Yacobi [HY17] introduced a category of quantum polynomial functors which quantizes the strict polynomial functors of Friedlander and Suslin [FS97]. The purpose of this paper is to introduce higher level categories of quantum polynomial functors that extend the construction of Hong and Yacobi and explain why they give a natural quantization of classical polynomial functors.…”

Section: Introductionmentioning

confidence: 99%

“…A natural question is whether one can deform the polynomial functors into quantum polynomial functors. A first such quantization is due to Hong and Yacobi [HY17]. They introduced a new category P d q that is a q-deformation of the category P d and showed that it enjoys many properties that P d has.…”

Section: Introductionmentioning

confidence: 99%

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Quantum polynomial functors from e-Hecke pairs

Buciumas

2018

Math. Z.

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We define a new category of quantum polynomial functors extending the quantum polynomials introduced by Hong and Yacobi. We show that our category has many properties of the category of Hong and Yacobi and is the natural setting in which one can define composition of quantum polynomial functors. Throughout the paper we highlight several key differences between the theory of classical and quantum polynomial functors.Proposition 2.2. The following diagram commutes:from which the commutativity of the diagram follows. Let (V, R V ) and (W, R W ) be Yang-Baxter spaces. We define the generalized (q-)Schur algebra S(V, W ; d) := (A(W, V ) d ) * as in [HY17]. The following is proved in [HY17]: Proposition 2.3. Let V, W be Yang-Baxter spaces. Then there is a natural isomorphism S(V, W ; d) ∼ = Hom B d (V ⊗d , W ⊗d ) Proof. See Proposition 2.7 in [HY17].By taking the dual of ∆ V,W,U we obtain a mapThere is a natural map m ′ U,W,V :The following Proposition shows they are the same map under the isomorphism in Proposition 2.3.Proposition 2.4. Given three Yang-Baxter spaces V, U, W , the following diagram commutes:Remark 2.5. If W = V , the quadratic relation (4) becomes the RTT relation due to Faddeev, Reshetikhin and Taktajan. The algebra A(V, V ) is then just the algebra denoted by A R V in [FRT88].

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Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Quantum Polynomial Functorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quantum polynomial functors from e-Hecke pairs

Buciumas

2018

Math. Z.

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Polynomial Functors and Two-Parameter Quantum Symmetric Pairs

Buciumas

2022

Transformation Groups

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We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups GLn, the two-parameter polynomial functors give a new interpretation of (polynomial) representations of the quantum symmetric pair ($$ {U}_{Q,q}^B $$ U Q , q B ($$ \mathfrak{gl} $$ gl n), Uq($$ \mathfrak{gl} $$ gl n)) which specializes to type AIII/AIV quantum symmetric pairs. The coideal subalgebra $$ {U}_{Q,q}^B $$ U Q , q B ($$ \mathfrak{gl} $$ gl n) appears in a Schur–Weyl duality with the type B Hecke algebra $$ {\mathcal{H}}_{Q,q}^B $$ H Q , q B (d). We endow two-parameter polynomial functors with a cylinder braided structure which we use to construct the two-parameter Schur functors. Our polynomial functors can be precomposed with the quantum polynomial functors of type A producing new examples of action pairs.

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Connectedness of cup products for polynomial representations of GLn and applications

Touzé

2018

Ann. K-Th.

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We find conditions such that cup products induce isomorphisms in low degrees for extensions between stable polynomial representations of the general linear group. We apply this result to prove generalizations and variants of the Steinberg tensor product theorem. Our connectedness bounds for cup product maps depend on numerical invariants which seem also relevant to other problems, for example to study the cohomological behavior of the Schur functor.

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Quantum polynomial functors

Cited by 4 publications

References 18 publications

Quantum polynomial functors from e-Hecke pairs

Quantum polynomial functors from e-Hecke pairs

Polynomial Functors and Two-Parameter Quantum Symmetric Pairs

Connectedness of cup products for polynomial representations of GLn and applications

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