First fundamental theorems of invariant theory for quantum supergroups

Abstract

Let U q (g) be the quantum supergroup of gl m|n or the modified quantum supergroup of osp m|2n over the field of rational functions in q, and let V q be the natural module for U q (g). There exists a unique tensor functor, associated with V q , from the category of ribbon graphs to the category of finite dimensional representations of U q (g), which preserves ribbon category structures. We show that this functor is full in the cases g = gl m|n or osp 2ℓ+1|2n . For g = osp 2ℓ|2n , we show that the space Hom Uq(g) (V ⊗r q , V ⊗s q ) is spanned by images of ribbon graphs if r + s < 2ℓ(2n + 1). The proofs involve an equivalence of module categories for two versions of the quantisation of U(g).

The first and second fundamental theorems of invariant theory for the quantum general linear supergroup

“…In this case, π r 19. We expect that the subspaces π r T λ have the same properties as in Proposition 5.8 and Proposition 5.9, though Ă M k|l r|s and M k|l r|s have different defining relations in general.…”

“…Another formulation of non-commutative invariant theory provides a description for the endomorphism algebra over quantum (super) groups. The paper [19] establishes a full tensor functor from the category of ribbon graphs to the category of finite dimensional representations of U q pgq with g " gl m|n , osppm|2nq, giving the FFT of invariant theory in this endomorphism algebra setting. However, very little was known previously about the non-commutative polynomial version of invariant theory for quantum supergroups.…”

On the Harish-Chandra Homomorphism for Quantum Superalgebras

“…Definition 2.2. [14,32,45] Let (A, {s}) be the Cartan matrix of a simple basic Lie superalgebra g in the distinguished root system. The quantum superalgebra U q (g) is defined over k in q generated by K ±1 i , E i , F i , i ∈ I (all generators are even except for E s and F s , which are odd), subject to the following relations:…”

“…They have been found applications in various areas, including in the study of the solution of quantum Yang-Baxter equation [18], construction of topological invariants of knots and 3-mainfolds [52,48,49] and so on. Quantum superalgebras have been investigated extensively by many authors from different perspectives in aspects such as Serre relations, PBW basis, universal R-matrix [45,46], crystal bases [30,31], invariant theory [32], highest weight representations [15,53,54] and so on.…”

The Quantum Isomeric Supercategory

“…We expect that these functors can be generalised to the super setting of . The generalisation of the first functor should follow from the results [LZZ20], and then the affine case follows from the general affinisation procedure of [MS21]. Once again, analogues exist in the orthosymplectic case, where the relevant categories are the Kauffman skein category , together with its affine analogue introduced in [GRS22].…”