Braided tensor products and polynomial invariants for the quantum queer superalgebra

2023/08/25 by Chang, Zhihua, Wang, Yongjie
#16T20 #17B37 #20G42 #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2308.13132

Abstract

The classical invariant theory for the queer Lie superalgebra \mathfrakq_n investigates its invariants in the supersymmetric algebra U_s,l^r,k:=Sym(V^{⊕ r}⊕ Π(V)^{⊕ k}⊕ V^{*⊕ s}⊕ Π(V^*)^{⊕ l} ), where V=ℂ^n|n is the natural supermodule, V^* is its dual and Π is the parity reversing functor. This paper aims to construct a quantum analogue B^r,k_s,l of U_s,l^r,k and to explore the quantum queer superalgebra U_q(\mathfrakq_n)-invariants in B^r,k_s,l. The strategy involves braided tensor products of the quantum analogues A_r,n, A_k,n^Π of the supersymmetric algebras Sym(V^{⊕ r}), Sym(Π(V)^{⊕ k}), and their dual partners A_s,n, and A_l,n^Π. These braided tensor products are defined using explicit braiding operator due to the absence of a universal R-matrix for U_q(\mathfrakq_n). Furthermore, we obtain an isomorphism between the braided tensor product A_r,n\otimesA_k,n and A_r+k,n, an isomorphism between A_k,n^Π and A_k,n, as well as the corresponding isomorphisms for their dual parts. Consequently, the U_q(\mathfrakq_n)-supermodule superalgebra B^r,k_s,l is identified with B^r+k,0_s+l,0. This allows us to obtain a set of generators of U_q(\mathfrakq_n)-invariants in B^r,k_s,l.

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