The first fundamental theorem of invariant theory for the quantum queer superalgebra

2022/09/02 by Chang, Zhihua, Wang, Yongjie
#FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2209.00811

Abstract

The classical invariant theory for the queer Lie superalgebra is an investigation of the U(\mathfrakq_n)-invariant sub-superalgebra of the symmetric superalgebra Sym(V^{⊕ r}⊕ V^{*⊕ s}) for V=ℂ^n|n. We establish the first fundamental theorem of invariant theory for the quantum queer superalgebra U_q(\mathfrakq_n). The key ingredient is a quantum analog O_r,s of the symmetric superalgebra Sym(V^{⊕ r}⊕ V^{*⊕ s}) that is created as a braided tensor product of a quantization A_r,n of Sym(V^{⊕ r}) and a quantization A_s,n of Sym(V^{*⊕ s}). Since the quantum queer superalgebra U_q(\mathfrakq_n) is not quasi-triangular, our braided tensor product is created via an explicit intertwining operator instead of the universal R-matrix.

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