2017/03/06 by Yang Zhang, Zhang, Yang
Mathematics · #17B35 #17B37 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1703.01848
openalex publication_date 2017/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop the non-commutative polynomial version of the invariant theory for the quantum general linear supergroup \rm Uq(\mathfrakglm|n). A non-commutative \rm Uq(\mathfrakglm|n)-module superalgebra Pk|l r|s is constructed, which is the quantum analogue of the supersymmetric algebra over ℂk|l⊗ ℂm|n⊕ ℂr|s⊗ (ℂm|n)∗. We analyse the structure of the subalgebra of \rm Uq(\mathfrakglm|n)-invariants in Pk|l r|s by using the quantum super analogue of Howe duality. The subalgebra of \rm Uq(\mathfrakglm|n)-invariants in Pk|l r|s is shown to be finitely generated. We determine its generators and establish a surjective superalgebra homomorphism from a braided supersymmetric algebra onto it. This establishes the first fundamental theorem of invariant theory for \rm Uq(\mathfrakglm|n). We show that the above mentioned superalgebra homomorphism is an isomorphism if and only if m≥ min\k,r\ and n≥ min\l,s\, and obtain a monomial basis for the subalgebra of invariants in this case. When the homomorphism is not injective, we give a representation theoretical description of the generating elements of the kernel associated to the partition ((m+1)n+1), producing the second fundamental theorem of invariant theory for \rm Uq(\mathfrakglm|n). We consider two applications of our results. A complete treatment of the non-commutative polynomial version of invariant theory for \rm Uq(\mathfrakglm) is obtained as the special case with n=0, where an explicit SFT is proved, which we believe to be new. The FFT and SFT of the invariant theory for the general linear superalgebra are recovered from the classical (i.e., q→ 1) limit of our results.