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

Abstract

We develop a non-commutative polynomial version of the invariant theory of the quantum general linear supergroup Uq(glm|n). A non-commutative Uq(glm|n)-module superalgebra Pr|sk|l 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 Uq(glm|n)-invariants in Pr|sk|l by using a quantum super analogue of Howe duality.The subalgebra of Uq(glm|n)-invariants in Pr|sk|l 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 Uq(glm|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 PBW 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. This way we obtain the relations obeyed by the generators of the subalgebra of invariants, producing the second fundamental theorem of invariant theory for Uq(glm|n).We consider the special case n=0 in greater detail, obtaining a complete treatment of the non-commutative polynomial version of the invariant theory of Uq(glm). In particular, the explicit SFT proved here is believed to be new. We also recover the FFT and SFT of invariant theory for the general linear superalgebra from the classical limit (i.e. q→1) of our results.