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

We develop a non-commutative polynomial version of the invariant theory of the quantum general linear supergroup U-q(gl(m vertical bar n)). A non-commutative U-q(gl(m vertical bar n))- module superalgebra P-r vertical bar s(k vertical bar l) is constructed, which is the quantum analogue of the supersymmetric algebra over C-k vertical bar l circle times C-m(vertical bar n) circle plus C-r vertical bar s circle times(C-m vertical bar n)* . We analyse the structure of the subalgebra of U-q(gl(m vertical bar n))-invariants in P(r vertical bar s)(k vertical bar l )by using a quantum super analogue of Howe duality. The subalgebra of U-q(gl(m vertical bar n))-invariants in P-r vertical bar s(k vertical bar 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 fundamen- tal theorem of invariant theory for U-q(gl(m vertical bar 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 U-q(gl(m vertical bar 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 U-q(gl(m)). 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. (C) 2020 Elsevier B.V. All rights reserved.