A First Fundamental Theorem of Invariant Theory for the Quantum Queer Superalgebra

The classical invariant theory for the queer Lie superalgebra is an investigation of the U(q(n))-invariant sub-superalgebra of the symmetric superalgebra Sym (V-circle plus r circle plus V*(circle plus s)) for V = C-n vertical bar n. We establish a first fundamental theorem of invariant theory in the case that the quantum queer superalgebra U-q(q(n)) acts on a quantum analogue O-r,O-s of the symmetric superalgebra Sym(V-circle plus r circle plus V*(circle plus s)). The superalgebra O-r,O-s is a braided tensor product of a quantum analogue A(r,n) of Sym (V-circle plus r) and a quantum analogue (A) over bar (s,n) of Sym (V*(circle plus s)). Since the quantum queer superalgebra U-q(q(n)) is not quasi-triangular, our braided tensor product is created via an explicit intertwining operator instead of the universal R-matrix.