A Quantum Analogue of the First Fundamental Theorem of Classical Invariant Theory

Abstract

Abstract. We establish a noncommutative analogue of the first fundamental theorem of classical invariant theory. For each quantum group associated with a classical Lie algebra, we construct a noncommutative associative algebra whose underlying vector space forms a module for the quantum group and whose algebraic structure is preserved by the quantum group action. The subspace of invariants is shown to form a subalgebra, which is finitely generated. We determine generators of this subalgebra of invariants and determine their commutation relations. In each case considered, the noncommutative modules we construct are flat deformations of their classical commutative analogues. Thus by taking the limit as q → 1, our results imply the first fundamental theorem of classical invariant theory, and therefore generalise them to the noncommutative case.

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

“…This result enables one to bring the invariant theory for classical supergroups developed in [7,31,32,33] into the quantum supergroup setting. One translates the first fundamental theorems (FFTs) for classical supergroups [7,31] to (modified) U(g; T ).…”

“…This statement is the first fundamental theorem (FFT) of invariant theory for these quantum supergroups in the categorical formulation of [30,31,32]. It implies that the endomorphism algebra End Uq(F) (V ⊗r q ) is the representation of the braid group on r-strings generated by the R-matrix of the natural U q (F)-module.…”

“…We will do this by following the categorical approach to invariant theory developed in [30,31,32] for the orthogonal group and orthosymplectic supergroup. The ground work for this program has already been done in [62,64] in the process of constructing invariants of knots and 3-manifolds using quantum supergroups.…”

“…One translates the first fundamental theorems (FFTs) for classical supergroups [7,31] to (modified) U(g; T ). The equivalence between U(g; T ) and U q (g; T ) as braided quasi Hopf superalgebras in turn enables us to obtain an FFT for U q (g; T ).…”

“…We mention the work [34], which is closely related to [28], on invariants of quantised coordinate rings of modules over ordinary quantum groups. A similar investigation on invariants of quantised coordinate rings of modules over the quantum supergroups will be carried out in a future publication.…”