A Quantum Deformation of Invariants of Higher Binary Forms

Abstract

We use the theory of the quantum group U q gl 2 to develop a quantum theory of invariants and show a decomposition of invariants into a Gordan-Capelli series. Higher binary forms are introduced on the basis of braided algebras. We define quantised invariants and give basic examples. We show that the symbolic method of Clebsch and Gordan works also in the quantised case. We discuss the deformed discriminant of the quadratic and the cubic form, the deformed invariants I 1 , I 2 of the quartic form and further invariants without a classical analog.

Transvectants, Modular Forms, and the Heisenberg Algebra

“…A very interesting generalization of classical invariant theory and the transvectant calculus to quantum groups appears in [28]. Our methods, which in themselves realize the classical theory as a deformation of the Heisenberg theory, should be particularly relevant.…”

Abel’s theorem in the noncommutative case

“…The simplest symbol (12) 3 gives the zero invariant. The Hessian covariant ∆ has the symbol (01)(02) (12) 2 . Up to a constant factor we obtain…”

“…Proposition 7 is not true for the quantum group U q (sl (2)) (cf. [12]), i.e. the commutation relations depend on the concrete type of form.…”

“…Another example is the form xA + yB := (02)(13) + (03) (12) corresponding to the fourth harmonic point with…”

“…An analogue of Lemma 11 is not true in the case of U q (sl(2))-symmetry. For the noncommutative coordinate algebra in [12] we can define polarisation operators but we have no analogue of G.…”