Multilinear Hirota operators, modular forms and the Heisenberg algebra

Abstract

We use the imbedding of the total differential operator D into a Heisenberg algebra to give a method to generate the transvectants and their multilinear generalizations using the coherent state method. This leads to tensor product decompositions where all the components play an equal role. 1. The Heisenberg algebra Consider the total differential operator D, given on the generators by Duk = uk+1 and obeying the Leibniz rule D(fg) = D(f)g + fD(g) on polynomials f, g ∈ P [u, u1, · · · ], where we denote ∂u ∂xk by uk. This defines D on P [u, u1, · · · ]. We now want to solve the following problem. Given a nonconstant f ∈ P [u, u1, · · · ], can we find f, f ∈ P [u, u1, · · · ] such that f = f +Df1, with f in a direct summand of ImD? We solve this problem by constructing a derivation F , such that KerF is a direct summand of ImD and such that D, F and E = [F ,D] form a Heisenberg algebra (cf. [SR94, SW97]). This last property allows us to find an algorithm to do the splitting over Ker E ⊕KerF ⊕ ImD in concrete cases. Since P [u, u1, · · · ] is generated by u, u1, · · · and u1, u2, · · · ∈ ImD, it seems natural to require Fu = 0 and Fuk 6= 0. Let us try Fuk = kuk−1 on the generators, and extend F by requiring it to be a derivation just like D, i.e. F(fg) = F(f)g + fF(g). Here one should remark that this is not just trial and error. The guess is inspired by looking at the standard finite dimensional irreducible representations of sl(2,R) and then taking the limit of n →∞, where n is the dimension of the representation space. This limiting behavior will explain the connection with classical invariant theory and modular functions, to be discussed in sections 5 and 6. 1See however section 8 for a quantized rule c ©0000 (copyright holder)