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Transvectants, modular forms, and the Heisenberg algebra. (English) Zbl 1041.11026
The main objects of the paper under review are binary forms and modular forms. The starting point is an observation due to D. Zagier [Proc. Indian Acad. Sci., Math. Sci. 104, 57–75 (1994; Zbl 0806.11022)] who noticed the similarity between the Rankin-Cohen brackets of modular forms and transvectants of binary forms. The first author discovered, in his earlier work, that if one regards the degree \(n\) of a binary form as \(-w\), minus the weight of a modular form, then, in fact, the two processes are identical. The authors’ goal is to develop this connection.
They show that the two theories of modular and binary forms have a common limiting theory as \(n=-w\to\infty\). Looking at the limit situation on the infinitesimal level, the authors arrive at the Heisenberg algebra. Going back to the classical situation, they present an explicit procedure in the form of a quantum deformation.
They also present connections with the theory of differential invariants arising from the view at transvectants as at relative differential invariants with respect to a certain action of \(SL(2)\). This leads to a theorem providing a basis for the algebra of transvectants of a binary form.
Yet other applications arise in connection with the theory of integrable systems and star products on the two-dimensional phase space.

MSC:
11E76 Forms of degree higher than two
11F11 Holomorphic modular forms of integral weight
17B80 Applications of Lie algebras and superalgebras to integrable systems
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
53D55 Deformation quantization, star products
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
81R30 Coherent states
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