# Transvectants, Modular Forms, and the Heisenberg Algebra

@article{Olver2000TransvectantsMF, title={Transvectants, Modular Forms, and the Heisenberg Algebra}, author={Peter J. Olver and Jan A. Sanders}, journal={Adv. Appl. Math.}, year={2000}, volume={25}, pages={252-283} }

We discuss the amazing interconnections between normal form theory, classical invariant theory and transvectants, modular forms and Rankin-Cohen brackets, representations of the Heisenberg algebra, differential invariants, solitons, Hirota operators, star products and Moyal brackets, and coherent states.

## 40 Citations

### Rankin–Cohen brackets, transvectants and covariant differential operators

- Mathematics
- 2010

We give a geometric construction of the transvectant on a Hermitian symmetric space (which in the case of the unit disk is also called the Rankin–Cohen bracket) in terms of the covariant…

### Generalized Transvectants and

- Mathematics
- 2005

We introduce a difierential operator invariant under the special linear group SL(2n; C), and, as a consequence, the symplectic group Sp(2n; C). Connections with generalized Rankin{Cohen brackets for…

### Rankin-Cohen Brackets and Invariant Theory

- Mathematics
- 2001

Using maps due to Ozeki and Broué-Enguehard between graded spaces of invariants for certain finite groups and the algebra of modular forms of even weight we equip these invariants spaces with a…

### Symplectic Transvectant and Siegel Modular Forms

- Mathematics
- 2005

We introduce a differential operator invariant under the symplectic group Sp(2n, C). A connection with a Rankin Cohen bracket for Siegel modular forms of genus n is sketched.

### Automorphic Lie algebras and modular forms

- Mathematics
- 2020

We introduce and study hyperbolic versions of automorphic Lie algebras for the modular group $SL(2,\mathbb Z)$ acting naturally on the upper half-plane $\mathbb H^2$ and on the Lie algebra $\mathfrak…

### Ring structures for holomorphic discrete series and Rankin-Cohen brackets

- Mathematics
- 2005

In the present note we discuss two different ring structures on the set of holomorphic discrete series of a causal symmetric space of Cayley type $G/H$ and we suggest a new interpretation of…

### Gauß–Manin from scratch: theme, variations and fantasia

- MathematicsProceedings of the Royal Society A
- 2022

We discuss the explicit construction of Gauß–Manin connections on the cohomology of families of low genus Riemann surfaces represented as curves with branch points in general position. The approach…

### Rankin–Cohen Brackets and Associativity

- Mathematics
- 2008

AbstractDon Zagier introduced and discussed in Zagier [Proc Indian Acad Sci (Math Sci) 104(1)
57–75, 1994] a particular algebraic structure of the graded ring of modular forms. In this note we…

### ALGEBRAIC HIROTA MAPS

- Mathematics
- 2006

We give definitions of Hirota maps acting as intertwining operators for representations of SLn(C). We show how these reduce to the conventional (generalized) Hirota derivatives in the limit of the…

## References

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### Rankin-Cohen Brackets and Invariant Theory

- Mathematics
- 2001

Using maps due to Ozeki and Broué-Enguehard between graded spaces of invariants for certain finite groups and the algebra of modular forms of even weight we equip these invariants spaces with a…

### A simple example of modular forms as tau-functions for integrable equations

- Mathematics
- 1992

We show how classical modular forms and functions appear as tau-functions for a certain integrable reduction of the self-dual Yang-Mills equations obtained by S. Chakravarty, M. Ablowitz, and P.…

### The Construction of Automorphic Forms From the Derivatives of a Given Form II

- MathematicsCanadian Mathematical Bulletin
- 1985

Abstract Explicit constructions of polynomials of preassigned degree and weight in the derivatives of a given automorphic form are described and studied, supplementing the results of an earlier…

### A quantum deformation of invariants of binary forms

- Mathematics
- 1998

The modern representation theory has its roots in the classical theory of invariants of binary forms of Caley, Silvester, Clebsch, Gordan, Capelli etc. We use the theory of the quantum group Uq(sl(2,…

### Modular forms and differential operators

- Mathematics
- 1994

In 1956, Rankin described which polynomials in the derivatives of modular forms are again modular forms, and in 1977, H Cohen defined for eachn ≥ 0 a bilinear operation which assigns to two modular…

### Rankin-Cohen type differential operators for Siegel modular forms

- Mathematics
- 1997

Let ℍn be the Siegel upper half space and let F and G be automorphic forms on ℍn of weights k and l, respectively. We give explicit examples of differential operators D acting on functions on ℍn × ℍn…

### AUTOMORPHIC PSEUDODIFFERENTIAL OPERATORS

- Mathematics
- 1997

The theme of this paper is the correspondence between classical modular forms and pseudodifferential operators (ΨDO’s) which have some kind of automorphic behaviour. In the simplest case, this…