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
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- 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
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- 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
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- 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…
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