RECORD
Modular Hecke Algebras and their Hopf Symmetry
Quantum Algebra · Submitted 9 Jan 2003
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Their underlying symmetry is shown to be encoded by the same Hopf algebra that controls the transverse geometry of codimension 1 foliations.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
We introduce and begin to analyse a class of algebras, associated to congruence subgroups, that extend both the algebra of modular forms of all levels and the ring of classical Hecke operators. At the intuitive level, these are algebras of `polynomial coordinates' for the `transverse space' of lattices modulo the action of the Hecke correspondences. Their underlying symmetry is shown to be encoded by the same Hopf algebra that controls the transverse geometry of codimension 1 foliations. Its action is shown to span the `holomorphic tangent space' of the noncommutative space, and each of its three basic Hopf cyclic cocycles acquires a specific meaning. The Schwarzian 1-cocycle gives an inner derivation implemented by the level 1 Eisenstein series of weight 4. The Hopf cyclic 2-cocycle representing the transverse fundamental class provides a natural extension of the first Rankin-Cohen bracket to the modular Hecke algebras. Lastly, the Hopf cyclic version of the Godbillon-Vey cocycle gives rise to a 1-cocycle on PSL(2, Q) with values in Eisenstein series of weight 2, which when coupled with the `period' cocycle yields a representative of the Euler class.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We introduce and begin to analyse a class of algebras, associated to congruence subgroups, that extend both the algebra of modular forms of all levels and the ring of classical Hecke operators. | contribution 1.00 | 0.93 | 0.22 | asserted 1.00 |
| s1 | At the intuitive level, these are algebras of `polynomial coordinates' for the `transverse space' of lattices modulo the action of the Hecke correspondences. | interpretation 0.93 | 0.31 | 0.06 | asserted 0.68 |
| s2 | Their underlying symmetry is shown to be encoded by the same Hopf algebra that controls the transverse geometry of codimension 1 foliations. | result 0.95 | 0.85 | 0.28 | asserted 1.00 |
| s3 | Its action is shown to span the `holomorphic tangent space' of the noncommutative space, and each of its three basic Hopf cyclic cocycles acquires a specific meaning. | result 0.94 | 0.88 | 0.18 | asserted 1.00 |
| s4 | The Schwarzian 1-cocycle gives an inner derivation implemented by the level 1 Eisenstein series of weight 4. | result 0.89 | 0.84 | 0.08 | asserted 1.00 |
| s5 | The Hopf cyclic 2-cocycle representing the transverse fundamental class provides a natural extension of the first Rankin-Cohen bracket to the modular Hecke algebras. | result 0.89 | 0.87 | 0.12 | asserted 1.00 |
| s6 | Lastly, the Hopf cyclic version of the Godbillon-Vey cocycle gives rise to a 1-cocycle on PSL(2, Q) with values in Eisenstein series of weight 2, which when coupled with the `period' cocycle yields a representative of the Euler class. | result 0.95 | 0.85 | 0.11 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v4
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
We introduce and begin to analyse a class of algebras, associated to congruence subgroups, that extend both the algebra of modular forms of all levels and the ring of classical Hecke operators. At the intuitive level, these are algebras of `polynomial coordinates' for the `transverse space' of lattices modulo the action of the Hecke correspondences. Their underlying symmetry is shown to be encoded by the same Hopf algebra that controls the transverse geometry of codimension 1 foliations. Its action is shown to span the `holomorphic tangent space' of the noncommutative space, and each of its three basic Hopf cyclic cocycles acquires a specific meaning. The Schwarzian 1-cocycle gives an inner derivation implemented by the level 1 Eisenstein series of weight 4. The Hopf cyclic 2-cocycle representing the transverse fundamental class provides a natural extension of the first Rankin-Cohen bracket to the modular Hecke algebras. Lastly, the Hopf cyclic version of the Godbillon-Vey cocycle gives rise to a 1-cocycle on PSL(2, Q) with values in Eisenstein series of weight 2, which when coupled with the `period' cocycle yields a representative of the Euler class.
Human review
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.