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132
Cyclic cohomology, quantum group symmetries and the local index formula for SUq(2
 J. Inst. Math. Jussieu
"... We analyse the NCspace underlying the quantum group SUq(2) from the spectral point of view which is the basis of noncommutative geometry, and show how the general theory developped in our joint work with H. Moscovici applies to the specific spectral triple defined by Chakraborty and Pal. This provi ..."
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Cited by 37 (2 self)
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We analyse the NCspace underlying the quantum group SUq(2) from the spectral point of view which is the basis of noncommutative geometry, and show how the general theory developped in our joint work with H. Moscovici applies to the specific spectral triple defined by Chakraborty and Pal. This provides the pseudodifferential calculus, the Wodzcikitype residue, and the local cyclic cocycle giving the index formula. The cochain whose coboundary is the difference between the original Chern character and the local one is given by the remainders in the rational approximation of the logarithmic derivative of the Dedekind eta function. This specific example allows to illustrate the general notion of locality in NCG. The formulas computing the residue are ”local”. Locality by stripping all the expressions from irrelevant details makes them computable. The key feature of this spectral triple is its equivariance, i.e. the SUq(2)symmetry. We shall explain how this leads naturally to the general concept of invariant cyclic cohomology in the framework of quantum group symmetries.
Renormalization automated by Hopf algebra
 J. Symb. Comput
, 1999
"... Renormalization automated by Hopf algebra ..."
RankinCohen brackets and the Hopf algebra of transverse geometry
 Moscow Math. J
"... We settle in this paper a question left open in our paper “Modular Hecke algebras and their Hopf symmetry”, by showing how to extend the RankinCohen brackets from modular forms to modular Hecke algebras. More generally, our procedure yields such brackets on any associative algebra endowed with an a ..."
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Cited by 31 (7 self)
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We settle in this paper a question left open in our paper “Modular Hecke algebras and their Hopf symmetry”, by showing how to extend the RankinCohen brackets from modular forms to modular Hecke algebras. More generally, our procedure yields such brackets on any associative algebra endowed with an action of the Hopf algebra of transverse geometry in codimension one, such that the derivation corresponding to the Schwarzian derivative is inner. Moreover, we show in full generality that these RankinCohen brackets give rise to associative deformations.
Modular Hecke algebras and their Hopf symmetry, Mosc
 Math. J
"... 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 lat ..."
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Cited by 31 (8 self)
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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 1cocycle gives an inner derivation implemented by the level 1 Eisenstein series of weight 4. The Hopf cyclic 2cocycle representing the transverse fundamental class provides a natural extension of the first RankinCohen bracket to the modular Hecke algebras. Lastly, the Hopf cyclic version of the GodbillonVey cocycle gives rise to a 1cocycle 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. Research supported by the National Science Foundation award no. DMS9988487.
Combinatorics of (perturbative) quantum field theory
, 2000
"... We review the structures imposed on perturbative QFT by the fact that its Feynman diagrams provide Hopf and Lie algebras. We emphasize the role which the Hopf algebra plays in renormalization by providing the forest formulas. We exhibit how the associated Lie algebra originates from an operadic oper ..."
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Cited by 30 (8 self)
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We review the structures imposed on perturbative QFT by the fact that its Feynman diagrams provide Hopf and Lie algebras. We emphasize the role which the Hopf algebra plays in renormalization by providing the forest formulas. We exhibit how the associated Lie algebra originates from an operadic operation of graph insertions. Particular emphasis is given to the connection with the Riemann–Hilbert problem. Finally, we outline how these structures relate to the numbers which we see in Feynman diagrams.
Combinatorics of rooted trees and Hopf algebras
, 2002
"... We begin by considering the graded vector space with a basis consisting of rooted trees, with grading given by the count of nonroot vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the op ..."
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Cited by 27 (3 self)
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We begin by considering the graded vector space with a basis consisting of rooted trees, with grading given by the count of nonroot vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the operator that multiplies a rooted tree by its number of vertices, and each operator naturally associates a multiplicity to each pair of rooted trees. By using symmetry groups of trees we define an inner product with respect to which the growth and pruning operators are adjoint, and obtain several results about the associated multiplicities. Now the symmetric algebra on the vector space of rooted trees (after a degree shift) can be endowed with a coproduct to make a Hopf algebra; this was defined by Kreimer in connection with renormalization. We extend the growth and pruning operators to Kreimer’s Hopf algebra and relate them via the inner product. On the other hand, the vector space of rooted trees itself can be given a noncommutative multiplication: with an appropriate coproduct, this leads to the Hopf algebra of Grossman and Larson. We show that the inner product on rooted trees leads to an isomorphism of the GrossmanLarson Hopf algebra with the graded dual of Kreimer’s Hopf algebra, correcting an earlier result of Panaite. 1
Insertion and elimination: the doubly infinite Lie algebra of Feynman graphs
 Ann. Henri Poincar
, 2002
"... The Lie algebra of Feynman graphs gives rise to two natural representations, acting as derivations on the commutative Hopf algebra of Feynman graphs, by creating or eliminating subgraphs. Insertions and eliminations do not commute, but rather establish a larger Lie algebra of derivations which we he ..."
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Cited by 26 (10 self)
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The Lie algebra of Feynman graphs gives rise to two natural representations, acting as derivations on the commutative Hopf algebra of Feynman graphs, by creating or eliminating subgraphs. Insertions and eliminations do not commute, but rather establish a larger Lie algebra of derivations which we here determine.
Structures in Feynman Graphs Hopf Algebras and Symmetries
, 2008
"... We review the combinatorial structure of perturbative quantum field theory with emphasis given to the decomposition of graphs into primitive ones. The consequences in terms of unique factorization of Dyson– Schwinger equations into Euler products are discussed. ..."
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Cited by 24 (10 self)
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We review the combinatorial structure of perturbative quantum field theory with emphasis given to the decomposition of graphs into primitive ones. The consequences in terms of unique factorization of Dyson– Schwinger equations into Euler products are discussed.