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48
Structure of the MalvenutoReutenauer Hopf algebra of permutations
 Adv. Math
"... Abstract. We analyze the structure of the MalvenutoReutenauer Hopf algebra SSym of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration, and show that it decomposes as a crossed product ..."
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Cited by 51 (15 self)
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Abstract. We analyze the structure of the MalvenutoReutenauer Hopf algebra SSym of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration, and show that it decomposes as a crossed product over the Hopf algebra of quasisymmetric functions. In addition, we describe the structure constants of the multiplication as a certain number of facets of the permutahedron. As a consequence we obtain a new interpretation of the product of monomial quasisymmetric functions in terms of the facial structure of the cube. The Hopf algebra of Malvenuto and Reutenauer has a linear basis indexed by permutations. Our results are obtained from a combinatorial description of the Hopf algebraic structure with respect to a new basis for this algebra, related to the original one via Möbius inversion on the weak order on the symmetric groups. This is in analogy with the relationship between the monomial and fundamental bases of the algebra of quasisymmetric functions. Our results reveal a close relationship between the structure of the MalvenutoReutenauer Hopf algebra and the weak order on the symmetric groups.
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.
Hopf algebra actions on strongly separable extensions of depth two
 Adv. in Math. 163
, 2001
"... Abstract. We bring together ideas in analysis of Hopf ∗algebra actions on II1 subfactors of finite Jones index [9, 24] and algebraic characterizations of Frobenius, Galois and cleft Hopf extensions [14, 13, 3] to prove a noncommutative algebraic analogue of the classical theorem: a finite field ex ..."
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Cited by 21 (20 self)
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Abstract. We bring together ideas in analysis of Hopf ∗algebra actions on II1 subfactors of finite Jones index [9, 24] and algebraic characterizations of Frobenius, Galois and cleft Hopf extensions [14, 13, 3] to prove a noncommutative algebraic analogue of the classical theorem: a finite field extension is Galois iff it is separable and normal. Suppose N ֒ → M is a separable Frobenius extension of kalgebras split as Nbimodules with a trivial centralizer CM(N). Let M1: = End(MN) and M2: = End(M1)M be the endomorphism algebras in the Jones tower N ֒ → M ֒ → M1 ֒ → M2. We show that under depth 2 conditions on the second centralizers A: = CM1 (N) and B: = CM2 (M) the algebras A and B are semisimple Hopf algebras dual to one another and such that M1 is a smash product of M and A, and that M is a BGalois extension of N. 1.
Structure of the LodayRonco Hopf algebra of trees
 J. Algebra
"... Abstract. Loday and Ronco defined an interesting Hopf algebra structure on the linear span of the set of planar binary trees. They showed that the inclusion of the Hopf algebra of noncommutative symmetric functions in the MalvenutoReutenauer Hopf algebra of permutations factors through their Hopf ..."
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Cited by 19 (2 self)
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Abstract. Loday and Ronco defined an interesting Hopf algebra structure on the linear span of the set of planar binary trees. They showed that the inclusion of the Hopf algebra of noncommutative symmetric functions in the MalvenutoReutenauer Hopf algebra of permutations factors through their Hopf algebra of trees, and these maps correspond to natural maps from the weak order on the symmetric group to the Tamari order on planar binary trees to the boolean algebra. We further study the structure of this Hopf algebra of trees using a new basis for it. We describe the product, coproduct, and antipode in terms of this basis and use these results to elucidate its Hopfalgebraic structure. We also obtain a transparent proof of its isomorphism with the noncommutative ConnesKreimer Hopf algebra of Foissy, and show that this algebra is related to noncommutative symmetric functions as the (commutative) ConnesKreimer Hopf algebra is related to symmetric functions.
Vertex operator algebras, generalized doubles and dual pairs
 Math. Z
"... Let V be a simple vertex operator algebra and G a finite automorphism group. Then there is a natural right Gaction on the set of all inequivalent irreducible Vmodules. Let S be a finite set of inequivalent irreducible Vmodules which is closed under the action of G. There is a finite dimensional s ..."
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Cited by 17 (4 self)
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Let V be a simple vertex operator algebra and G a finite automorphism group. Then there is a natural right Gaction on the set of all inequivalent irreducible Vmodules. Let S be a finite set of inequivalent irreducible Vmodules which is closed under the action of G. There is a finite dimensional semisimple associative algebra Aα(G, S) for a suitable 2cocycle naturally determined by the Gaction on S such that Aα(G, S) and the vertex operator algebra V G form a dual pair on the sum of Vmodules in S in the sense of Howe. In particular, every irreducible Vmodule is completely reducible V Gmodule. 1
Quantum Groups on Fibre Bundles
 Commun. Math. Phys
, 1994
"... It is shown that the principle of locality and noncommutative geometry can be connnected by a sheaf theoretical method. In this framework quantum spaces are introduced and examples in mathematical physics are given. With the language of quantum spaces noncommutative principal and vector bundles are ..."
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Cited by 14 (1 self)
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It is shown that the principle of locality and noncommutative geometry can be connnected by a sheaf theoretical method. In this framework quantum spaces are introduced and examples in mathematical physics are given. With the language of quantum spaces noncommutative principal and vector bundles are defined and their properties are studied. Important constructions in the classical theory of principal fibre bundles like associated bundles and differential calculi are carried over to the quantum case. At the end qdeformed instanton models are
Lazy cohomology: an analogue of the Schur multiplier for arbitrary Hopf algebra
 J. Pure Appl. Algebra
"... We propose a detailed systematic study of a group H2 L (A) associated, by elementary means of lazy 2cocycles, to any Hopf algebra A. This group was introduced by Schauenburg in order to generalize G.I. Kac’s exact sequence. We study the various interplays of lazy cohomology in Hopf algebra theory: ..."
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Cited by 13 (3 self)
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We propose a detailed systematic study of a group H2 L (A) associated, by elementary means of lazy 2cocycles, to any Hopf algebra A. This group was introduced by Schauenburg in order to generalize G.I. Kac’s exact sequence. We study the various interplays of lazy cohomology in Hopf algebra theory: Galois and biGalois objects, Brauer groups and projective representations. We obtain a KacSchauenburgtype sequence for double crossed products of possibly infinitedimensional Hopf algebras. Finally the explicit computation of H2 L (A) for monomial Hopf algebras and for a class of cotriangular Hopf algebras is performed. Key words: Hopf 2cocycle, Galois objects, biGalois objects.
Cocycle twisting of E(n)module algebras and applications to the Brauer group
"... We classify the orbits of coquasitriangular structures for the Hopf algebra E(n) under the action of lazy cocycles and the Hopf automorphism group. This is applied to detect subgroups of the Brauer group BQ(k,E(n)) of E(n) that are isomorphic. For a triangular structure R on E(n) we prove that the ..."
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Cited by 10 (5 self)
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We classify the orbits of coquasitriangular structures for the Hopf algebra E(n) under the action of lazy cocycles and the Hopf automorphism group. This is applied to detect subgroups of the Brauer group BQ(k,E(n)) of E(n) that are isomorphic. For a triangular structure R on E(n) we prove that the subgroup BM(k,E(n),R) of BQ(k,E(n)) arising from R is isomorphic to a direct product of BW(k), the BrauerWall group of the ground field k, and Symn(k), the group of n × n symmetric matrices under addition. For a general quasitriangular structure R ′ on E(n) we construct a split short exact sequence having BM(k,E(n),R ′ ) as a middle term and as kernel a central extension of the group of symmetric matrices of order r < n (r depending on R ′). We finally describe how the image of the Hopf automorphism group inside BQ(k,E(n)) acts on Symn(k).
Cleft extensions of Hopf algebroids
 Appl. Categor. Struct
"... ABSTRACT. The notions of a cleft extension and a cross product with a Hopf algebroid are introduced and studied. In particular it is shown that an extension (with a Hopf algebroid H = (HL,HR)) is cleft if and only if it is HRGalois and has a normal basis property relative to the base ring L of HL. ..."
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Cited by 8 (7 self)
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ABSTRACT. The notions of a cleft extension and a cross product with a Hopf algebroid are introduced and studied. In particular it is shown that an extension (with a Hopf algebroid H = (HL,HR)) is cleft if and only if it is HRGalois and has a normal basis property relative to the base ring L of HL. Cleft extensions are identified as crossed products with invertible cocycles. The relationship between the equivalence classes of crossed products and gauge transformations is established. Strong connections in cleft extensions are classified and sufficient conditions are derived for the ChernGalois characters to be independent on the choice of strong connections. The results concerning cleft extensions and crossed product are then extended to the case of weak cleft extensions of Hopf algebroids hereby defined. 1.
Polynomial identities and non commutative versal torsors
 Adv. Math
"... Abstract. To any cleft Hopf Galois object, i.e., any algebra αH obtained from a Hopf algebra H by twisting its multiplication with a twococycle α, we attach two “universal algebras ” Aα H and Uα H. The algebra Aα H is obtained by twisting the multiplication of H with the most general twococycle σ ..."
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Cited by 5 (5 self)
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Abstract. To any cleft Hopf Galois object, i.e., any algebra αH obtained from a Hopf algebra H by twisting its multiplication with a twococycle α, we attach two “universal algebras ” Aα H and Uα H. The algebra Aα H is obtained by twisting the multiplication of H with the most general twococycle σ formally cohomologous to α. The cocycle σ takes values in the field of rational functions on H. By construction, Aα H is a cleft HGalois extension of a “big” commutative algebra Bα H. Any “form ” of αH can be obtained from Aα H by a specialization of Bα H and vice versa. If the algebra αH is simple, then Aα H is an Azumaya algebra with center Bα H. The algebra Uα H is constructed using a general theory of polynomial identities that we set up for arbitrary comodule algebras; it is the universal comodule algebra in which all comodule algebra identities of αH are satisfied. We construct an embedding of Uα H into Aα H; this embedding maps the center Zα H of Uα H into Bα H when the algebra αH is simple. In this case, under an additional assumption, A α H ∼ = B α H ⊗Z α H Uα H, thus turning Aα H into a central localization of Uα H. We completely work out these constructions in the case of the fourdimensional Sweedler algebra.