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Combinatorial Hopf algebras in quantum field theory I
 Reviews of Mathematical Physics
, 2005
"... This manuscript collects and expands for the most part a series of lectures on the interface between combinatorial Hopf algebra theory (CHAT) and renormalization theory, delivered by the secondnamed author in the framework of the joint mathematical physics seminar of the Universités d’Artois and Li ..."
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Cited by 35 (3 self)
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This manuscript collects and expands for the most part a series of lectures on the interface between combinatorial Hopf algebra theory (CHAT) and renormalization theory, delivered by the secondnamed author in the framework of the joint mathematical physics seminar of the Universités d’Artois and Lille 1, from late January till midFebruary 2003. The plan is as follows: Section 1 is the introduction, and Section 2 contains an elementary invitation to the subject. Sections 3–7 are devoted to the basics of Hopf algebra theory and examples, in ascending level of complexity. Section 8 contains a first, direct approach to the Faà di Bruno Hopf algebra. Section 9 gives applications of that to quantum field theory and Lagrange reversion. Section 10 rederives the Connes– Moscovici algebras. In Section 11 we turn to Hopf algebras of Feynman graphs. Then in Section 12 we give an extremely simple derivation of (the properly combinatorial part of) Zimmermann’s method, in its original diagrammatic form. In Section 13 general incidence algebras are introduced. In the next section the Faà di Bruno bialgebras
Strong Connections And ChernConnes Pairing In The HopfGalois Theory
, 2000
"... We reformulate the concept of connection on a HopfGalois extension B ⊆ P in order to apply it in computing the ChernConnes pairing between the cyclic cohomology HC 2n (B) and K0(B). This reformulation allows us to show that a HopfGalois extension admitting a strong connection is projective and le ..."
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Cited by 16 (4 self)
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We reformulate the concept of connection on a HopfGalois extension B ⊆ P in order to apply it in computing the ChernConnes pairing between the cyclic cohomology HC 2n (B) and K0(B). This reformulation allows us to show that a HopfGalois extension admitting a strong connection is projective and left faithfully flat. It also enables us to conclude that a strong connection is a CuntzQuillentype bimodule connection. To exemplify the theory, we construct a strong connection (super Dirac monopole) to find out the ChernConnes pairing for the super line bundles associated to super Hopf fibration.
Mixable Shuffles, Quasishuffles and Hopf Algebras
 J. Alg. Combinatorics
"... Abstract. The quasishuffle product and mixable shuffle product are both generalizations of the shuffle product and have both been studied quite extensively recently. We relate these two generalizations and realize quasishuffle product algebras as subalgebras of mixable shuffle product algebras. As ..."
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Cited by 15 (8 self)
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Abstract. The quasishuffle product and mixable shuffle product are both generalizations of the shuffle product and have both been studied quite extensively recently. We relate these two generalizations and realize quasishuffle product algebras as subalgebras of mixable shuffle product algebras. As an application, we obtain Hopf algebra structures in free RotaBaxter algebras. 1.
The Classification of Semisimple Hopf Algebras of dimension 16
 J. of Algebra
"... Abstract. In this paper we completely classify nontrivial semisimple Hopf algebras of dimension 16. We also compute all the possible structures of the Grothendieck ring of semisimple noncommutative Hopf algebras of dimension 16. Moreover, we prove that noncommutative semisimple Hopf algebras of di ..."
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Cited by 13 (1 self)
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Abstract. In this paper we completely classify nontrivial semisimple Hopf algebras of dimension 16. We also compute all the possible structures of the Grothendieck ring of semisimple noncommutative Hopf algebras of dimension 16. Moreover, we prove that noncommutative semisimple Hopf algebras of dimension p n, p is prime, cannot have a cyclic group of grouplikes. 1. Introduction. Recently various classification results were obtained for finitedimensional semisimple Hopf algebras over an algebraically closed field of characteristic 0. The smallest dimension, for which the question was still open, was 16. In this paper we completely classify all nontrivial (i.e. noncommutative and noncocommutative) Hopf algebras of dimension 16. Moreover, we consider all
A COMBINATORIAL APPROACH TO THE QUANTIFICATION OF LIE ALGEBRAS
"... We propose a notion of a quantumuniversal enveloping algebra for any Lie algebra defined by generators and relations which is based on the quantumLie operation concept. This enveloping algebra has a PBW basis that admits a monomial crystallization by means of the Kashiwara idea. We describe all skew ..."
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Cited by 12 (3 self)
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We propose a notion of a quantumuniversal enveloping algebra for any Lie algebra defined by generators and relations which is based on the quantumLie operation concept. This enveloping algebra has a PBW basis that admits a monomial crystallization by means of the Kashiwara idea. We describe all skew primitive elements of the quantum universal enveloping algebras for the classical nilpotent algebras of the infinite series defined by the Serre relations and prove that the above set of PBWgenerators for each of these enveloping algebras coincides with the Lalonde–Rambasis of the ground Lie algebra with a skew commutator in place of the Lie operation. The similar statement is valid for Hall–Shirshov basis of any Lie algebra defined by one relation, but it is not so in the general case. 1. Introduction.
Duality and Rational Modules in Hopf Algebras over Commutative Rings 1
, 2000
"... Let A be an algebra over a commutative ring R. IfR is noetherian and A ◦ is pure in R A, then the categories of rational left Amodules and right A ◦comodules are isomorphic. In the Hopf algebra case, we can also strengthen the Blattner– Montgomery duality theorem. Finally, we give sufficient condi ..."
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Cited by 12 (8 self)
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Let A be an algebra over a commutative ring R. IfR is noetherian and A ◦ is pure in R A, then the categories of rational left Amodules and right A ◦comodules are isomorphic. In the Hopf algebra case, we can also strengthen the Blattner– Montgomery duality theorem. Finally, we give sufficient conditions to get the purity of A ◦ in R A. © 2001 Academic Press Key Words: rational module; comodule; duality.
Computing the FrobeniusSchur indicator for abelian extensions of Hopf algebras
"... Let H be a finitedimensional semisimple Hopf algebra. Recently it was shown in [LM] that a version of the FrobeniusSchur theorem holds for Hopf algebras, and thus that the Schur indicator ν(χ) of the character χ of a simple Hmodule is welldefined; this fact for the special case of Kac algebras w ..."
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Cited by 10 (3 self)
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Let H be a finitedimensional semisimple Hopf algebra. Recently it was shown in [LM] that a version of the FrobeniusSchur theorem holds for Hopf algebras, and thus that the Schur indicator ν(χ) of the character χ of a simple Hmodule is welldefined; this fact for the special case of Kac algebras was shown in [FGSV]. In this paper we
Hilbert space representations of cross product algebras
 J. Funct. Anal
"... In this paper, we study and classify Hilbert space representations of cross product ∗algebras of the quantized enveloping algebra Uq(e2) with the coordinate algebras O(Eq(2)) of the quantum motion group and O(Cq) of the complex plane, and of the quantized enveloping algebra Uq(su1,1) with the coord ..."
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Cited by 8 (4 self)
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In this paper, we study and classify Hilbert space representations of cross product ∗algebras of the quantized enveloping algebra Uq(e2) with the coordinate algebras O(Eq(2)) of the quantum motion group and O(Cq) of the complex plane, and of the quantized enveloping algebra Uq(su1,1) with the coordinate algebras O(SUq(1,1)) of the quantum group SUq(1,1) and O(Uq) of the quantum disc. Invariant positive functionals and the corresponding Heisenberg representations are explicitely described.
Quasishuffles, Mixable Shuffles and Hopf Algebras
 J. Algebraic Combinatorics
"... Abstract. The quasishuffle product and mixable shuffle product are both generalizations of the shuffle product and have both been studied quite extensively recently. We relate these two generalizations and realize quasishuffle product algebras as subalgebras of mixable shuffle product algebras. Th ..."
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Cited by 6 (6 self)
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Abstract. The quasishuffle product and mixable shuffle product are both generalizations of the shuffle product and have both been studied quite extensively recently. We relate these two generalizations and realize quasishuffle product algebras as subalgebras of mixable shuffle product algebras. This allows us to extend a previous result of Hopf algebra structure on Baxter algebras. 1.
A COMMUTING PAIR IN HOPF ALGEBRAS
"... Abstract. We prove that if H is a semisimple Hopf algebra, then the action of the Drinfeld double D(H) onHand the action of the character algebra on H form a commuting pair. This result and a result of G. I. Kats imply that the dimension of every simple D(H)submodule of H is a divisor of dim (H). L ..."
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Cited by 5 (1 self)
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Abstract. We prove that if H is a semisimple Hopf algebra, then the action of the Drinfeld double D(H) onHand the action of the character algebra on H form a commuting pair. This result and a result of G. I. Kats imply that the dimension of every simple D(H)submodule of H is a divisor of dim (H). Let H be a finite dimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0, D(H) be the Drinfeld double of H, andC(H)be the character algebra of H. C(H) is spanned by the characters of Hmodules and is an associative subalgebra of H ∗.ItisknownthatD(H)actsonHand that C(H) acts on H by the restriction of the action “⇀ ” ofH∗on H (these actions will be recalled below). The purpose of this note is to prove that these two actions form a commuting pair. Using this result, we prove that the dimension of every simple D(H)submodule of H is a divisor of dim(H). It would be interesting if there exists an analog of this commuting pair in the context of Poisson Lie groups. We first recall the construction of the Drinfeld double (cf. [D], [M]) and fix necessary notations. Let H be a finite dimensional Hopf algebra over a field k (here we do not need any additional assumptions on H and k). The Drinfeld double of H, denoted by D(H), as a vector space, is the tensor space H ∗ ⊗ H. The comultiplication of D(A) isgivenby ∆(f ⊗ a) = � � � � � f(2) ⊗ a (1) ⊗ f(1) ⊗ a (2) ∈ D(H) ⊗ D(H), where ∆f = f (1) ⊗ f (2), ∆a=a (1) ⊗ a (2) are comultiplictions in H and H ∗ respectively. The multiplication in D(H) is defined as follows: for f ⊗ a and g ⊗ b in D(H), (f ⊗ a)(g ⊗ b) = � (1) f(a (1) ⊲g (2)) ⊗ (a (2) ⊳g (1))b, where a⊲g is the action of H on H ∗ given by a⊲g=a (1) ⇀g↼S −1 a (2) and a⊳g is the right action of H ∗ on H given by a⊳g=S −1 g (1) ⇀a↼g (2). The notations ⇀ and ↼ mean the usual left and right actions of H on H ∗ , i.e., for a ∈ H and g ∈ H ∗, a⇀g = � g (1)〈g (2),a〉∈H ∗ , g ↼ a = � g (2)〈g (1),a〉.