Showing papers on "Hopf algebra published in 1989"
TL;DR: In this paper, the authors define and investigate a multiparametric deformation of the general linear supergroup, which is the simplest example of some general constructions described in [5, 6].
Abstract: In the work L. D. Faddeev and his collaborators, and subsequently V. G. Drinfeld, M. Jimbo, S. L. Woronowicz, a new class of Hopf algebras was constructed. They can be considered as one-parametric deformations of either group ring or the universal enveloping algebra of a simple algebraic group. In this paper we define and investigate a multiparametric deformation of the general linear supergroup. This is the simplest example of some general constructions described in [5, 6].
393 citations
241 citations
TL;DR: In this paper, the authors describe Hopf algebras which are associated with certain families of trees and give new proofs of enumerations of such objects as rooted trees and ordered rooted trees.
228 citations
171 citations
01 Jan 1989
TL;DR: The notion of quasitriangular quasi-Hopf algebras was introduced in this article, which is an abstract version of the notion of R-matrix and can be used for knot invariants.
Abstract: This paper is a brief exposition of [6] In §1 we remind the notion of quasitriangular Hopf algebra which is an abstract version of the notion of R-matrix In §2 the notion of quasitriangular quasi-Hopf algebra is introduced (coassociativity is replaced by a weaker axiom) In §3 we construct a class of quasitriangular quasi-Hopf algebras using the differential equations for n-point functions in the WZW theory introduced by VGKnizhnik and ABZamolodchikov Theorem 1 asserts that within perturbation theory with respect to Planck’s constant essentially all quasitriangular quazi-Hopf algebras belong to this class A natural proof of Kohno’s theorem on the equivalence of two kinds of braid group representations is given In §4 we discuss applications to knot invariants In §5 the classical limit of various quantum notions is discussed
155 citations
TL;DR: In this article, it was shown that the rational singular chain complex on a topological monoid is the enveloping algebra of a Lie algebra and that pth powers vanish in H * (QX; Zp) if A is generated as an R-algebra in dimensions below rp.
Abstract: Let (A, d) denote a free r-reduced differential graded R-algebra, where R is a commutative ring containing n for 1 a "diagonal" yi: A -+ A 0 A exists which satisfies the Hopf algebra axioms, including cocommutativity and coassociativity, up to homotopy. We show that (A, d) must equal U(L, 3) for some free differential graded Lie algebra (L, 6) This content downloaded from 157.55.39.203 on Sat, 27 Aug 2016 06:23:15 UTC All use subject to http://about.jstor.org/terms HOPF ALGEBRAS UP TO HOMOTOPY 453 if A is generated as an R-algebra in dimensions below rp . As a consequence, the rational singular chain complex on a topological monoid is seen to be the enveloping algebra of a Lie algebra. We also deduce, for an r-connected CW complex X of dimension enveloping algebra and that pth powers vanish in H* (QX; Zp) . DEPARTMENT OF MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE, MASSACHUSETTS 02139 This content downloaded from 157.55.39.203 on Sat, 27 Aug 2016 06:23:15 UTC All use subject to http://about.jstor.org/terms
105 citations
93 citations
TL;DR: In this article, a non-commutative version of Rivano and Deligne-Milne's result was shown for finite Hopf algebras over a commutative ring.
Abstract: Let A be a commutative Hopf algebra over a field k; the k-valued fibre functors on the category of finite dimensional A-comodules correspond to Spec(A)-torsors over k as was shown by Saavedra Rivano and Deligne-Milne. We prove a non-commutative version of this result by using methods developed in a previous paper [5] for the case of finite Hopf algebras over a commutative ring. We also exhibit right adjoints for fibre functors under the assumption that the antipode is bijective.
50 citations
TL;DR: In this paper, a Hopf algebraic approach simplifies and generalizes the Picard-Vessiot theory for positive characteristic, for fields with not necessarily commuting derivations, and even for fields having (a set of) higher derivations.
42 citations
01 Mar 1989
TL;DR: In this paper, a new class of examples of non-commutative and non-cocommutative Hopf algebras is presented. But the latter are obtained from a canonical solution of the classical Yang-Baxter Equations.
Abstract: We construct a new class of examples of non-commutative and non-cocommutative Hopf algebras. There is essentially one for every Lie bialgebra, and several other examples including one for every simple real Lie algebra. The latter are obtained from a canonical solution of the Classical Yang-Baxter Equations. Physically, these Hopf algebras arise as the algebra of observables of quantum mechanics on homogeneous spacetimes.
01 Jan 1989
TL;DR: In this paper, the exterior product operation is used to define lambda operations on the homology of the Lie algebra of matrices gl(A) when A is a commutative algebra.
Abstract: The exterior product operation permits us to define lambda operations on the homology of the Lie algebra of matrices gl(A), when A is a commutative algebra. By the Loday-Quillen theorem the primitive part of this homology is cyclic homology, which, therefore, inherits lambda operations. The aim of this paper is to give an explicit formula for these lambda operations on cyclic homology. It turns out that the classical Euler partition of the symmetric group is involved.
TL;DR: In this article, the authors consider infinite dimensional hopf algebras over grouplike subalgebrains and show that they are infinite-dimensional hopf over subalgebra.
Abstract: (1989). Freeness of infinite dimensional hopf algebras over grouplike subalgebras. Communications in Algebra: Vol. 17, No. 2, pp. 413-424.
TL;DR: The Hopf algebra structure of H * (Ω G, F 2 ) and the action of the dual Steenrod algebra are completely and explicitly determined when G is one of the connected, simply connected, exceptional, simple Lie groups as mentioned in this paper.
Abstract: The Hopf algebra structure of H * (Ω G , F 2 ) and the action of the dual Steenrod algebra are completely and explicitly determined when G isone of the connected, simply connected, exceptional, simple Lie groups. The approach is homological, using connected coverings and spectral sequences.
01 Jan 1989
TL;DR: The quantum inverse problem method allows for a systematic treatment of a variety of exactly solvable models of quantum field theory in 1 + 1 dimensions and vertex type lattice models in statistical physics as discussed by the authors.
Abstract: The quantum inverse problem method allows for a systematic treatment of a variety of exactly solvable models of quantum field theory in 1 + 1 dimensions and vertex type lattice models in statistical physics [cf. e.g. 1,2]. Moreover, it enables one to construct new integrable models and leads also to new mathematical structures [3,4] known as quantum Lie groups and algebras, quadratic algebras, quantum Hopf algebras. Recently these concepts entered into numerous publications on the operator solution to the quantum Liouville equation, rational conformal field theory, representation of braid groups, knot and link invariants, quantum topological field theory etc. Unfortunately, quantum groups and algebras appear as an artifact, and the efforts to give them a physical meaning or interpretation are far from being convincing [cf. e.g. 5].
TL;DR: Local Hopf algebras as discussed by the authors is a notion of Hopf co-algebra which is motivated by the introduction of an integrable dynamics in dimension d = 3, which leads to the notion of gauging of a quantum group.
TL;DR: In this paper, a finite-dimensional Hopf algebra over a field k, and a subgroup of the group of grouplikes of H is considered, and every left (H, kG )-Hopf module is free as a left kG -module.
TL;DR: In this article, a generalization of the mod [rgr] Steenrod algebra A ∗ to G -equivariant cohomology theory for a finite group G is presented.
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TL;DR: Rational conformal field theories can be interpreted as defining quasi-triangular Hopf algebras The Hopf algebra is determined by the duality properties of the conformal theory.
Abstract: Rational conformal field theories can be interpreted as defining quasi-triangular Hopf algebras The Hopf algebra is determined by the duality properties of the conformal theory
TL;DR: In this paper, the authors use the results known about the homology and cohomology of the symmetric groups and the operations they induce in mod-p cohology to write down two (additive) bases of the bigraded vector space of primitives of the above Hopf algebra.
Abstract: The bigraded group {H, (En, Z/p)} becomes a Hopf algebra, if multiplication is induced by restriction, and comultiplication is induced by transfer. Using Steenrod's method of considering elements of this bigraded group as mod-p cohomology operations, the primitives of this Hopf algebra correspond to additive cohomology operations. In this paper we use the results known about the homology and cohomology of the symmetric groups and the operations they induce in mod -p cohomology to write down two (additive) bases of the bigraded vector space of primitives of the above Hopf algebra.
01 Jan 1989
TL;DR: For finite dimensional Hopf-algebras with nilpotent augmentation ideal over the field of p elements, the upper and lower Loewy-series coincide as discussed by the authors.
Abstract: Abstract An easy proof will be given to show that for finite dimensional Hopf-algebras with nilpotent augmentation ideal over the field of p elements, the upper and lower Loewy-series coincide. In particular, this holds for the restricted universal envelope of nilpotent Lie-p-algebras with nilpotent p-map.
TL;DR: In this paper, the homology of infinite loop spaces has been studied in both the Dyer-Lashof algebra, R, and the opposite of the Steenrod algebra, A∗.
Abstract: This paper arises out of joint work with F. R. Cohen and F. P. Peterson [5, 2, 3] on the joint structure of infinite loop spaces QX. The homology of such a space is operated on by both the Dyer-Lashof algebra, R, and the opposite of the Steenrod algebra A∗. We describe a convenient summary of these actions; let M be the algebra which is R ⊗ A∗ as a vector space and where multiplication Q1 ⊗ PJ. Q1’ ⊗ PJ’ ∗ is given by applying the Nishida relations in the middle and then the appropriate Adem relations on the ends. Then M is a Hopf algebra which acts on the homology of infinite loop spaces.