Cocommutative Hopf Algebra Research Articles

Rota-Baxter operators on cocommutative Hopf algebras

In this paper, we introduce and study the notion of a Rota-Baxter operator on a cocommutative Hopf algebra that generalizes notions of a Rota-Baxter operator on a group and a Rota-Baxter operator of weight 1 on a Lie algebra. We show that every Rota-Baxter operator (of weight 1) on a Lie algebra g (resp., on a group G) can be uniquely extended to a Rota-Baxter operator on the universal enveloping algebra U(g) (resp., on the group algebra F[G]).

Algebraic properties of quantum quasigroups

Quantum quasigroups provide a self-dual framework for the unification of quasigroups and Hopf algebras . This paper furthers the transfer program, investigating extensions to quantum quasigroups of various algebraic features of quasigroups and Hopf algebras. Part of the difficulty of the transfer program is the fact that there is no standard model-theoretic procedure for accommodating the coalgebraic aspects of quantum quasigroups. The linear quantum quasigroups, which live in categories of modules under the direct sum, are a notable exception. They form one of the central themes of the paper. From the theory of Hopf algebras, we transfer the study of grouplike and setlike elements, which form separate concepts in quantum quasigroups. From quasigroups, we transfer the study of conjugate quasigroups, which reflect the triality symmetry of the language of quasigroups. In particular, we construct conjugates of cocommutative Hopf algebras. Semisymmetry, Mendelsohn , and distributivity properties are formulated for quantum quasigroups. We classify distributive linear quantum quasigroups that furnish solutions to the quantum Yang-Baxter equation. The transfer of semisymmetry is designed to prepare for a quantization of web geometry.

On the First Hochschild Cohomology of Cocommutative Hopf Algebras of Finite Representation Type

Abstract Let $\mathscr{B}_0({\mathcal{G}})\subseteq k\,{\mathcal{G}}$ be the principal block algebra of the group algebra $k\,{\mathcal{G}}$ of an infinitesimal group scheme ${\mathcal{G}}$ over an algebraically closed field $k$ of characteristic ${\operatorname{char}}(k)=:p\geq 3$. We calculate the restricted Lie algebra structure of the first Hochschild cohomology ${\mathcal{L}}:={\operatorname{H}}^1(\mathscr{B}_0({\mathcal{G}}),\mathscr{B}_0({\mathcal{G}}))$ whenever $\mathscr{B}_0({\mathcal{G}})$ has finite representation type. As a consequence, we prove that the complexity of the trivial ${\mathcal{G}}$-module $k$ coincides with the maximal toral rank of ${\mathcal{L}}$.

Brauer–Clifford Group of ($$\varvec{S,{{\mathcal {G}}},H}$$)-Azumaya Algebras

In this paper we extend the notion of the Brauer–Clifford group to the case of an $$(S,{{\mathcal {G}}},H)$$ -algebra, when H is a cocommutative Hopf algebra, $${{\mathcal {G}}}$$ is a Lie algebra in the symmetric monoidal category of left H-modules, and S is a commutative algebra which is an H-module algebra, a $${\mathcal G}$$ -module algebra and the H-action is compatible with the $${\mathcal {G}}$$ -action. This Brauer–Clifford group turns out to be an example of the Brauer group of a symmetric monoidal category.

Brauer–Clifford group of Azumaya–Poisson (S, H)-algebras

In this paper we extend the notion of the Brauer–Clifford group to the case of an Azumaya–Poisson (S, H)-algebra, when H is a cocommutative Hopf algebra and S is an H-module Poisson algebra. This is the situation that arises in applications with connections to differential geometry, algebraic geometry and quantum groups. These Brauer–Clifford groups turn out to be an example of the Brauer group of a symmetric monoidal category.

Cohomology Operations andπ-Strongly Homotopy Commutative Hopf Algebra

Steenrod operations are cohomology operations that are themselves natural transformations between cohomology functors. There are two distinct types of steenrod operations initially constructed by Norman Steenrod and called Steenrod squares and reduced p-th power operations usually denoted <i>Sq</i> and <i>pⁱ</i> respectively. Since their creation, it has been proved that these operations can be constructed in the cohomology of many algebraic structures, for instance in the cohomology of simplicial restricted Lie algebras, the cohomology of cocommutative Hopf algebras and the homology of infinite loop space. Later on J. P. May developped a general algebraic setting in which all the above cases can be studied. In this work we consider a cyclic group π of oder a fixed prime p and combine theπ-strongly homotopy commutative Hopf algebra structure to the May’s approach with the aim to build these natural transformations on the Hochschild cohomology groups. Moreover we give under some conditions a link of these natural transformations with the Gerstenhaber algebra structure.

Actions of cocommutative Hopf algebras

Let H be a cocommutative Hopf algebra acting on an algebra A. Assuming the base field to be algebraically closed and the H-action on A to be integral, that is, it is given by a coaction of some Hopf subalgebra of the finite dual H∘ that is an integral domain, we stratify the prime spectrum Spec A in terms of the prime spectra of certain commutative algebras. For arbitrary H-actions in characteristic 0, we show that the largest H-stable ideal of A that is contained in a given semiprime ideal of A is semiprime as well.

The functor of singular chains detects weak homotopy equivalences

The normalized singular chains of a path connected pointed space X X may be considered as a connected E ∞ E_{\infty } -coalgebra C ∗ ( X ) \mathbf {C}_*(X) with the property that the 0 0 th homology of its cobar construction, which is naturally a cocommutative bialgebra, has an antipode; i.e., it is a cocommutative Hopf algebra. We prove that a continuous map of path connected pointed spaces f : X → Y f: X\to Y is a weak homotopy equivalence if and only if C ∗ ( f ) : C ∗ ( X ) → C ∗ ( Y ) \mathbf {C}_*(f): \mathbf {C}_*(X)\to \mathbf {C}_*(Y) is an Ω \mathbf {\Omega } -quasi-isomorphism, i.e., a quasi-isomorphism of dg algebras after applying the cobar functor Ω \mathbf {\Omega } to the underlying dg coassociative coalgebras. The proof is based on combining a classical theorem of Whitehead together with the observation that the fundamental group functor and the data of a local system over a space may be described functorially from the algebraic structure of the singular chains.

Hopf Ore Extensions

Brown, O'Hagan, Zhang, and Zhuang gave a set of conditions on an automorphism $\sigma$ and a $\sigma$-derivation $\delta$ of a Hopf $k$-algebra $R$ for when the skew polynomial extension $T=R[x, \sigma, \delta]$ of $R$ admits a Hopf algebra structure that is compatible with that of $R$. In fact, they gave a complete characterization of which $\sigma$ and $\delta$ can occur under the hypothesis that $\Delta(x)=a\otimes x +x\otimes b +v(x\otimes x) +w$, with $a, b\in R$ and $v, w\in R\otimes_k R$, where $\Delta: R\to R\otimes_k R$ is the comultiplication map. In this paper, we show that after a change of variables one can in fact assume that $\Delta(x)=\beta^{-1}\otimes x +x\otimes 1 +w$, with $\beta $ is a grouplike element in $R$ and $w\in R\otimes_k R,$ when $R\otimes_k R$ is a domain and $R$ is noetherian. In particular, this completely characterizes skew polynomial extensions of a Hopf algebra that admit a Hopf structure extending that of the ring of coefficients under these hypotheses. We show that the hypotheses hold for domains $R$ that are noetherian cocommutative Hopf algebras of finite Gelfand-Kirillov dimension.

Hall's criterion for nilpotence in semi-abelian categories

A well-known theorem of P. Hall, usually called Hall's criterion for nilpotence, states: a group G is nilpotent whenever it has a normal subgroup N such that G/[N,N] and N are nilpotent. We widely generalize this result, replacing groups with objects in an abstract semi-abelian category satisfying suitable conditions. In particular, these conditions are satisfied in any algebraically coherent semi-abelian category, and hence in many categories of classical algebraic structures (including a few where Hall's theorem is already known). Note that the categories of crossed modules, crossed Lie algebras, n-cat groups, and cocommutative Hopf algebras over fields are also algebraically coherent. We give, however, a counter-example showing that Hall's theorem does not hold in the (semi-abelian) variety of non-associative rings.

Cohomology for partial actions of Hopf algebras

In this work, the cohomology theory for partial actions of co-commutative Hopf algebras on commutative algebras is formulated. This theory generalizes the cohomology theory for Hopf algebras introduced by Sweedler and the cohomology theory for partial group actions, introduced by Dokuchaev and Khrypchenko. Some nontrivial examples, not coming from groups are constructed. Given a partial action of a co-commutative Hopf algebra H over a commutative algebra A, we prove that there exists a new Hopf algebra A˜, over a commutative ring E(A), upon which H still acts partially and which gives rise to the same cohomologies as the original algebra A. We also study the partially cleft extensions of commutative algebras by partial actions of cocommutative Hopf algebras and prove that these partially cleft extensions can be viewed as a cleft extensions by Hopf algebroids.

Hopf algebra structure of symmetric and quasisymmetric functions in superspace

We show that the ring of symmetric functions in superspace is a cocommutative and self-dual Hopf algebra. We provide formulas for the action of the coproduct and the antipode on various bases of that ring. We introduce the ring sQSym of quasisymmetric functions in superspace and show that it is a Hopf algebra. We give explicitly the product, coproduct and antipode on the basis of monomial quasisymmetric functions in superspace. We prove that the Hopf dual of sQSym, the ring sNSym of noncommutative symmetric functions in superspace, has a multiplicative basis dual to the monomial quasisymmetric functions in superspace.

Barbasch-Sahi algebras and Dirac cohomology

We define a class of algebras which are distinguished by a PBW property and an orthogonality condition, and which we call Hopf-Hecke algebras, since they generalize the Drinfeld Hecke algebras defined by Drinfeld. In the course of studying the orthogonality condition and in analogy to the orthogonal group we show the existence of a pin cover for cocommutative Hopf algebras over $\mathbb{C}$ with an orthogonal module or, more generally, pointed cocommutative Hopf algebras over a field of characteristic $0$ with an orthogonal module. Following the suggestion of Dan Barbasch and Siddhartha Sahi, we define a Dirac operator and Dirac cohomology for modules of Hopf-Hecke algebras, generalizing those concepts for connected semisimple Lie groups, graded affine Hecke algebras and symplectic reflection algebras. Using the pin cover, we prove a general theorem for a class of Hopf-Hecke algebras which we call Barbasch-Sahi algebras, which relates the central character of an irreducible module with non-vanishing Dirac cohomology to the central characters occurring in its Dirac cohomology, generalizing a result called Vogan's conjecture for connected semisimple Lie groups, analogous results for graded affine Hecke algebras and for symplectic reflection algebras and Drinfeld Hecke algebras.

The Brauer-Clifford group for locally finite (S, H)-Azumaya algebras

In this article, we define the notion of Brauer-Clifford group for H-locally finite (S, H)-Azumaya algebras, when H is a cocommutative Hopf algebra and S is an H-locally finite commutative H-module algebra over a commutative noetherian ring k. This is the situation that arises in applications with connections to algebraic geometry. This Brauer-Clifford group turns out to be an example of a Brauer group of a symmetric monoidal category.

Rota-Baxter H-operators and pre-Lie H-pseudoalgebras over a cocommutative Hopf algebra H

ABSTRACT The aim of this paper is to study the Rota-Baxter H-operators and H-pseudoalgebras of different types over a cocommutative Hopf algebra H. Firstly, we introduce the concept of a Rota-Baxter H-operator on an H-pseudoalgebra, and give some basic properties and examples. Then, we obtain a large number of pre-Lie (resp. associative) H-pseudoalgebras from the ordinary Rota-Baxter algebras. Finally, the annihilation algebras of the left pre-Lie H-pseudoalgebras are discussed.

A semi-abelian extension of a theorem by Takeuchi

We prove that the category of cocommutative Hopf algebras over a field is a semi-abelian category. This result extends a previous special case of it, based on the Milnor–Moore theorem, where the field was assumed to have zero characteristic. Takeuchi's theorem asserting that the category of commutative and cocommutative Hopf algebras over a field is abelian immediately follows from this new observation. We also prove that the category of cocommutative Hopf algebras over a field is action representable. We make some new observations concerning the categorical commutator of normal Hopf subalgebras, and this leads to the proof that two definitions of crossed modules of cocommutative Hopf algebras are equivalent in this context.

Enriched model categories in equivariant contexts

We give a general framework of equivariant model category theory. Our groups G, called Hopf groups, are suitably defined group objects in any well-behaved symmetric monoidal category V. For any V, a discrete group G gives a Hopf group, denoted I[G]. When V is cartesian monoidal, the Hopf groups are just the group objects in V. When V is the category of modules over a commutative ring R, I[G] is the group ring R[G] and the general Hopf groups are the cocommutative Hopf algebras over R. We show how all of the usual constructs of equivariant homotopy theory, both categorical and model theoretic, generalize to Hopf groups for any V. This opens up some quite elementary unexplored mathematical territory, while systematizing more familiar terrain.

Split extension classifiers in the category of cocommutative Hopf algebras

We describe the split extension classifiers in the semi-abelian category of cocommutative Hopf algebras over an algebraically closed field of characteristic zero. The categorical notions of centralizer and of center in the category of cocommutative Hopf algebras is then explored. We show that the categorical notion of center coincides with the one that is considered in the theory of general Hopf algebras.

Brackets in representation algebras of Hopf algebras

For any graded bialgebras A and B , we define a commutative graded algebra A_B representing the functor of B -representations of A . When A is a cocommutative graded Hopf algebra and B is a commutative ungraded Hopf algebra, we introduce a method deriving a Gerstenhaber bracket in A_B from a Fox pairing in A and a balanced biderivation in B . Our construction is inspired by Van den Bergh’s non-commutative Poisson geometry, and may be viewed as an algebraic generalization of the Atiyah–Bott–Goldman Poisson structures on moduli spaces of representations of surface groups.

The Hopf algebra of skew shapes, torsion sheaves on [formula omitted], and ideals in Hall algebras of monoid representations

We study ideals in Hall algebras of monoid representations on pointed sets corresponding to certain conditions on the representations. These conditions include the property that the monoid act via partial permutations, that the representation possess a compatible grading, and conditions on the support of the module. Quotients by these ideals lead to combinatorial Hopf algebras which can be interpreted as Hall algebras of certain sub-categories of modules. In the case of the free commutative monoid on n generators, we obtain a co-commutative Hopf algebra structure on n-dimensional skew shapes, whose underlying associative product amounts to a “stacking” operation on the skew shapes. The primitive elements of this Hopf algebra correspond to connected skew shapes, and form a graded Lie algebra by anti-symmetrizing the associative product. We interpret this Hopf algebra as the Hall algebra of a certain category of coherent torsion sheaves on A/F1n supported at the origin, where F1 denotes the field of one element. This Hopf algebra may be viewed as an n-dimensional generalization of the Hopf algebra of symmetric functions, which corresponds to the case n=1.