Noncocommutative Hopf Algebra Research Articles

Double Ore extensions of anti-angle type for Hopf algebras

<abstract><p>The aim of this article is to extend the structure of a bialgebra (Hopf algebra) which is connected graded as an algebra and generated in degree $ 1 $ to its double Ore extensions of anti-angle type. We construct two non-commutative and non-cocommutative Hopf algebras of infinite dimension and investigate the lifting of homological properties.</p></abstract>

Algebraic Structures on Typed Decorated Rooted Trees

Typed decorated trees are used by Bruned, Hairer and Zambotti to give a description of a renormalisation process on stochastic PDEs. We here study the algebraic structures on these objects: multiple pre-Lie algebras and related operads (generalizing a result by Chapoton and Livernet), noncommutative and cocommutative Hopf algebras (generalizing Grossman and Larson's construction), commutative and noncocommutative Hopf algebras (generalizing Connes and Kreimer's construction), bialgebras in cointeraction (generalizing Calaque, Ebrahimi-Fard and Manchon's result). We also define families of morphisms and in particular we prove that any Connes-Kreimer Hopf algebra of typed and decorated trees is isomorphic to a Connes-Kreimer Hopf algebra of non-typed and decorated trees (the set of decorations of vertices being bigger), through a contraction process, and finally obtain the Bruned-Hairer-Zambotti construction as a subquotient.

Hopf algebras arising from partial (co)actions

In this paper, extending the idea presented by Takeuchi in [M. Takeuchi, Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra 9 (1981) 841–882.] and more generally by Majid in [S. Majid, Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction, J. Algebra 130(1) (1990) 17–64.], we introduce the notion of partial matched pair [Formula: see text] involving the concepts of partial action and partial coaction between two bialgebras [Formula: see text] and [Formula: see text]. Furthermore, we present sufficient conditions for the corresponding bismash product [Formula: see text] to generate a new Hopf algebra and, as illustration, a family of examples is provided. Moreover, under some hypotheses such sufficient conditions are also necessary conditions.

Tensor Products and Support Varieties for Some Noncocommutative Hopf Algebras

We explore questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. We construct many classes of examples in which tensor powers of nonprojective modules are projective and tensor products of modules in one order are projective but in the other order are not. Our examples are smash coproducts with duals of group algebras, some having algebra and coalgebra structures twisted by cocycles. We apply support variety theory for these Hopf algebras as a tool in our investigations.

A combinatorial Hopf algebra for nonlinear output feedback control systems

In this work a combinatorial description is provided of a Faà di Bruno type Hopf algebra which naturally appears in the context of Fliess operators in nonlinear feedback control theory. It is a connected graded commutative and non-cocommutative Hopf algebra defined on rooted circle trees. A cancellation free forest formula for its antipode is given.

Modular quantizations of Lie algebras of Cartan type K via Drinfeld twists of Jordanian type

We construct explicit Drinfel'd twists of Jordanian type for the generalized Cartan type K Lie algebras in characteristic 0 and obtain the corresponding quantizations, especially their integral forms. By making modular reductions including modulo p and modulo p-restrictedness reduction, and base changes, we derive certain modular quantizations of the restricted universal enveloping algebra u(K(2n+1;1_)) for the restricted simple Lie algebra of Cartan type K in characteristic p. They are new families of noncommutative and noncocommutative Hopf algebras of dimension pp2n+1+1 (if 2n+4≢0(modp)) or pp2n+1 (if 2n+4≡0(modp)) over a truncated p-polynomials ring, which also contain the well-known Radford algebras (see [20]) as Hopf subalgebras. Some open questions are proposed.

Quantizations of the Extended Affine Lie Algebra $\widetilde{\frak{sl}_2(\mathbb{C}_q)}$

In this paper, the extended affine Lie algebra [Formula: see text] is quantized from three different points of view, which produces three non-commutative and non-cocommutative Hopf algebra structures, and yields other three quantizations by an isomorphism of [Formula: see text] correspondingly. Moreover, two of these quantizations can be restricted to the extended affine Lie algebra [Formula: see text].

The Hopf algebra of finite topologies and T-partitions

A noncommutative and noncocommutative Hopf algebra on finite topologies HT is introduced and studied (freeness, cofreeness, self-duality…). Generalizing Stanley's definition of P-partitions associated to a special poset, we define the notion of T-partitions associated to a finite topology, and deduce a Hopf algebra morphism from HT to the Hopf algebra of packed words WQSym. Generalizing Stanley's decomposition by linear extensions, we deduce a factorization of this morphism, which induces a combinatorial isomorphism from the shuffle product to the quasi-shuffle product of WQSym. It is strongly related to a partial order on packed words, here described and studied.

COASSOCIATIVE LIE ALGEBRAS

AbstractA coassociative Lie algebra is a Lie algebra equipped with a coassociative coalgebra structure satisfying a compatibility condition. The enveloping algebra of a coassociative Lie algebra can be viewed as a coalgebraic deformation of the usual universal enveloping algebra of a Lie algebra. This new enveloping algebra provides interesting examples of non-commutative and non-cocommutative Hopf algebras and leads to the classification of connected Hopf algebras of Gelfand–Kirillov dimension four in Wang et al. (Trans. Amer. Math. Soc., to appear).

TWISTED DRINFELD DOUBLES AND REPRESENTATIONS OF A HOPF ALGEBRA

In this paper, we consider a class of non-commutative and non-cocommutative Hopf algebras Hp(α, q, m) and then show that these Hopf algebras can be realized as a quantum double of certain Hopf algebras with respect to Hopf skew pairings (Ap(q, m), Bp(q, m), τα). Furthermore, using the Hopf skew pairing with appropriate group homomorphisms: ϕ : π → Aut (Ap(q, m)) and ψ : π → Aut (Bp(q, m)), we construct a twisted Drinfeld double D(Ap(q, m), Bp(q, m), τ; ϕ, ψ) which is a Turaev [Formula: see text]-coalgebra, where the group [Formula: see text] is a twisted semi-direct square of a group π. Then we obtain its quasi-triangular Turaev [Formula: see text]-coalgebra structure. We also study irreducible representations of Hp(1, q, m) and construct a corresponding R-matrix. Finally, we introduce the notion of a left Yetter–Drinfeld category over a Turaev group coalgebra and show that such a category is a Turaev braided group category by a direct proof, without center construction. As an application, we consider the case of the quasi-triangular Turaev [Formula: see text]-coalgebra structure on our twisted Drinfeld double.

Extensions of quantum enveloping algebras

In this paper, we define a class of extended quantum enveloping algebras Uq(f(K, J)) and some new Hopf algebras, which are certain extensions of quantum enveloping algebras by a Hopf algebra H. This construction generalizes some well-known extensions of quantum enveloping algebras by a Hopf algebra and provides a large of new noncommutative and noncocommutative Hopf algebras.

Graded Associative Conformal Algebras of Finite Type

In this paper, we consider graded associative conformal algebras. The class of these objects includes pseudo-algebras over non-cocommutative Hopf algebras of regular functions on some linear algebraic groups. In particular, an associative conformal algebra which is graded by a finite group $\Gamma $ is a pseudo-algebra over the coordinate Hopf algebra of a linear algebraic group $G$ such that the identity component $G^0$ is the affine line and $G/G^0\simeq \Gamma $. A classification of simple and semisimple graded associative conformal algebras of finite type is obtained.

QUANTUM ISOMETRY GROUPS OF SYMMETRIC GROUPS

We identify the quantum isometry groups of spectral triples built on the symmetric groups with length functions arising from the nearest-neighbor transpositions as generators. It turns out that they are isomorphic to certain "doubling" of the group algebras of the respective symmetric groups. We discuss the doubling procedure in the context of regular multiplier Hopf algebras. In the last section we study the dependence of the isometry group of Snon the choice of generators in the case n = 3. We show that two different choices of generators lead to nonisomorphic quantum isometry groups which exhaust the list of noncommutative noncocommutative semisimple Hopf algebras of dimension 12. This provides noncommutative geometric interpretation of these Hopf algebras.

Quantization of Schrödinger-Virasoro Lie algebra

In this paper, we use the general quantization method by Drinfel’d twists to quantize the Schrodinger-Virasoro Lie algebra whose Lie bialgebra structures were recently discovered by Han-Li-Su. We give two different kinds of Drinfel’d twists, which are then used to construct the corresponding Hopf algebraic structures. Our results extend the class of examples of noncommutative and noncocommutative Hopf algebras.

Some Graded Bialgebras Related to Borcherds Superalgebras

In present paper we define a new kind of weak quantized enveloping algebra of Borcherds superalgebras. We denote this algebra by $wU_{q}^{\tau}(\mathcal{G})$ . It is a noncommutative and noncocommutative weak graded Hopf algebra under some additional condition. It has a homomorphic image which is isomorphic to the usual quantum enveloping algebra $U_{q}(\mathcal{G})$ of $\mathcal{G}$ .

A Class of Weak Hopf Algebras

We introduce a class of noncommutative and noncocommutative weak Hopf algebras with infinite Ext quivers and study their structure. We decompose them into a direct sum of two algebras. The coalgebra structures of these weak Hopf algebras are described by their Ext quiver. The weak Hopf extension of Hopf algebra Hn has a quotient Hopf algebra and a sub‐Hopf algebra which are isomorphic to Hn.

EXTENDED QUANTUM ENVELOPING ALGEBRAS OF (2)

AbstractIn present paper we define a new kind of quantized enveloping algebra of (2). We denote this algebra by Ur,t, where r, t are two non-negative integers. It is a non-commutative and non-cocommutative Hopf algebra. If r = 0, then the algebra Ur,t is isomorphic to a tensor product of the algebra of infinite cyclic group and the usual quantum enveloping algebra of (2) as Hopf algebras. The representation of this algebra is studied.

Weak Quantum Enveloping Algebras of Borcherds Superalgebras

In present paper we define a new kind of weak quantized enveloping algebra \(wU_{q}^{\tau}({\mathcal{G}})\) of Borcherds superalgebras \({\mathcal{G}}\) . It is a noncommutative and noncocommutative weak graded Hopf algebra. Using localizing with some Ore set, we obtain a different kind of quantized enveloping algebras of Borcherds superalgebras \({\mathcal{U}}_{q}({\mathcal{G}})\) . It has a homomorphic image which is isomorphic to the usual quantum enveloping algebra \(U_{q}({\mathcal{G}})\) of \({\mathcal{G}}\) . Moreover, \({\mathcal{U}}_{q}({\mathcal{G}})\) is isomorphic to a direct sum of \(U_{q}({\mathcal{G}})\) and an other algebra as algebras.

Varieties for Modules of Quantum Elementary Abelian Groups

We define a rank variety for a module of a noncocommutative Hopf algebra $A = \Lambda \rtimes G$ where $\Lambda = k[X_1, \dots, X_m]/(X_1^{\ell}, \dots, X_m^{\ell}), G = (\mathbb{Z}/\ell\mathbb{Z})^m$ and char k does not divide l, in terms of certain subalgebras of A playing the role of “cyclic shifted subgroups”. We show that the rank variety of a finitely generated module M is homeomorphic to the support variety of M defined in terms of the action of the cohomology algebra of A. As an application we derive a theory of rank varieties for the algebra Λ. When l=2, rank varieties for Λ-modules were constructed by Erdmann and Holloway using the representation theory of the Clifford algebra. We show that the rank varieties we obtain for Λ-modules coincide with those of Erdmann and Holloway.

Quantum fields, nonlocality and quantum group symmetries

We study the action of space-time symmetries on quantum fields in the presence of small departures from locality determined by dynamical gravity. It is shown that, under such relaxation of locality the symmetries of the theory cannot be described within the usual framework of Lie algebras but rather in terms of noncocommutative Hopf algebras or 'quantum groups'. Similar 'quantizations' of space-time symmetries are expected to emerge in the low-energy limit of certain quantum gravity models and have been used to describe the symmetries of various noncommutative space-times. Our result provides an intuitive characterization of the mechanism that could lead to the emergence of deformed coproducts in models of quantum relativistic symmetries.