Braided Bialgebras Research Articles

Duality for infinite-dimensional braided bialgebras and their (co)modules

Duality for infinite-dimensional braided bialgebras and their (co)modules

Diagonal reduction algebra and the reflection equation

We describe the diagonal reduction algebra D(gl n ) of the Lie algebra gl n in the R-matrix formalism. As a byproduct we present two families of central elements and the braided bialgebra structure of D(gl n ).

The Fundamental Theorem for weak braided bimonads

The theories of (Hopf) bialgebras and weak (Hopf) bialgebras have been introduced for vector space categories over fields and make heavily use of the tensor product. As first generalisations, these notions were formulated for monoidal categories, with braidings if needed. The present authors developed a theory of bimonads and Hopf monads H on arbitrary categories A , employing distributive laws, allowing for a general form of the Fundamental Theorem for Hopf algebras. For τ -bimonads H , properties of braided (Hopf) bialgebras were captured by requiring a Yang–Baxter operator τ : H H → H H . The purpose of this paper is to extend the features of weak (Hopf) bialgebras to this general setting including an appropriate form of the Fundamental Theorem. This subsumes the theory of braided Hopf algebras (based on weak Yang–Baxter operators) as considered by Alonso Álvarez and others.

From Braided Infinitesimal Bialgebras to Braided Lie Bialgebras

The present paper is a continuation of [1], where we considered braided infinitesimal Hopf algebras (i.e., infinitesimal Hopf algebras in the Yetter-Drin feld category for any Hopf algebra H), and constructed their Drinfeld double as a generalization of Aguiar’s result. In this paper we mainly investigate the necessary and sufficient condition for a braided infinitesimal bialgebra to be a braided Lie bialgebra (i.e., a Lie bialgebra in the category ).

On rigidity of Nichols algebras

We study deformations of graded braided bialgebras using cohomological methods. In particular, we show that many examples of Nichols algebras, including the finite-dimensional ones arising in the Andruskiewitsch–Schneider program of classification of pointed Hopf algebras, are rigid. This result can be regarded as nonexistence of “braided Lie algebras” with nontrivial bracket.

Adjunctions and braided objects

In this paper, we investigate the categories of braided objects, algebras and bialgebras in a given monoidal category, some pairs of adjoint functors between them and their relations. In particular, we construct a braided primitive functor and its left adjoint, the braided tensor bialgebra functor, from the category of braided objects to the one of braided bialgebras. The latter is obtained by a specific elaborated construction introducing a braided tensor algebra functor as a left adjoint of the forgetful functor from the category of braided algebras to the one of braided objects. The behavior of these functors in the case when the base category is braided is also considered.

UNIVERSAL ENVELOPING ALGEBRAS OF PBW TYPE

AbstractWe continue our investigation of the general notion of universal enveloping algebra introduced in [A. Ardizzoni, A Milnor–Moore type theorem for primitively generated braided Bialgebras, J. Algebra 327(1) (2011), 337–365]. Namely, we study a universal enveloping algebra when it is of Poincaré–Birkhoff–Witt (PBW) type, meaning that a suitable PBW-type theorem holds. We discuss the problem of finding a basis for a universal enveloping algebra of PBW type: as an application, we recover the PBW basis both of an ordinary universal enveloping algebra and of a restricted enveloping algebra. We prove that a universal enveloping algebra is of PBW type if and only if it is cosymmetric. We characterise braided bialgebra liftings of Nichols algebras as universal enveloping algebras of PBW type.

Quadratic Lie Algebras

In this article, the notion of universal enveloping algebra introduced in Ardizzoni [4] is specialized to the case of braided vector spaces whose Nichols algebra is quadratic as an algebra. In this setting, a classification of universal enveloping algebras for braided vector spaces of dimension not greater than 2 is handled. As an application, we investigate the structure of primitively generated connected braided bialgebras whose braided vector space of primitive elements forms a Nichols algebra, which is a quadratic algebra.

Additive deformations of braided Hopf algebras

Additive deformations of bialgebras in the sense of J. Wirth, i.e. deformations of the multiplication map fulfilling a certain compatibility condition w.r.t. the coalgebra structure, can be generalized to braided bialgebras. The theorems for additive deformations of Hopf algebras can also be carried over to that case. We consider *-structures and prove a general Schoenberg correspondence in this context. Finally we give some examples.

On the combinatorial rank of a graded braided bialgebra

Let B be a graded braided bialgebra. Let S ( B ) denote the algebra obtained dividing out B by the two sided ideal generated by homogeneous primitive elements in B of degree at least two. We prove that S ( B ) is indeed a graded braided bialgebra quotient of B . It is then natural to compute S ( S ( B ) ) , S ( S ( S ( B ) ) ) and so on. This process yields a direct system whose direct limit comes out to be a graded braided bialgebra which is strongly N -graded as a coalgebra. Following V.K. Kharchenko, if the direct system is stationary exactly after n steps, we say that B has combinatorial rank n and we write κ ( B ) = n . We investigate conditions guaranteeing that κ ( B ) is finite. In particular, we focus on the case when B is the braided tensor algebra T ( V , c ) associated to a braided vector space ( V , c ) , providing meaningful examples such that κ ( T ( V , c ) ) ≤ 1 .

On Primitively Generated Braided Bialgebras

The main aim of this paper is to investigate the structure of primitively generated braided bialgebras A with respect to the braided vector space P consisting of their primitive elements. When the Nichols algebra of P is obtained dividing out the tensor algebra T(P) by the two-sided ideal generated by its primitive elements of degree at least two, we show that A can be recovered as a sort of universal enveloping algebra of P. One of the main applications of our construction is the description, in terms of universal enveloping algebras, of braided bialgebras whose associated graded coalgebra is a quadratic algebra.

A Milnor–Moore type theorem for primitively generated braided bialgebras

A braided bialgebra is called primitively generated if it is generated as an algebra by its space of primitive elements. We prove that any primitively generated braided bialgebra is isomorphic to the universal enveloping algebra of its infinitesimal braided Lie algebra, notions hereby introduced. This result can be regarded as a Milnor-Moore type theorem for primitively generated braided bialgebras and leads to the introduction of a concept of braided Lie algebra for an arbitrary braided vector space.

WEAK BRAIDED BIALGEBRAS AND WEAK ENTWINING STRUCTURES

AbstractIn this paper we clarify and improve the notion of weak braided bialgebra using weak entwining structures. As a main result we show that the notion of weak braided bialgebra can be rewritten in terms of weak entwining structures.

Representation theory of (modified) reflection equation algebra of $GL(m|n)$ type

Let R : V 2 → V 2 be a Hecke type solution of the quantum Yang-Baxter equation (a Hecke symmetry). Then, the Hilbert-Poincre series of the associated R-exterior algebra of the space V is a ratio of two polynomials of degree m (numerator) and n (denominator). Assuming R to be skew-invertible, we define a rigid quasitensor category SW(V(m|n)) of vector spaces, generated by the space V and its dual V � , and compute certain numerical char- acteristics of its objects. Besides, we introduce a braided bialgebra structure in the modified Reflection Equation Algebra, associated with R, and equip objects of the category SW(V(m|n)) with an action of this algebra. In the case related to the quantum group Uq(sl(m)), we con- sider the Poisson counterpart of the modified Reflection Equation Algebra and compute the semiclassical term of the pairing, defined via the categorical (or quantum) trace.

Braided bialgebras of Hecke-type

The paper is devoted to prove a version of Milnor–Moore Theorem for connected braided bialgebras that are infinitesimally cocommutative. Namely in characteristic different from 2, we prove that, for a given connected braided bialgebra ( A , c A ) which is infinitesimally λ -cocommutative for some element λ ≠ 0 that is not a root of one in the base field, then the infinitesimal braiding of A is of Hecke-type of mark λ and A is isomorphic as a braided bialgebra to the symmetric algebra of the braided subspace of its primitive elements.

Braided Bialgebras of Type One

Braided bialgebras of type one in abelian braided monoidal categories are characterized as braided graded bialgebras which are strongly ℕ-graded both as an algebra and as a coalgebra.

The braided structures for ω-smash coproduct Hopf algebras

Braided bialgebras were introduced by Larson and Towber by considering a bilinear form with certain properties. Here we study the braided structures of ω -smash coproduct Hopf algebras B ω ⋈ H as constructed by Caenepeel, Ion, Militaru and Zhu. Necessary and sufficient conditions for a class of ω -smash coproduct Hopf algebras to be braided Hopf algebras are given in terms of properties of their components. As applications of our results some special cases are discussed and explicit examples are given.

On a universal solution to the reflection equation

For a given quasi-triangular Hopf algebra \(\mathcal{H}\), we study relations between the braided group \(\tilde {\mathcal{H}}^* \) and Drinfeld's twist. We show that the braided bialgebra structure of \(\tilde {\mathcal{H}}^* \) is naturally described by means of twisted tensor powers of \(\mathcal{H}\) and their module algebras. We introduce a universal solution to the reflection equation (RE) and deduce a fusion prescription for RE-matrices.

A Pairing Theorem between a Braided Bialgebra and Its Dual Bialgebra

This paper constructs a dual bialgebra for the FRT-bialgebra related to a Hayashi's one parameter R -matrix by using the skew-derivation method. The natural pairing between them is non-degenerate.

Braided coaddition on the quantized braided matrices

An additive braided coproduct (also be called braided coaddition) is introduced on the quantized braided matrices (QBMs). It gives QBMs another braided-Hopf algebra structure. The coaddition is also shown to be compatible with the existing coproduct such that they together form a so-called quantized-braided ring. And some quantized braided differential operator bialgebras (Hopf algebras) relating to this braided coaddition are constructed. These give a unification and generalization of the known results about braided and quantum matrices. Moreover, the coaddition construction and the related differential calculi on the QBMs are further extended to a kind of more general quantum matrix algebraic systems. Some examples are also given.