Braided Vector Space Research Articles

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.

Endomorphism algebras and q-traces

For a braided vector space ( V , σ ) with braiding σ of Hecke type, we introduce three associative algebra structures on the space ⊕ p = 0 M End S σ p ( V ) of graded endomorphisms of the quantum symmetric algebra S σ ( V ) . We use the second product to construct a new trace. This trace is an algebra morphism with respect to the third product. In particular, when V is the fundamental representation of U q s l N + 1 and σ is the action of the R -matrix, this trace is a scalar multiple of the quantum trace of type A .

Quantum symmetric algebras II

M. Rosso has generalized G. Lusztig's construction of the Drinfeld–Jimbo quantum group [G. Lusztig, in: Progr. Math., Vol. 110, Birkhäuser, Boston, 1993], by associating to any (finite-dimensional) braided vector space ( V, τ) two “twisted” bialgebras: namely the “quantum shuffle algebra” T( V), and a subalgebra S( V) of T( V), called the “quantum symmetric algebra.” When τ is Lusztig's braiding, S( V) is isomorphic to the “plus part” U + of a certain quantum group U q ( g). Rosso [Invent. Math. 133 (1998) 399–416] generalizes this by using a braiding determined by a family of parameters t ij , and our paper [D. Flores, J.A. Green, Algebr. Represent. Theory 4 (2001) 55–76] was concerned with this case. In the present work we study twisted bialgebras in more general context, and get results on Rosso's S( V) for arbitrary braidings. It is useful to consider a pair ( T( U), T( V)) of twisted bialgebras which are adjoint under a given non-degenerate bilinear form. We give explicit formulae for the comultiplication of T( V), and also for its antipode. Under suitable conditions, S( V) is self-adjoint: this generalizes the part of Lusztig's construction which allows the “Drinfeld double” construction.

From racks to pointed Hopf algebras

A fundamental step in the classification of finite-dimensional complex pointed Hopf algebras is the determination of all finite-dimensional Nichols algebras of braided vector spaces arising from groups. The most important class of braided vector spaces arising from groups is the class of braided vector spaces ( CX,c q) , where X is a rack and q is a 2-cocycle on X with values in C × . Racks and cohomology of racks appeared also in the work of topologists. This leads us to the study of the structure of racks, their cohomology groups and the corresponding Nichols algebras. We will show advances in these three directions. We classify simple racks in group-theoretical terms; we describe projections of racks in terms of general cocycles; we introduce a general cohomology theory of racks containing properly the existing ones. We introduce a “Fourier transform” on racks of certain type; finally, we compute some new examples of finite-dimensional Nichols algebras.

QUANDLE KNOT INVARIANTS ARE QUANTUM KNOT INVARIANTS

We prove that the knot invariants given by quandles can be recovered by invariants given by traces on braided vector spaces, i.e. quantum invariants.

Integrals of Vertex Operators and uantum Shuffles

Given a braided vector space \(\left( {V,\sigma } \right)\) , we show that iterated integrals of operator-valued functions satisfying a certain exchange relation give rise to representations of the quantum shuffle algebra built on \(\left( {V,\sigma } \right)\). Using the quantum shuffle construction of the 'upper triangular part' \(U_q n_{\text{ + }}\) of a quantum shuffle, this provides a simple proof of the result of Bouwknegt, MacCarthy and Pilch saying that integrals of vertex operators acting on certain Fock modules give rise to representations of \(U_q n_{\text{ + }}\).

Q-epsilon tensor for quantum and braided spaces

The machinery of braided geometry introduced previously is used now to construct the ε ‘‘totally antisymmetric tensor’’ on a general braided vector space determined by R-matrices. This includes natural q-Euclidean and q-Minkowski spaces. The formalism is completely covariant under the corresponding quantum group such as SOq(4) or SOq(1,3). The Hodge * operator and differentials are also constructed in this approach.