Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group (original) (raw)

Journal of Geometry and Physics, 1994

We show that such an object has an enveloping braided-bialgebra U (L). We show that every generic R-matrix leads to such a braided Lie algebra with [ , ] given by structure constants c IJ K determined from R. In this case U (L) = B(R) the braided matrices introduced previously. We also introduce the basic theory of these braided Lie algebras, including the natural right-regular action of a braided-Lie algebra L by braided vector fields, the braided-Killing form and the quadratic Casimir associated to L. These constructions recover the relevant notions for usual, colour and super-Lie algebras as special cases. In addition, the standard quantum deformations U q (g) are understood as the enveloping algebras of such underlying braided Lie algebras with [ , ] on L ⊂ U q (g) given by the quantum adjoint action.

A new quantum group associated with a ?nonstandard? braid group representation

Letters in Mathematical Physics, 1991

A new quantum group is derived from a 'nonstandard' braid group representation by employing the Faddeev-Reshetlkhin-Takhtajan constructive method. The classical limit is not a Lie superalgebra, despite relations like .,c 2 = )'~ = 0. We classify all finite-dimensional irreducible representations of the new Hopf algebra and find only one-and two-dimensional ones.

Beyond supersymmetry and quantum symmetry (an introduction to braided groups and braided matrices)

1992

This is a systematic introduction for physicists to the theory of algebras and groups with braid statistics, as developed over the last three years by the author. There are braided lines, braided planes, braided matrices and braided groups all in analogy with superlines, superplanes etc. The main idea is that the bose-fermi ±1 statistics between Grassmannn coordinates is now replaced by a general braid statistics Ψ, typically given by a Yang-Baxter matrix R. Most of the algebraic proofs are best done by drawing knot and tangle diagrams, yet most constructions in supersymmetry appear to generalise well. Particles of braid statistics exist and can be expected to be described in this way. At the same time, we find many applications to ordinary quantum group theory: how to make quantum-group covariant (braided) tensor products and spin chains, action-angle variables for quantum groups, vector addition on q-Minkowski space and a semidirect product q-Poincaré group are among the main applications so far. Every quantum group can be viewed as a braided group, so the theory contains quantum group theory as well as supersymmetry. There also appears to be a rich theory of braided geometry, more general than supergeometry and including aspects of quantum geometry. Braided-derivations obey a braided-Leibniz rule and recover the usual Jackson q-derivative as the 1-dimensional case.

Beyond Supersymmetry and Quantum Symmetry (An Introduction to Braided-Groups and Braided-Matrices) 1

1992

Examples of quantum braided groups

1993

summary:Summary: The author gives the defining relations of a new type of bialgebras that generalize both the quantum groups and braided groups as well as the quantum supergroups. The relations of the algebras are determined by a pair of matrices (R,Z)(R, Z)(R,Z) that solve a system of Yang-Baxter-type equations. The matrix coproduct and counit are of standard matrix form, however, the multiplication in the tensor product of the algebra is defined by virtue of the braiding map given by the matrix ZZZ. Besides simple solutions of the system of the Yang-Baxter-type equations that generate either quantum groups or braided groups, we have found several solutions that generate genuine quantum braided groups that by a choice of parameters give quantum or braided groups as a special case

Braided groups and algebraic quantum field theories

Letters in Mathematical Physics, 1991

We introduce the notion of a braided group. This is analogous to a supergroup with Bose-Fermi statistics __+ l replaced by braid statistics. We show that every algebraic quantum field theory in two dimensions leads to a braided group of internal symmetries. Every quantum group can be viewed as a braided group.

Quantum groups and cylinder braiding

Forum Mathematicum, 1998

The purpose of this paper is to introduce a new structure into the representation theory of quantum groups. The structure is motivated by braid and knot theory. Represen¬ tations of quantum groups associated to classical Lie algebras have an additional symmetry which cannot be seen in the classical limit. We first explain the general formalism of these symmetries (called cylinder forms) in the context of comodules. Basic ingredients are tensor representations of braid groups of type B derived from standard R-matrices associated to socalled four braid pairs. These are applied to the Faddeev-Reshetikhin-Takhtadjian construc¬ tion of bialgebras from R-matrices. As a consequence one obtains four braid pairs on all representations of the quantum group. In the second part of the paper we study in detail the dual situation of modules over the quantum enveloping algebra Uq(sl2). The main result here is the computation of the universal cylinder twist.

Quantized braided groups

Journal of Mathematical Physics, 1994

Algebras that represent a generalization of both quantum groups, quantum supergroups, and braided groups are defined. They are given by a pair of solutions of the Yang–Baxter equation that satisfy some additional conditions. Several examples are presented.

Modified braid equations, Baxterizations and noncommutative spaces for the quantum groups GL sub q, SOsub q, and Spsub q

Journal of Mathematical Physics, 2003

Modified braid equations satisfied by generalizedR matrices ( for a given set of group relations obeyed by the elements of T matrices ) are constructed for q-deformed quantum groups GL q (N ), SO q (N ) and Sp q (N ) with arbitrary values of N . The Baxterization ofR matrices, treated as an aspect complementary to the modification of the braid equation, is obtained for all these cases in particularly elegant forms. A new class of braid matrices is discovered for the quantum groups SO q (N ) and Sp q (N ). TheR matrices of this class, while being distinct from restrictions of the universalR matrix to the corresponding vector representations, satisfy the standard braid equation. The modified braid equation and the Baxterization are obtained for this new class ofR matrices. Diagonalization of the generalizedR matrices is studied. The diagonalizers are obtained explicitly for some lower dimensional cases in a convenient way, giving directly the eigenvalues of the correspondingR matrices. Applications of such diagonalization are then studied in the context of associated covariantly quantized noncommutative spaces. *

Braid representations from quantum groups of exceptional Lie type

2010

We study the problem of determining if the braid group representations obtained from quantum groups of types E,FE, FE,F and GGG at roots of unity have infinite image or not. In particular we show that when the fusion categories associated with these quantum groups are not weakly integral, the braid group images are infinite. This provides further evidence for a

Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group (original) (raw)

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