Related papers: Quantum Lorentz and braided Poincare groups
A quantum groups of type $A$ is defined in terms of a Hecke symmetry. We show in this paper that the representation category of such a quantum group is uniquely determined as an abelian braided monoidal category by the bi-rank of the Hecke…
We define a finite-dimensional cubic quotient of the group algebra of the braid group, endowed with a (essentially unique) Markov trace which affords the Links-Grould invariant of knots and links. We investigate several of its properties,…
Braided quantum field theories proposed by Oeckl can provide a framework for defining quantum field theories having Hopf algebra symmetries. In quantum field theories, symmetries lead to non-perturbative relations among correlation…
Quantum matrices $A(R)$ are known for every $R$ matrix obeying the Quantum Yang-Baxter Equations. It is also known that these act on `vectors' given by the corresponding Zamalodchikov algebra. We develop this interpretation in detail,…
We study braided Hochschild and cyclic homology of ribbon algebras in braided monoidal categories, as introduced by Baez and by Akrami and Majid. We compute this invariant for several examples coming from quantum groups and braided groups.
This is the first paper in a series where we generalize the Categorical Quantum Mechanics program (due to Abramsky, Coecke, et al) to braided systems. In our view a uniform description of quantum information for braided systems has not yet…
We construct a braid group action on quantum covering groups. We further use this action to construct a PBW basis for the positive half in finite type which is pairwise-orthogonal under the inner product. This braid group action is induced…
Quantum groups lead to an algebraic structure that can be realized on quantum spaces. These are noncommutative spaces that inherit a well defined mathematical structure from the quantum group symmetry. In turn such quantum spaces can be…
We investigate the braid group representations arising from categories of representations of twisted quantum doubles of finite groups. For these categories, we show that the resulting braid group representations always factor through finite…
We initiate a general approach to the relative braid group symmetries on (universal) $\imath$quantum groups, arising from quantum symmetric pairs of arbitrary finite types, and their modules. Our approach is built on new intertwining…
We describe geometrically the classical and quantum inhomogeneous groups $G_0=(SL(2, \BbbC)\triangleright \BbbC^2)$ and $G_1=(SL(2, \BbbC)\triangleright \BbbC^2)\triangleright \BbbC$ by studying explicitly their shape algebras as a spaces…
We introduce the notion of a braided Lie algebra consisting of a finite-dimensional vector space $\CL$ equipped with a bracket $[\ ,\ ]:\CL\tens\CL\to \CL$ and a Yang-Baxter operator $\Psi:\CL\tens\CL\to \CL\tens\CL$ obeying some axioms. We…
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…
Suppose $D$ is a finite dimensional C*-algebra carrying a continuous action $\overline{\Pi}$ of the circle group $\mathbb{T}$. We study the quantum symmetry group of $D$, taking $\overline{\Pi}$ into account. We show that they are braided…
It is shown that every quantum principal bundle is braided, in the sense that there exists an intrinsic braid operator twisting the functions on the bundle. A detailed algebraic analysis of this operator is performed. In particular, it…
We construct some braided quantum groups over the circle group. These are analogous to the free orthogonal quantum groups and generalise the braided quantum SU(2) groups for complex deformation parameter. We describe their irreducible…
We present a systematic introduction to the geometry of linear braided spaces. These are versions of $\R^n$ in which the coordinates $x_i$ have braid-statistics described by an R-matrix. From this starting point we survey the author's…
We construct a series of finite-dimensional quantum groups as braided Drinfeld doubles of Nichols algebras of type Super A, for an even root of unity, and classify ribbon structures for these quantum groups. Ribbon structures exist if and…
We extend the main result of [N. Andruskiewitsch and H.-J. Schneider, A characterization of quantum groups], see math/0201095, to braided vector spaces of generic diagonal type using results of Heckenberger.
We reconstruct a quantum group associated with any Lie algebra together with its representation theory from twisted homologies of generalized configuration spaces of disks. Along the way it brings new combinatorics to the theory, but our…