Related papers: Free Braided Differential Calculus, Braided Binomi…
In braided tensor categories we show the Maschke's theorem and give the necessary and sufficient conditions for double cross biproducts and crossbiproducts and biproducts to be bialgebras. We obtain the factorization theorem for braided…
Quantum monodromy matrices coming from a theory of two coupled (m)KdV equations are modified in order to satisfy the usual Yang-Baxter relation. As a consequence, a general connection between braided and {\it unbraided} (usual) Yang-Baxter…
This paper explores of the role of unitary braiding operators in quantum computing. We show that a single specific solution R (the Bell basis change matrix) of the Yang-Baxter Equation is a universal gate for quantum computing, in the…
On any Reflection Equation algebra corresponding to a skew-invertible Hecke symmetry (i.e. a special type solution of the Quantum Yang-Baxter Equation) we define analogs of the partial derivatives. Together with elements of the initial…
We construct a noncommutative Cartan calculus on any braided commutative algebra and study its applications in noncommutative geometry. The braided Lie derivative, insertion and de Rham differential are introduced and related via graded…
In this paper we discuss a new type of 4-dimensional representation of the braid group. The matrices of braid operations are constructed by q-deformation of Hamiltonians. One is the Dirac Hamiltonian for free electron with mass m, the…
Braided algebras are algebraic structures consisting of an algebra endowed with a Yang-Baxter operator, satisfying some compatibility conditions.Yang-Baxter Hochschild cohomology was introduced by the authors to classify infinitesimal…
A generalization of the Yang-Baxter algebra is found in quantizing the monodromy matrix of two (m)KdV equations discretized on a space lattice. This braided Yang-Baxter equation still ensures that the transfer matrix generates operators in…
Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. They are a class of algebras with triangular decomposition, arising from a deformation problem, the…
A new approach is suggested to quantum differential calculus on certain quantum varieties. It consists in replacing quantum de Rham complexes with differentials satisfying Leibniz rule by those which are in a sense close to Koszul complexes…
The homotopy algebraic formalism of braided noncommutative field theory is used to define the explicit example of braided electrodynamics, that is, $\mathsf{U}(1)$ gauge theory minimally coupled to a Dirac fermion. We construct the braided…
In this paper we investigate a multi-parameter deformation $\mathfrak{B}_{r,s}^n(a,\lambda,\delta)$ of the walled Brauer algebra which was previously introduced by Leduc (\cite{leduc}). We construct an integral basis of…
In the framework of (vector valued) quantized holomorphic functions defined on non-commutative spaces, ``quantized hermitian symmetric spaces'', we analyze what the algebras of quantized differential operators with variable coefficients…
We introduce "noninvertible" generalization of statistics - semistatistics replacing condition when double exchanging gives identity to "regularity" condition. Then in categorical language we correspondingly generalize braidings and the…
We introduce an addition law for the usual quantum matrices $A(R)$ by means of a coaddition $\underline{\Delta} t=t\otimes 1+1\otimes t$. It supplements the usual comultiplication $\Delta t=t\otimes t$ and together they obey a…
In this paper, we first introduce a weighted derivation on algebras over an operad $\cal P$, and prove that for the free $\cal P$-algebra, its weighted derivation is determined by the restriction on the generators. As applications, we…
We construct a braiding operator in terms of the quantum dilogarithm function based on the quantum cluster algebra. We show that it is a q-deformation of the R-operator for which hyperbolic octrahedron is assigned. Also shown is that, by…
The main goal of this review is to compare different approaches to constructing geometry associated with a Hecke type braiding (in particular, with that related to the quantum group U_q(sl(n))). We make an emphasis on affine braided…
Braid theories are applied to quantum computation processes, where to each crossing in the Braid diagram a unitary Yang-Baxter operator R is associated, representing either a Braiding matrix or a universal quantum gate. By operating with…
In order to construct solutions of the braid equation we consider bijective left non-degenerate set-theoretic type solutions, which correspond to regular q-cycle coalgebras. We obtain a partial classification of the different q-cycle…