Related papers: Null-plane Quantum Universal $R$-matrix
Using Cartan's equivalence method for point transformations we obtain from first principles the conformal geometry associated with third order ODEs and a special class of PDEs in two dimensions. We explicitly construct the null tetrads of a…
The FRT quantum group and space theory is reformulated from the standard mathematical basis to an arbitrary one. The $N$-dimensional quantum vector Cayley-Klein spaces are described in Cartesian basis and the quantum analogs of…
In this paper, using a Hopf-algebraic method, we construct deformed Poincar\'e SUSY algebra in terms of twisted (Hopf) algebra. By adapting this twist deformed super-Poincar\'e algrebra as our fundamental symmetry, we can see the…
We consider two different types of deformations for the linear group $ GL(n)$ which correspond to using of a general diagonal R-matrix. Relations between braided and quantum deformed algebras and their coactions on a quantum plane are…
Lead by examples we introduce the notions of Hopf algebra and quantum group. We study their geometry and in particular their Lie algebra (of left invariant vectorfields). The examples of the quantum sl(2) Lie algebra and of the quantum…
Star products on the classical double group of a simple Lie group and on corresponding symplectic grupoids are given so that the quantum double and the "quantized tangent bundle" are obtained in the deformation description. "Complex"…
A review of the multiparametric linear quantum group GL_qr(N), its real forms, its dual algebra U(gl_qr(N)) and its bicovariant differential calculus is given in the first part of the paper. We then construct the (multiparametric) linear…
Smooth threefold flops with irreducible centres are classified by the length invariant, which takes values 1, 2, 3, 4, 5 or 6. This classification by Katz and Morrison identifies 6 possible partial resolutions of Kleinian singularities that…
Classical r-matrices of the three-dimensional real Lie bialgebras are obtained. In this way all three-dimensional real coboundary Lie bialgebras and their types (triangular, quasitriangular or factorizable) are classified. Then, by using…
For each simple Lie algebra $\mathfrak{g}$, we construct an algebra embedding of the quantum group $U_q(\mathfrak{g})$ into certain quantum torus algebra $D_\mathfrak{g}$ via the positive representations of split real quantum group. The…
We categorify a class of quantum groups associated with quivers, possibly with loops, by constructing the corresponding Khovanov-Lauda-Rouquier algebras (KLR) algebras $R$. We prove that the indecomposable projective $R$-modules realize the…
In these notes, we introduce formal hom-associative deformations of the quantum planes and the universal enveloping algebras of the two-dimensional non-abelian Lie algebras. We then show that these deformations induce formal hom-Lie…
We consider the deformation of the Poincar\'e group in 2+1 dimensions into the quantum double of the Lorentz group and construct Lorentz-covariant momentum-space formulations of the irreducible representations describing massive particles…
We consider fundamental facts from the theory of Hopf superalgebras. We use them to construct the quantum double of the quantum superalgebra $sl(2|1)$ at roots of unity. Thus we obtain a multiplicative formula for universal $R$-matrix. Next…
We consider two new classes of twisted D=4 quantum Poincar\'{e} symmetries described as the dual pairs of noncocommutative Hopf algebras. Firstly we investigate a two-parameter class of twisted Poincar\'{e} algebras which provide the…
Quantum deformations of (anti-)de Sitter algebras in (2+1) dimensions are revisited, and several features of these quantum structures are reviewed. In particular, the classification problem of (2+1) (A)dS Lie bialgebras is presented and the…
The quantum Fourier transform (QFT) is a powerful tool in quantum computing. The main ingredients of QFT are formed by the Walsh-Hadamard transform H and phase shifts P(.), both of which are 2x2 unitary matrices as operators on the…
Given any pair of positive integers m and n, we construct a new Hopf algebra, which may be regarded as a degenerate version of the quantum group of gl(m+n). We study its structure and develop a highest weight representation theory. The…
One particular approach to quantum groups (matrix pseudo groups) provides the Manin quantum plane. Assuming an appropriate set of non-commuting variables spanning linearly a representation space one is able to show that the endomorphisms on…
The Heisenberg double of a Hopf algebra may be regarded as a quantum analogue of the cotangent bundle of a Lie group. Quantum duality principle describes relations between a Hopf algebra, its dual, and their Heisenberg double in a way which…