相关论文: Realization of Vector fields for Quantum Groups as…
We show that a quantum deformation of quantum mechanics given in a previous work is equivalent to quantum mechanics on a nonlinear lattice with step size $\Delta x=~(1-q)x$. Then, based on this, we develop the basic formalism of quantum…
Quantum mechanical operators and quantum fields are interpreted as realizations of timespace manifolds. Such causal manifolds are parametrized by the classes of the positive unitary operations in all complex operations, i.e. by the…
The representation theory of the quantum group su$_q(2)$ is used to introduce $q$-analogues of the Wigner rotation matrices, spherical functions, and Legendre polynomials. The method amounts to an extension of variable separation from…
Suppose q is a complex number of modulus one and different from 1,-1. Let O(R^2_q) be the *-algebra with two hermitean generators x and y satisfying the relation xy=qyx. Using operator representations of the *-algebra O(R^2_q) on Hilbert…
We start with the observation that the quantum group SL_q(2), described in terms of its algebra of functions has a quantum subgroup, which is just a usual Cartan group. Based on this observation we develop a general method of constructing…
The q-deformed traces and orbits for the two parametric quantum groups $GL_{qp}(2)$ and $GL_{qp}(1|1)$ are defined. They are subsequently used in the construction of $q$-orbit invariants for these groups. General $qp$-(super)oscillator…
We construct differential calculi on multiparametric quantum orthogonal planes in any dimension N. These calculi are bicovariant under the action of the full inhomogeneous (multiparametric) quantum group ISO_{q,r}(N), and do contain…
A brief proof of Lie's classification of solvable algebras of vector fields on the plane is given. The proof uses basic representation theory and PDEs.
The definitions and some properties (e.g. the Wigner-Eckart theorem, the fusion procedure) of covariant and contravariant q-tensor operators for quasitriangular quantum Lie algebras are formulated in the R-matrix language. The case of…
We first give a pedagogical introduction to the differential calculus on q-groups and analize the relation between differential calculus and q-Lie algebra. Equivalent definitions of bicovariant differential calculus are studied and their…
For a quasi-split Satake diagram, we define a modified $q$-Weyl algebra, and show that there is an algebra homomorphism between it and the corresponding $\imath$quantum group. In other words, we provide a differential operator approach to…
The representations of dimension vector $\alpha$ of the quiver Q can be parametrised by a vector space $R(Q,\alpha)$ on which an algebraic group $\Gl(\alpha)$ acts so that the set of orbits is bijective with the set of isomorphism classes…
A class of quantum field theories invariant with respect to the action of an odd vector field Q on a source supermanifold $\Sigma$ is considered. We suppose that Q satisfies the conditions under which an integral of any Q-invariant function…
Let $M$ be a smooth manifold, $\cal S$ the space of polynomial on fibers functions on $T^*M$ (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, $Vect(M)$, of vector fields on $M$ with…
Using the corepresentation of the quantum group $ SL_q(2)$ a general method for constructing noncommutative spaces covariant under its coaction is developed. The method allows us to treat the quantum plane and Podle\'s' quantum spheres in a…
Quantum algebras U_q(su_n) used as the algebras of flavour symmetry (usually described by SU(n)) to study static properties of hadrons lead to intriguing results. In this contribution we focus on the peculiar properties manifested by…
We classify the compact quantum groups acting on 4 points. These are the quantum subgroups of the quantum permutation group $\mathcal Q_4$. Our main tool is a new presentation for the algebra $\rm C(\mathcal Q_4)$, corresponding to an…
We extend the Ruzhansky-Turunen theory of pseudo differential operators on compact Lie groups into a tool that can be used to investigate group-valued Markov processes in the spirit of the work in Euclidean spaces of N.Jacob and…
This article surveys the application of the representation theory of loop groups to simple models in quantum field theory and to certain integrable systems. The common thread in the discussion is the construction of quantum fields using…
Three-dimensional bicovariant differential calculus on the quantum group SU_q(2) is constructed using the approach based on global covariance under the action of the stabilizing subgroup U(1). Explicit representations of possible q-deformed…