The Dirac operator and gamma matrices for quantum Minkowski spaces

Gamma matrices for quantum Minkowski spaces are found. The invariance of the corresponding Dirac operator is proven. We introduce momenta for spin 1/2 particles and get (in certain cases) formal solutions of the Dirac equation.Comment: 25 pages, LaTeX fil

Quasitriangularity and enveloping algebras for inhomogeneous quantum groups

Coquasitriangular universal ${\cal R}$ matrices on quantum Lorentz and quantum Poincar\'e groups are classified. The results extend (under certain assumptions) to inhomogeneous quantum groups of [10]. Enveloping algebras on those objects are described.Comment: 18 pages, LaTeX file, minor change

Representations of Uh(su(N))U_h(su(N)) derived from quantum flag manifolds

A relationship between quantum flag and Grassmann manifolds is revealed. This enables a formal diagonalization of quantum positive matrices. The requirement that this diagonalization defines a homomorphism leads to a left \Uh -- module structure on the algebra generated by quantum antiholomorphic coordinate functions living on the flag manifold. The module is defined by prescribing the action on the unit and then extending it to all polynomials using a quantum version of Leibniz rule. Leibniz rule is shown to be induced by the dressing transformation. For discrete values of parameters occuring in the diagonalization one can extract finite-dimensional irreducible representations of \Uh as cyclic submodules.Comment: LaTeX file, JMP (to appear

Quantum Chains with GLq(2)GL_q(2) Symmetry

Usually quantum chains with quantum group symmetry are associated with representations of quantized universal algebras $U_q(g)$ . Here we propose a method for constructing quantum chains with $C_q(G)$ global symmetry , where $C_q(G)$ is the algebra of functions on the quantum group. In particular we will construct a quantum chain with $GL_q(2)$ symmetry which interpolates between two classical Ising chains.It is shown that the Hamiltonian of this chain satisfies in the generalised braid group algebra.Comment: 7 pages,latex,this is the completely revised version of my paper which is submitted for publicatio

Wigner-Eckart theorem for tensor operators of Hopf algebras

We prove Wigner-Eckart theorem for the irreducible tensor operators for arbitrary Hopf algebras, provided that tensor product of their irreducible representation is completely reducible. The proof is based on the properties of the irreducible representations of Hopf algebras, in particular on Schur lemma. Two classes of tensor operators for the Hopf algebra U$_{t}$(su(2)) are considered. The reduced matrix elements for the class of irreducible tensor operators are calculated. A construction of some elements of the center of U$_{t}$(su(2)) is given.Comment: 14 pages, late

Symplectic and orthogonal Lie algebra technology for bosonic and fermionic oscillator models of integrable systems

To provide tools, especially L-operators, for use in studies of rational Yang-Baxter algebras and quantum integrable models when the Lie algebras so(N) (b_n, d_n) or sp(2n) (c_n) are the invariance algebras of their R matrices, this paper develops a presentation of these Lie algebras convenient for the context, and derives many properties of the matrices of their defining representations and of the ad-invariant tensors that enter their multiplication laws. Metaplectic-type representations of sp(2n) and so(N) on bosonic and on fermionic Fock spaces respectively are constructed. Concise general expressions (see (5.2) and (5.5) below) for their L-operators are obtained, and used to derive simple formulas for the T operators of the rational RTT algebra of the associated integral systems, thereby enabling their efficient treatment by means of the algebraic Bethe ansatz.Comment: 24 pages, LaTe

The Gell-Mann-Okubo and Colemann-Glashow relations for octet and decuplet baryons in the SUq(3)SU_q(3) quantum algebra

The q-deformed Clebsch-Gordan coefficients corresponding to the \lrpy{3}\times\lrpy{21} reduction of the $SU_q(3)$ quantum algebra are computed. From these results and using the quantum Clebsch-Gordan coefficients for the \lrpy{21}\times\lrpy{21} reduction found by Zhong Qi Ma, the q-deformed Gell-Mann-Okubo mass relations for octet and decuplet baryons are determined by generalizing the procedure used for the SU(3) algebra. We also determine the Coleman-Glashow relations for octet and decuplet baryons in the $SU_q(3)$ algebra. Finally, by using the experimental particle masses of the octet and decuplet baryons, two values of the $q$-parameter are found and adjusted for the predicted masses expressions (one for the Gell-Mann-Okubo mass relations and the other for the Coleman-Glashow relations) and a possible physical interpretation is given.Comment: 19 pages, 4 figures. Corrected typo

Quantum Jacobi-Trudi and Giambelli Formulae for Uq(Br(1))U_q(B_r^{(1)}) from Analytic Bethe Ansatz

Analytic Bethe ansatz is executed for a wide class of finite dimensional $U_q(B^{(1)}_r)$ modules. They are labeled by skew-Young diagrams which, in general, contain a fragment corresponding to the spin representation. For the transfer matrix spectra of the relevant vertex models, we establish a number of formulae, which are $U_q(B^{(1)}_r)$ analogues of the classical ones due to Jacobi-Trudi and Giambelli on Schur functions. They yield a full solution to the previously proposed functional relation ($T$-system), which is a Toda equationComment: Plain Tex(macro included), 18 pages. 7 figures are compressed and attache

Representations of Super Yangian

We present in detail the classification of the finite dimensional irreducible representations of the super Yangian associated with the Lie superalgebra $gl(1|1)$.Comment: 14 pages, plain latex, no figur