[en] Shift operators are constructed in product form for the Lie algebra O(p,q). Their normalization is found and they are used to construct discrete series representations of the algebra. Certain other representations are also constructed for the case q = 1

[en] We describe the generators of κ-conformal transformations, leaving invariant the κ-deformed d'Alembert equation. In such a way one obtains the conformal extension of-shell spin spin zero realization of κ-deformed Poincare algebra. Finally the algebraic structure of κ-deformed conformal algebra is discussed. (author)

[en] Eigenvalues of the invariant operators of the orthogonal and symplectic groups have been obtained in closed form. All semisimple Lie groups, the unitary, orthogonal, and symplectic groups, are treated in a systematic way by modifying Perelomov and Popov's method. The eigenvalues of the invariant operators for the orthogonal and symplectic groups are then calculated with reference to the unitary group. (U.S.)

[en] In the same way the folding of the Dynkin diagram of A_2n (respectively A_2n+1) produces the B_n (respectively C_n) Dynkin diagram, the symmetry algebra W of a Toda model based on B_n (respectively C_n) can be seen as resulting from the folding of a W-algebra based on A_2n (respectively A_2n-1). More generally, W algebras related to the B-C-D algebra series can appear from W algebras related to the unitary ones. Such an approach is in particular well adapted to obtain fusion rules of W algebras based on non simply laced algebras from fusion rules corresponding to the A_n case. Analogously, super W algebras associated to orthosymplectic superalgebras are deduced from those relative to the unitary A(m, n) series. (orig.)

[en] Lie algebra contractions from o(n + 1) to e(n) are used to obtain asymptotic limits of interbases expansions between bases corresponding to different subgroup chains for the group O(n + 1). The contractions lead to interbases expansions for different subgroup chains of the Euclidean group E(n). The article is restricted to the low dimensional cases n = 2 and n = 3

[en] The two-dimensional dense O(n) loop model for n = 1 is equivalent to the bond percolation and for n = 0 to the dense polymers or spanning trees. We consider the boundary correlations on the half space and calculate the probability P_b that a cluster of bonds has a single common point with the boundary. In the limit n → 0, we find an analytical expression for P_b using the generalized Kirchhoff theorem. (paper)

[en] We study the new boundary condition of the O(n) model proposed by Jacobsen and Saleur using the matrix model. The spectrum of boundary operators and their conformal weights are obtained by solving the loop equations. Using the diagrammatic expansion of the matrix model as well as the loop equations, we make an explicit correspondence between the new boundary condition of the O(n) model and the 'alternating height' boundary conditions in RSOS model.

[en] In this paper, we explore the relationship between integrability and the discrete holomorphicity of a class of complex lattice observables in the context of the Potts dense loop model and the O(n) dilute loop model. It is shown that the conditions for integrability, namely, the inversion and Yang–Baxter relations, can be derived from the condition of holomorphicity of the observables. Furthermore, the Z-invariance of the models is shown to result in the invariance of the observables on the boundary of a sublattice under reshuffling of the rhombuses of its planar rhombic embedding. (paper)

[en] Suppose that q is not a root of unity. We classify all bicovariant differential calculi of dimension greater than one on the quantum groups GL_q(N), O_q(N) and Sp_q(N) for which the differentials duⁱ_j of the matrix entries uⁱ_j generate the left module of first order forms. Our first classification theorem asserts that there are precisely two one-parameter families of such calculi on GL_q(N) for N≥3. In the limit q → 1 only two of these calculi give the ordinary differential calculus on GL(N). Our second main theorem states that apart from finitely many q there exist precisely two differential calculi with these properties on O_q(N) and Sp_q(N) for N≥4. There are four such calculi on O_q(3). We introduce two new 4-dimensional bicovariant differential calculi on O_q(3). (orig.)

[en] It is suggested that a dynamical space-time symmetry group of a given non-linear relativistic theory is able to generate an internal symmetry group. It is shown that only 0(2N-1,1) and 0(2N,1) groups are admitted as simple dynamical symmetry groups, and they imply U(2sup(N-2)) as the largest internal symmetry groups. The boson and fermion content of the considered models is determined and a new dynamical symmetry breaking mechanism, different from the spontaneous one, is introduced. (author)