Abstract:
Using the multi-parametric deformation of the algebra of functions on $ \GL{n+1} $ and the universal enveloping algebra $ \U{\igl{n+1}} $, we construct the multi-parametric quantum groups $ \IGLq{n} $ and $ \Uq{\igl{n}} $.

Abstract:
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 inhomogeneous quantum group IGL_qr(N) as a projection from GL_qr(N+1), or equivalently, as a quotient of GL_qr(N+1) with respect to a suitable Hopf algebra ideal. A bicovariant differential calculus on IGL_qr(N) is explicitly obtained as a projection from the one on GL_qr(N+1). Our procedure unifies in a single structure the quantum plane coordinates and the q-group matrix elements T^a_b, and allows to deduce without effort the differential calculus on the q-plane IGL_qr(N) / GL_qr(N). The general theory is illustrated on the example of IGL_qr(2).

Abstract:
Kashaev algebra associated to a surface is a noncommutative deformation of the algebra of rational functions of Kashaev coordinates. For two arbitrary complex numbers, there is a generalized Kashaev algebra. The relationship between the shear coordinates and Kashaev coordinates induces a natural relationship between the quantum Teichm\"uller space and the generalized Kashaev algebra.

Abstract:
We discuss quantum deformation of the affine transformation algebra. It is shown that the quantum algebra has a non-cocommutative Hopf algebra structure, simple realizations and quantum tensor operators.

Abstract:
A quantum deformation of the Virasoro algebra is defined. The Kac determinants at arbitrary levels are conjectured. We construct a bosonic realization of the quantum deformed Virasoro algebra. Singular vectors are expressed by the Macdonald symmetric functions. This is proved by constructing screening currents acting on the bosonic Fock space.

Abstract:
We investigate the $q$-deformation of the BRST algebra, the algebra of the ghost, matter and gauge fields on one spacetime point using the result of the bicovariant differential calculus. There are two nilpotent operations in the algebra, the BRST transformation $\brs$ and the derivative $d$. We show that one can define the covariant commutation relations among the fields and their derivatives consistently with these two operation as well as the $*$-operation, the antimultiplicative inner involution.

Abstract:
We proceed to study infinite-dimensional symmetries in two-dimensional squashed Wess-Zumino-Novikov-Witten (WZNW) models at the classical level. The target space is given by squashed S^3 and the isometry is SU(2)_L x U(1)_R. It is known that SU(2)_L is enhanced to a couple of Yangians. We reveal here that an infinite-dimensional extension of U(1)_R is a deformation of quantum affine algebra, where a new deformation parameter is provided with the coefficient of the Wess-Zumino term. Then we consider the relation between the deformed quantum affine algebra and the pair of Yangians from the viewpoint of the left-right duality of monodromy matrices. The integrable structure is also discussed by computing the r/s-matrices that satisfy the extended classical Yang-Baxter equation. Finally two degenerate limits are discussed.

Abstract:
We review the W_N algebra and its quantum deformation, based on free field realizations. The (quantum deformed) W_N algebra is defined through the (quantum deformed) Miura transformation, and its singular vectors realize the Jack (Macdonald) polynomials. (Talk at the Nankai-CRM joint meeting on ``Extended and Quantum Algebras and their Applications to Physics'', Tianjin, China, August 19-24, 1996.)

Abstract:
We review several procedures of quantization formulated in the framework of (classical) phase space M. These quantization methods consider Quantum Mechanics as a "deformation" of Classical Mechanics by means of the "transformation" of the commutative algebra of smooth functions on M in a new non-commutative algebra. These ideas lead in a natural way to Quantum Groups as deformation (or quantization, in a broad sense) of Poisson-Lie groups, which is also analysed here.

Abstract:
In order to realize supersymmetric quantum mechanics methods on a four dimensional classical phase-space, the complexified Clifford algebra of this space is extended by deforming it with the Moyal star-product in composing the components of Clifford forms. Two isospectral matrix Hamiltonians having a common bosonic part but different fermionic parts depending on four real-valued phase space functions are obtained. The Hamiltonians are doubly intertwined via matrix-valued functions which are divisors of zero in the resulting Moyal-Clifford algebra. Two illustrative examples corresponding to Jaynes-Cummings-type models of quantum optics are presented as special cases of the method. Their spectra, eigen-spinors and Wigner functions as well as their constants of motion are also obtained within the autonomous framework of deformation quantization.