arXiv subjects

Mathijs S. Dijkhuizen

Publications and source records attributed to Mathijs S. Dijkhuizen.

4 recordsLinked to original sources

arXiv1998

Some limit transitions between BC type orthogonal polynomials interpreted on quantum complex Grassmannians

The quantum complex Grassmannian U_q/K_q of rank l is the quotient of the quantum unitary group U_q=U_q(n) by the quantum subgroup K_q=U_q(n-l)xU_q(l). We show that (U_q,K_q) is a quantum Gelfand pair and we express the zonal spherical functions, i.e. K_q-biinvariant matrix coefficients of finite- dimensional irreducible representations of U_q, as multivariable little q-Jacobi polynomials depending on one discrete parameter. Another type of biinvariant matrix coefficients is identified as multivariable big q-Jacobi polynomials. The proof is based on earlier results by Noumi, Sugitani and the first author relating Koornwinder polynomials to a one-parameter family of quantum complex Grassmannians, and certain limit transitions from Koornwinder polynomials to multivariable big and little q-Jacobi polynomials studied by Koornwinder and the second author.

math.QA↗

arXiv1998

Quantized flag manifolds and irreducible *-representations

We study irreducible *-representations of a certain quantization of the algebra of polynomial functions on a generalized flag manifold regarded as a real manifold. All irreducible *-representations are classified for a subclass of flag manifolds containing in particular the irreducible compact Hermitian symmetric spaces. For this subclass it is shown that the irreducible *-representations are parametrized by the symplectic leaves of the underlying Poisson bracket. We also discuss the relation between the quantized flag manifolds studied in this paper and the quantum flag manifolds studied by Soibelman, Lakshimibai and Reshetikhin, Jurco and Stovicek, and Korogodsky.

math.QA↗

arXiv1995

Some remarks on the construction of quantum symmetric spaces

We present a general survey of some recent developments regarding the construction of compact quantum symmetric spaces and the analysis of their zonal spherical functions in terms of $q$-orthogonal polynomials. In particular, we define a one-parameter family of two-sided coideals in $\Uq(\gog\gol(n,\CC))$ and express the zonal spherical functions on the corresponding quantum projective spaces as Askey-Wilson polynomials containing two continuous and one discrete parameter.

math.QA↗

arXiv1994

CQG algebras: a direct algebraic approach to compact quantum groups

The purely algebraic notion of CQG algebra (algebra of functions on a compact quantum group) is defined. In a straightforward algebraic manner, the Peter-Weyl theorem for CQG algebras and the existence of a unique positive definite Haar functional on any CQG algebra are established. It is shown that a CQG algebra can be naturally completed to a $C^\ast$-algebra. The relations between our approach and several other approaches to compact quantum groups are discussed.

hep-th↗