Restrictions and expansions of holomorphic representations

We compute tensor products of representations of the holomorphic discrete series of a Lie group G, or restrictions to some subgroup G′. A detailed study is done for the case of the conformal group O(4, 2).     

The Exponential Nature and Positivity

In the present article, a basis of the coordinate algebra of the multi-parameter quantized matrix is constructed by using an elementary method due to Lusztig. The construction depends heavily on an anti-automorphism, the bar action. The exponential nature of the bar action is derived which provides an inductive way to compute the basis elements. By embedding the basis into the dual basis of Lusztig's canonical basis of , the positivity properties of the basis as well as the positivity properties of the canonical basis of the modified quantum enveloping algebra of type , which has been conjectured by Lusztig, are proved.Presented by A. Verschoren.     

Tensor products,reproducing kernels,and power series

Tensor products of holomorphic discrete series representations in reproducing kernel Hilbert spaces are decomposed by considering power series expansions of functions in the direction perpendicular to the diagonal in $\text{D}$ × $\text{D}$.     

Quantized Dirac operators

We determine what should correspond to the Dirac operator on certain quantized hermitian symmetric spaces and what its properties are. A new insight into the quantized wave operator is obtained. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

Painleve analysis for a semi-linear parabolic equation arising in smectic liquid crystals

This article presents a systematic method, employing a Painlevéanalysis for obtaining static and travelling wave solutionsto a semi-linear parabolic equation which frequently arisesin the smectic liquid crystal literature. The equation consideredhas sinusoidal non-linearities and shares some of the featuresof the double sine-Gordon equation; a brief discussion of therelationship of this equation to liquid crystals is also given.     

Quantized Heisenberg Space

We investigate the algebra F_q(N) introduced by Faddeev, Reshetikhin and Takhadjian. In the case where q is a primitive root of unity, the degree, the center, and the set of irreducible representations are found. The Poisson structure is determined and the De Concini–Kac–Procesi Conjecture is proved for this case.