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On a cubic Hecke algebra associated with the quantum group U q ( 2 )

Janusz Wysoczański — 2010

Banach Center Publications

We define an operator α on ℂ³ ⊗ ℂ³ associated with the quantum group U q ( 2 ) , which satisfies the Yang-Baxter equation and a cubic equation (α² - 1)(α + q²) = 0. This operator can be extended to a family of operators h j : = I j α I n - 2 - j on ( ³ ) n with 0 ≤ j ≤ n - 2. These operators generate the cubic Hecke algebra q , n ( 2 ) associated with the quantum group U q ( 2 ) . The purpose of this note is to present the construction.

bm-independence and central limit theorems associated with symmetric cones

Janusz Wysoczański — 2007

Banach Center Publications

We present a generalization of the classical central limit theorem to the case of non-commuting random variables which are bm-independent and indexed by a partially ordered set. As the set of indices I we consider discrete lattices in symmetric positive cones, with the order given by the cones. We show that the limit measures have moments which satisfy recurrences generalizing the recurrence for the Catalan numbers.

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