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Factor representations of diffeomorphism groups

Robert P. Boyer (2003)

Studia Mathematica

We give a new construction of semifinite factor representations of the diffeomorphism group of euclidean space. These representations are in canonical correspondence with the finite factor representations of the inductive limit unitary group. Hence, many of these representations are given in terms of quasi-free representations of the canonical commutation and anti-commutation relations. To establish this correspondence requires a generalization of complete positivity as developed in operator algebras....

Formula for unbiased bases

Maurice R. Kibler (2010)

Kybernetika

The present paper deals with mutually unbiased bases for systems of qudits in d dimensions. Such bases are of considerable interest in quantum information. A formula for deriving a complete set of 1 + p mutually unbiased bases is given for d = p where p is a prime integer. The formula follows from a nonstandard approach to the representation theory of the group S U ( 2 ) . A particular case of the formula is derived from the introduction of a phase operator associated with a generalized oscillator algebra. The case...

Free dynamical quantum groups and the dynamical quantum group S U Q d y n ( 2 )

Thomas Timmermann (2012)

Banach Center Publications

We introduce dynamical analogues of the free orthogonal and free unitary quantum groups, which are no longer Hopf algebras but Hopf algebroids or quantum groupoids. These objects are constructed on the purely algebraic level and on the level of universal C*-algebras. As an example, we recover the dynamical S U q ( 2 ) studied by Koelink and Rosengren, and construct a refinement that includes several interesting limit cases.

From double Lie groups to quantum groups

Piotr Stachura (2005)

Fundamenta Mathematicae

It is shown, using geometric methods, that there is a C*-algebraic quantum group related to any double Lie group (also known as a matched pair of Lie groups or a bicrossproduct Lie group). An algebra underlying this quantum group is the algebra of a differential groupoid naturally associated with the double Lie group.

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