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Entwining Yang-Baxter maps and integrable lattices

Theodoros E. Kouloukas, Vassilios G. Papageorgiou (2011)

Banach Center Publications

Yang-Baxter (YB) map systems (or set-theoretic analogs of entwining YB structures) are presented. They admit zero curvature representations with spectral parameter depended Lax triples L₁, L₂, L₃ derived from symplectic leaves of 2 × 2 binomial matrices equipped with the Sklyanin bracket. A unique factorization condition of the Lax triple implies a 3-dimensional compatibility property of these maps. In case L₁ = L₂ = L₃ this property yields the set-theoretic quantum Yang-Baxter equation, i.e. the...

Exponentiations over the quantum algebra U q ( s l 2 ( ℂ ) )

Sonia L’Innocente, Françoise Point, Carlo Toffalori (2013)

Confluentes Mathematici

We define and compare, by model-theoretical methods, some exponentiations over the quantum algebra U q ( s l 2 ( ℂ ) ) . We discuss two cases, according to whether the parameter q is a root of unity. We show that the universal enveloping algebra of s l 2 ( ℂ ) embeds in a non-principal ultraproduct of U q ( s l 2 ( ℂ ) ) , where q varies over the primitive roots of unity.

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.

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