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Contractions of Poisson-Lie groups, Lie bialgebras and quantum deformations

Angel Ballesteros, Mariano del Olmo (1997)

Banach Center Publications

Contractions of Poisson-Lie groups are introduced by using Lie bialgebra contractions. As an application, contractions of SL(2,R) Poisson-Lie groups leading to (1+1) Poincaré and Heisenberg structures are analysed. It is shown how the method here introduced allows a systematic construction of the Poisson structures associated to non-coboundary Lie bialgebras. Finally, it is sketched how contractions are also implemented after quantization by using the Lie bialgebra approach.

Control affine systems on solvable three-dimensional Lie groups, I

Rory Biggs, Claudiu C. Remsing (2013)

Archivum Mathematicum

We seek to classify the full-rank left-invariant control affine systems evolving on solvable three-dimensional Lie groups. In this paper we consider only the cases corresponding to the solvable Lie algebras of types II, IV, and V in the Bianchi-Behr classification.

CR-structures on a real Lie algebra

Giuliana Gigante, Giuseppe Tomassini (1991)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Given the notion of C ⁢ R -structures without torsion on a real 2 ⁢ n + 1 dimensional Lie algebra L 0 we study the problem of their classification when L 0 is a reductive algebra.

Crystal bases for the quantum queer superalgebra

Dimitar Grantcharov, Ji Hye Jung, Seok-Jin Kang, Masaki Kashiwara, Myungho Kim (2015)

Journal of the European Mathematical Society

In this paper, we develop the crystal basis theory for the quantum queer superalgebra U q ( 𝔮 ( n ) ) . We define the notion of crystal bases and prove the tensor product rule for U q ( 𝔮 ( n ) ) -modules in the category 𝒪 int ≥ 0 . Our main theorem shows that every U q ( 𝔮 ( n ) ) -module in the category 𝒪 int ≥ 0 has a unique crystal basis.

Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras

Peng Shan (2011)

Annales scientifiques de l'École Normale Supérieure

We define the i -restriction and i -induction functors on the category 𝒪 of the cyclotomic rational double affine Hecke algebras. This yields a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space.

Cuadrados especiales en la categoría de álgebras de Lie.

Daniel Tarazona (1982)

Stochastica

In this paper the concepts of mixed cartesian square and quasi-cocartesian square, already known in the category of groups, are adapted to the category of Lie algebras. These concepts can be used in the study of the obstructions of Lie algebra extensions in the same way that Wu has studied the obstructions of group extensions.

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