Classification-of6–dimensional-real-Drinfelddoubles.pdfVIP

Classification of 6–dimensional real Drinfeld doubles ∗ ˇ L. Snobl and L. Hlavat´y Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Bˇrehov´a 7, 115 19 Prague 1, Czech Republic March 27, 2002 Abstract Starting from the classification of real Manin triples we look for those that are isomorphic as 6–dimensional Drinfeld doubles i.e. Lie algebras with the ad–invariant form used for construction of the Manin triples. We use several invariants of the Lie algebras to distinguish the non–isomorphic structures and give explicit form of maps between Manin triples that are decompositions of isomorphic Drinfeld doubles. The result is a complete list of 6–dimensional real Drinfeld doubles. It consists of 22 classes of non–isomorphic Drinfeld doubles. PACS codes: 02.20.Qs, 02.20.Sv, 11.25.Hf ∗E–mail: snobl@newton.fjfi.cvut.cz, hlavaty@br.fjfi.cvut.cz 1 1 Introduction In recent years, the study of T–duality in string theory has led to ˇ discovery of Poisson–Lie T–dual sigma models. Klimˇc´ık and Severa have found a procedure allowing to construct the dual models from ˜ Manin triples (D , G , G ) , i.e. a decompositions of a Lie algebra D (it must be even–dimensional) into two maximally isotropic subalgebras ˜ G , G w.r.t. a bilinear form. The construction of the Poisson–Lie T– dual sigma models is described in [1] and [2]. The Lie group possessing a Lie algebra that can be written as a Manin triple