Nonstandard Deformation U′ q (so n ): The Imbedding U′ q (so n ) ⊂ U q (sl n ) and Representations

A. M. Gavrilik, N. Z. Iorgov and A. U. Klimyk
Additional contact information
A. M. Gavrilik: Ukrainian National Academy of Sciences, Bogolibov Institute for Theoretical Physics
N. Z. Iorgov: Ukrainian National Academy of Sciences, Bogolibov Institute for Theoretical Physics
A. U. Klimyk: Ukrainian National Academy of Sciences, Bogolibov Institute for Theoretical Physics

A chapter in Symmetries in Science X, 1998, pp 121-133 from Springer

Abstract: Abstract Quantum orthogonal groups, quantum Lorentz group and their corresponding quantum algebras are of special interest for modern physics [1–4]. M. Jimbo [5] and V. Drinfeld [6] defined q-deformations (quantum algebras) U q (g)for all simple complex Lie algebras gby means of Cartan subalgebras and root subspaces. Reshetikhin, Takhtajan and Faddeev [7] defined quantum algebras U q (g)in terms of the universal R-matrix. However, none of these approaches is able to give a satisfactory presentation of the quantum algebra U q (so(n,ℂ)) from a viewpoint of some problems in quantum physics and representation theory. In fact, several important problems of theoretical physics demand to define an action of the quantum Lorentz group SO q (n,1) and of the corresponding quantum algebra (the real forms of the quantum group SO q (n+ 1, ℂ) and of the quantum algebra U q (so(n+ 1, ℂ)) respectively) upon the quantum (n + 1)-dimensional linear space. The approaches mentioned above are not satisfactory for such definition. Besides, when considering representations of the quantum groups SO q (n +1) and SO q (n,1) we are interested in reducing them onto the quantum subgroup SO q (n). This reduction would give the analog of the Gel’fand-Tsetlin basis for these representations. However, definitions of quantum algebras mentioned above do not allow the inclusions SO q (n+ 1) ⊃SO q (n)and U q (so(n,l)) ⊃U q (so(n)). To be able to exploit such reductions we have to consider q-deformations of the Lie algebra so(n+ 1, ℂ) defined in terms of the generators I k,k−1= E k,k−1−E k−1,k (where E is is the matrix with elements (E is ) rt = δirδ st ) rather than by means of Cartan subalgebras and root elements.

Keywords: Irreducible Representation; Hopf Algebra; Quantum Group; Quantum Algebra; Dimensional Linear Space (search for similar items in EconPapers)
Date: 1998
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4899-1537-5_6

Ordering information: This item can be ordered from
http://www.springer.com/9781489915375

DOI: 10.1007/978-1-4899-1537-5_6

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().