Further results on q-Lie groups, q-Lie algebras and q-homogeneous spaces
| Autor: | Ernst Thomas |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Special Matrices, Vol 9, Iss 1, Pp 119-148 (2021) |
| Druh dokumentu: | article |
| ISSN: | 2300-7451 2020-0129 |
| DOI: | 10.1515/spma-2020-0129 |
| Popis: | We introduce most of the concepts for q-Lie algebras in a way independent of the base field K. Again it turns out that we can keep the same Lie algebra with a small modification. We use very similar definitions for all quantities, which means that the proofs are similar. In particular, the quantities solvable, nilpotent, semisimple q-Lie algebra, Weyl group and Weyl chamber are identical with the ordinary case q = 1. The computations of sample q-roots for certain well-known q-Lie groups contain an extra q-addition, and consequently, for most of the quantities which are q-deformed, we add a prefix q in the respective name. Important examples are the q-Cartan subalgebra and the q-Cartan Killing form. We introduce the concept q-homogeneous spaces in a formal way exemplified by the examples SUq(1,1)SOq(2){{S{U_q}\left( {1,1} \right)} \over {S{O_q}\left( 2 \right)}} and SOq(3)SOq(2){{S{O_q}\left( 3 \right)} \over {S{O_q}\left( 2 \right)}} with corresponding q-Lie groups and q-geodesics. By introducing a q-deformed semidirect product, we can define exact sequences of q-Lie groups and some other interesting q-homogeneous spaces. We give an example of the corresponding q-Iwasawa decomposition for SLq(2). |
| Databáze: | Directory of Open Access Journals |
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