What a Classical r-Matrix Really Is

DOI

10.2991/jnmp.1999.6.4.5How to use a DOI?

Abstract

The notion of classical r-matrix is re-examined, and a definition suitable to differential (-difference) Lie algebras, ­ where the standard definitions are shown to be deficient, ­ is proposed, the notion of an O-operator. This notion has all the natural properties one would expect form it, but lacks those which are artifacts of finite-dimensional isomorpisms such as not true in differential generality relation End (V ) V V for a vector space V . Examples considered include a quadratic Poisson bracket on the dual space to a Lie algebra; generalized symplectic-quadratic models of such brackets (aka Clebsch representations); and Drinfel'd's 2-cocycle interpretation of nondegenate classical r-matrices.

Copyright

© 2006, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Boris A. Kupershmidt
PY  - 1999
DA  - 1999/11/01
TI  - What a Classical r-Matrix Really Is
JO  - Journal of Nonlinear Mathematical Physics
SP  - 448
EP  - 488
VL  - 6
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.1999.6.4.5
DO  - 10.2991/jnmp.1999.6.4.5
ID  - Kupershmidt1999
ER  -