1 research outputs found
What a classical r-matrix really is
The notion of classical -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 -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 \mbox{End}\, (V) \simeq V^*
\otimes V for a vector space . 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 -matrices