i) [GAMMA](E) is endowed with a

Lie algebra structure [,] over R,

Let U be the complexified universal enveloping algebra of the real

Lie algebra g of G; which is canonically isomorphic onto the algebra of all distributions on G supported by {0} , where 0 is the identity element of G.

The structure of Hopf co-Poisson algebra on the universal enveloping algebra U(ST(2)) of

Lie algebra ST(2) is determined with the help of a solution of the Yang-Baxter equation.

angle](G) is the

Lie algebra of the group G, rad G is the radical of G, and [G.

For example for the semisimple complex finite-dimensional

Lie algebra g we have [U.

In [LP], Lam and Pylyavskyy introduced the electrical

Lie algebra [el.

It is not about mathematics, he says, but about the use of symmetries, mainly described by the techniques of Lie groups and

Lie algebra.

C) Compact case: g is the

Lie algebra of a compact simple Lie group.

lambda]](q) is the graded character of a simple

Lie algebra coming from tensor products of KR modules.

n] satisfying Poisson brackets or as a quantum system with the generators of the

Lie algebra sl (2) and commutation relations

Lie-Rinehart algebras, also known as Lie algebroids, give rise to Hopf algebroids by a universal enveloping algebra construction, much as the universal enveloping algebra of an ordinary

Lie algebra gives a Hopf algebra, of infinite dimension.

By the classical Lie theory, the

Lie algebra of a compact Lie group is a direct product of an abelian

Lie algebra and some simple

Lie algebras.