Quantum group

As a quantum group is referred to in the mathematical group theory, a certain class of Hopf algebras, namely quantization (ie non-trivial deformations ) of the enveloping Hopf algebras of semisimple Lie algebras. Alternatively, one can consider quantum groups as deformations of the algebra of regular functions on algebraic groups.

The term was coined in the context of the International Congress of Mathematicians in 1986 in Berkeley by the Ukrainian mathematician Vladimir Drinfeld Gershonovich. Independently, they were found by Michio Jimbo at the same time.

Example

The simplest quantum group. This is the algebra, which is generated by the variable, and in which the relations and

. apply

The Hopf algebra structure is given by

And are therefore wrong primitive, and group and are similar.

Universal enveloping algebra

Is not defined in this form, since one would have to divide it by 0. However, it is possible to formulate the definition by means of another variable so that it is possible.

In this form is well defined and is closely related to the universal enveloping algebra. It is namely

Being on, on and on mapped.