The Quantum Universal Enveloping Algebra

We start from the universal enveloping algebra of , and then deform it to get the quantum universal enveloping algebra .
Recall from the example that is the Lie algebra of complex traceless matrices, and it has a standard basis with Lie brackets , and . Its universal enveloping algebra is a unital associative algebra generated by .

-Number

Suppose is a field, and is a nonzero element that is not either or . Then the -number of an integer is defined as

Consider . The quantum universal enveloping algebra is the unital associative algebra over generated by subject to the relations:

is deformed from

When , this definition reduces to the usual universal enveloping algebra with and identified with and (up to a sign), respectively.
We don’t actually need the generator if . In fact, if , then . But we keep in the definition to easily compare with the classical case .

Proposition

has a Hopf algebra structure with comultiplication , counit and antipode defined by

Universal -Matrix

Highest Weight Representations