## Quantization of Lie groups and Lie algebras.(English)Zbl 0677.17010

Algebraic analysis, Pap. Dedicated to Prof. Mikio Sato on the Occas. of his Sixtieth Birthday, Vol. 1, 129-139 (1989).
[For the entire collection see Zbl 0665.00008.]
An algebraic definition of an algebra of functions $$A(R)$$ on the quantum formal group corresponding to the matrix $$R$$ is given, based on two algebraic matrix equations. The first one correlates the matrix $$R$$ of “quantum” structure constants with the matrix $$T$$ of generators of $$A(R)$$, the second one generalizes the Jacobi identity for the quantum structure constants. This definition is predual to the definition of quantum groups and hence a subalgebra of the dual $$A'(R)$$ is characterized as a quantization of the universal enveloping algebra with respect to $$R$$ and denoted by $$U(R)$$. It is stated (Theorems 1 and 2), that $$A(R)$$ as well as $$U(R)$$ are bialgebras.
A finite dimensional example with prescribed matrix $$R$$ is considered, leading to the algebra of functions $$\text{Fun}_ q G$$ on the $$q$$-deformation of the groups $$\text{GL}(n,\mathbb C)$$ and $$\text{SL}(n,{\mathbb C})$$. It is stated (Theorem 4), that $$\text{Fun}_q \text{SL}(n,{\mathbb C})$$ has an antipode and hence is a Hopf algebra. Moreover, the equality $$U(R)=U_q(\mathfrak{sl}(n,\mathbb C))$$ holds in this case (Theorem 5). There is an infinite dimensional example sketched, stating that $$U(R)$$ is connected with the $$q$$-deformation of Kac-Moody algebras.
In the last part on deformation theory and quantum groups the Poisson structure on the Lie algebra $$\mathfrak G$$ of a Lie group $$G$$ defined by $$A(R)=\text{Fun}_q G$$ in the limit $$q\to 1$$ is considered, while the analogous contraction of the corresponding $$U(R)$$ comes down to the Lie bialgebra structure on the dual of $$\mathfrak G$$. There are no proofs at all.

### MSC:

 17B37 Quantum groups (quantized enveloping algebras) and related deformations 16T20 Ring-theoretic aspects of quantum groups 16T25 Yang-Baxter equations 16T05 Hopf algebras and their applications 20G42 Quantum groups (quantized function algebras) and their representations 53D17 Poisson manifolds; Poisson groupoids and algebroids 53D55 Deformation quantization, star products

Zbl 0665.00008