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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

Citations:

Zbl 0665.00008