# Classical R-Matrix and Semi-Simple Lie Algebras

• Zhangju Liu
• Min Qian
Conference paper
Part of the Research Reports in Physics book series (RESREPORTS)

## Abstract

Let g be a Lie algebra. A linear operator R ε End(g) is called a r-matrix (see [5]) if the bracket given by
$${\left[ {X,Y} \right]_R} = \left[ {RX,Y} \right] + \left[ {X,RY} \right]$$
(1)
is also a Lie bracket on g. Such a pair (g,R) is called a double Lie algebra. Moreover, if there is a nondegenerate invariant bilinear form on g and R is skew-symmetric (g,R) becomes a Lie bialgebra ([1], [5]). It is known that (g.R) is a double Lie algebra iff the following bilinear map BR:gxg→g given by
$${B_R}(X,Y) = \left[ {RX,RY} \right] - R({\left[ {X,Y} \right]_R})$$
(2)
is ad-invariant, i.e., the equation
$$\left[ {X,{B_R}(X,Y)} \right] + \left[ {Y,{B_R}(Z,X)} \right] + \left[ {Z,{B_R}(X,Y)} \right] = 0$$
(3)
holds for all X,Y,Z ∈ g. Particularly, the equation
$${B_R}(X,Y) = 0,\forall X,Y \in g$$
is the Yang-Baxter equation. The modified Yang-Baxter equation was defined in [5] as follows:
$${B_R}(X,Y) = - \left[ {X,Y} \right]$$
(4)
Obviously, the equation (4) means (3) hold.

### Keywords

Manifold Weinstein Wallach

## Preview

Unable to display preview. Download preview PDF.

### References

1. [1]
V.G. Drinfel’d, Quantum groups, Proc. ICM, Berkeley 1906, Vol. I, 709–020.Google Scholar
2. [2]
R. Goodman and N.R. Wallach, Commun. Math. Phys. 94, 177–217 (1904).
3. [3]
V. Guillemin and S. Sternberg, Symplectic techiniques in physics, Cambridge University press, (1904).Google Scholar
4. [4]
d.H.Lu and At Weinstein, Poisson Lie group, dressing transformation and Bruhat decompositions, Preprint.Google Scholar
5. [5]
Semenov-Tyan-Shansky, What is a classical r-matrix? Funct. Anal. Appl. 17 (4) (1903) 259–272.