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.

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).CrossRefADSMathSciNetGoogle Scholar
  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.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1990

Authors and Affiliations

  • Zhangju Liu
    • 1
  • Min Qian
    • 1
  1. 1.Department of MathematicsBeijing UniversityBeijingPeople’s Rep. of China

Personalised recommendations