Advertisement

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

Toda Lattice Cartan Decomposition Periodic Toda Lattice Involution Theorem Nondegenerate Invariant Bilinear Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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