Graded Poisson Lie structures on classical complex Lie groups (Q1918115)

The connection between graded Poisson Lie structures and bicovariant differential calculi on quantum groups is the motivation for the author to investigate possible Poisson Lie structures in the classical nonquantized case. The author defines a bilinear operation, called bracket, on the external algebra \(\Omega\) over the cotangent bundle of a complex Lie group \(G\). Lifting right (left) translations by elements of \(G\) to \(\Omega\) he investigates the bicovariant condition. Theorem 1 states an explicit form for the restriction of a bicovariant bracket to the subalgebra of \(\Omega\) generated by the matrix elements of the adjoint representation of \(G\) and a basis of right invariant 1-forms in the case that \(G\) is simple and connected. Obstructions to the definition of a graded Poisson Lie structure on \(\Omega\) by a bicovariant bracket arise from the Jacobi identity. Necessary conditions that the Jacobi identity holds true and the homogeneous bicovariant bracket defines a graded Poisson Lie structure on \(\Omega\) are formulated in theorem 2 using a solution of the modified Yang-Baxter equation. In theorem 3 the graded Poisson Lie structures on \(\Omega\) defined by homogeneous bicovariant brackets are classified in the case of complex simple Lie groups of type \(A_{n - 1}, B_n, C_n, D_n\). If \(G\) is of type \(A_{n - 1} (n > 2)\), \(\Omega\) admits 4 different Poisson Lie structures. \(\Omega\) admits two different Poisson Lie structures if \(G\) is of type \(A_1\), and there are no Poisson Lie structures in the other cases.