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I am considering the set $$ \mathbb{L} = \lbrace X \in \mathbb{C}^{2n \times 2n} \; ; \; J^{-1}X^HJ = -X \rbrace, \quad J = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix},$$ of complex Hamiltonian matrices (where superscript $H$ means conjugate transposition). This is a $\mathbb{R}$-linear subspace of matrices and, in case I'm not mistaken, an $\mathbb{R}$-Lie-algebra with Lie bracket $[X,Y]=XY-YX$ (unfortunately it is not a complex Lie algebra).

My primary research area is matrix computations and I'm not very familiar with Lie algebras. What I found out so far is:

  • As a real Lie algebra of complex matrices, $\mathbb{L}$ is reductive (According to A. Knapp, Lie Groups - Beyond an Introduction, Prop. 1.56., $\mathbb{L}$ is reductive if it is closed under the operation conjugate transpose - which is the case).
  • The real Lie algebra $\mathbb{L}$ is semisimple (again according to A. Knapp, Lie Groups - Beyond an Introduction, $\mathbb{L}$ is semisimple as a real Lie algebra if and only if its center is 0 - which is the case).

Is there a relation between (1) $X \in \mathbb{L}$ being semisimple as an element of the $\mathbb{R}$-Lie-algebra $\mathbb{L}$ and (2) $X$ being semisimple as a complex matrix (in the classical sense of being diagonalizable over $\mathbb{C}$)? In particular, does (1) imply (2)? I'm a little confused about the definition of semisimplicity in Lie algebras (e.g. via the adjoint action) and I was wondering whether these concepts in the sense (1) and (2) above are actually the same for matrices in $\mathbb{L}$? I would very much appreciate any help or any references.

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