Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

A Lie algebra $\mathfrak{g}$ is said to be semisimple if its radical is zero. An element $x \in \mathfrak{g}$ is said to be semisimple if $\text{ad} x$ is diagonalizable.

A complex semisimple Lie algebra must contain non-zero semisimple elements. But is there any deeper connection underlying the common names?

(For instance, in the theory of algebraic groups, a separable element of a matrix group (one with distinct eigenvalues) is one that generates a separable algebra.)

share|cite|improve this question
up vote 3 down vote accepted

In each name, the word "semisimple" means "a direct sum of simple objects" in the appropriate sense. It can be shown that a complex Lie algebra is semisimple (has radical zero) if and only if it is a direct sum of simple Lie algebras. On the other hand, saying that $\operatorname{ad} x$ is diagonalizable is the same as saying that $\mathfrak{g}$ decomposes into a direct sum of eigenspaces for $\operatorname{ad} x$, on each of which $\operatorname{ad} x$ acts as a scalar multiple of the identity; in other words, $\operatorname{ad} x$ is a "direct sum of scalars".

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.