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Let $sp(2n)$ be the symplectic algebra. I have to prove that $sp(2n)$ is a simple algebra and its type is $C_{n}$.

In order to prove the semisimplicity we can consider the this theorem: Let $V$ be a finite dimensional vector space and $L \subset sl(V)$ a Lie algebra. If the action of $L$ on $V$ is irreducible then $L$ is semisimple algebra. So we can take $V= \mathbb{C}^{2n}$ and look at $sp(n)$ as a subalgebra of $sl(2n)$, and we can prove that $sp(2n)$ is semisimple algebra. Then we can find an irreducible system of roots and verify the simplicity and the type of $sp(2n)$.

But is there an easier way to prove semisimplicity, simplicity and type?

rschwieb
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ArthurStuart
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1 Answers1

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You can also do the following (which may or may not be easier): Given any root system, you get a Cartan matrix. This matrix in turn gives you a set of generators and relations (see for example Humphreys' Introduction to Lie Algebras and Representation Theory chapter 18).

If you can find a set of generators for the Lie algebra which satisfy the relations given there, then you will have shown that the Lie algebra is simple and has the correct root system (Theorem 18.3 in the above reference).

Tobias Kildetoft
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