Is the symplectic group $\operatorname{Sp}(2n,\mathbb{R})$ simple?

Wikipedia states that the Lie algebra $\mathfrak{sp}(2n,\mathbb{R})$ is simple.

However it only lists projective groups in its list of simple Lie groups.

up vote 1 down vote accepted

There is a bijective correspondence between real connected simple Lie groups and real simple Lie algebras. Since the Lie algebras ${\frak{sp}}(2n,\mathbb{R})$ are simple, every connected Lie group with Lie algebra ${\frak{sp}}(2n,\mathbb{R})$ is simple.

On the other hand, there is no generally accepted definition of a simple Lie group. Sometimes it is required in addition that the center should be trivial. I think, that also $Sp(2n,\mathbb{R})$ should be called "simple". The projective group $PSp(2n,F)$ is usually used for the case where $F$ is a finite field.

  • 1
    Ah! The definition of simplicity is debatable so, to err on the side of caution, Wikipedia only lists those groups with trivial centre. However it is very common to allow non-trivial centres in the definition of simplicity. – Matta Dec 4 '14 at 15:00

Your Answer


By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

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