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

Wikipedia states that the Lie algebra $\mathfrak{sp}(2n,\mathbb{R})$ is simple. http://en.wikipedia.org/wiki/Table_of_Lie_groups

However it only lists projective groups in its list of simple Lie groups. http://en.wikipedia.org/wiki/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

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