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Symplectic Lie Algebra

The symplectic Lie algebra sp_(2n)(F) is the Lie algebra of the symplectic group. For the standard symplectic matrix

J=[0 I_n; -I_n 0],

it is the vector space of 2n×2n matrices X satisfying

X^TJ+JX=0,

with Lie bracket [X,Y]=XY-YX. It has dimension n(2n+1) and, over the complex numbers, is the simple Lie algebra of type C_n.

See also

Lie Algebra, Simple Lie Algebra, Symplectic Form, Symplectic Group

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References

Humphreys, J. E. Introduction to Lie Algebras and Representation Theory. New York: Springer-Verlag, 1972.

Cite this as:

Weisstein, Eric W. "Symplectic Lie Algebra." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SymplecticLieAlgebra.html

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