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Killing Form


The Killing form is an inner product on a finite dimensional Lie algebra g defined by

 B(X,Y)=Tr(ad(X)ad(Y))
(1)

in the adjoint representation, where ad(X) is the adjoint representation of X. (1) is adjoint-invariant in the sense that

 B(ad(X)Y,Z)=-B(Y,ad(X)Z).
(2)

When g is a semisimple Lie algebra, the Killing form is nondegenerate.

For example, the special linear Lie algebra sl_2(C) has three basis vectors {X,Y,H}, where [X,Y]=2H:

X=[ 0  1;  1  0]
(3)
Y=[ 0  -1;  1  0]
(4)
H=[ 1  0;  0  -1].
(5)

The other brackets are given by [X,H]=2Y and [Y,H]=2X. In the adjoint representation, with the ordered basis {X,Y,H}, these elements are represented by

ad(X)=[0  0 0;  0 0 2;  0 2 0]
(6)
ad(Y)=[0  0 -2;  0 0 0; 2 0 0]
(7)
ad(H)=[0 -2 0; -2 0 0;  0 0 0],
(8)

and so B(u,v)=u^(T)Bv where

 B=[8  0 0; 0 -8 0; 0  0 8].
(9)

See also

Cartan Matrix, Inner Product, Killing's Equation, Killing Vectors, Lie Algebra, Matrix Signature, Semisimple Lie Algebra, Trace Form, Weyl Group

This entry contributed by Todd Rowland

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References

Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, pp. 23-26, 1996.

Referenced on Wolfram|Alpha

Killing Form

Cite this as:

Rowland, Todd. "Killing Form." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/KillingForm.html

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