TOPICS
Search

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, Special Linear Lie Algebra, Trace Form, Weyl Group

This entry contributed by Todd Rowland

Explore with Wolfram|Alpha

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

Subject classifications