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000 κ(x, x) = 0 κ(x, y ) = 4 κ(x, h) = 0
κ(y, y ) = 0 κ(y, h) = 0
κ(h, h) = 8
The matrix of the form κ is therefore 040
4 0 0
008 MATH 223A NOTES 2011 LIE ALGEBRAS 19 Since this matrix is invertible, κ is nondegenerate and L = sl(2, R) is semisimple. Since
this matrix has negative determinant and positive trace its signature (#+ eigenvalues −
#− eigenvalues) is 1.
Exercise 5.2.7. Compute the Killing form of the real cross product algebra. Conclude
that this algebra is semisimple but not isomorphic to sl(2, R).
5.3. Product of simple ideals.
Theorem 5.3.1. Suppose that L is a ﬁnite dimensional semisimple Lie algebra over any
subﬁeld F ⊆ C. Then...
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This document was uploaded on 03/09/2014 for the course MATH 223a at Brandeis.
- Fall '11