Killing form
The Killing form
Proof of symmetric bilinearity
is linear in the second argument, since πΎ π ( π , πΌ π + π½ π ) = t r β‘ ( a d π β a d πΌ π + π½ π ) = t r β‘ ( a d π β ( πΌ a d π + π½ a d π ) ) = t r β‘ ( πΌ a d π β a d π + π½ a d π β a d π ) = πΌ t r β‘ ( a d π β a d π ) + π½ t r β‘ ( a d π β a d π ) = πΌ π ( π , π ) + π½ π ( π , π ) and symmetric since
π ( π , π ) β π ( π , π ) = t r β‘ ( a d π β‘ a d π ) β t r β‘ ( a d π β‘ a d π ) = t r β‘ ( a d π β‘ a d π β a d π β‘ a d π ) = t r β‘ ( [ a d π , a d π ] ) = t r β‘ ( a d [ π , π ] ) = 0 by Properties.
Properties
- Let
π β΄ π€ π€ π π
Proof
Relation to Lie groups
If
π ( A d π β‘ ( π ) , A d π β‘ ( π ) ) = π ( π , π ) π , π β π€ π β πΊ
Proof of 1
Since
is a Lie algebra automorphism, for all A d π and π , π β π€ , π β πΊ a d A d π β‘ ( π ) β‘ ( π ) = [ A d π β‘ ( π ) , π ] = A d π β‘ A d π β 1 β‘ [ A d π β‘ ( π ) , π ] = A d π β‘ [ π , A d π β 1 β‘ ( π ) ] = A d π β‘ a d π β‘ A d π β 1 β‘ ( π ) hence
πΎ ( A d π β‘ ( π ) , A d π β‘ ( π ) ) = t r β‘ ( a d A d π β‘ ( π ) β‘ a d A d π β‘ ( π ) ) = t r β‘ ( A d π β‘ a d π β‘ A d π β 1 β‘ A d π β‘ a d π β‘ A d π β 1 ) = t r β‘ ( A d π β‘ a d π β‘ a d π β‘ A d π β 1 ) = t r β‘ ( a d π β‘ a d π ) as required.
Footnotes
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1972. Introduction to Lie Algebras and Representation Theory, Β§5.1, p. 21 β©