I am studying Helgason's Differential Geometry and Symmetric Spaces, trying to understand real forms of...

antennense

antennense

Answered

2022-07-01

I am studying Helgason's Differential Geometry and Symmetric Spaces, trying to understand real forms of Lie Algebras.
My problem is related to Lemma 6.1 (1st edition):
Let K 0 be the Killing form of a Lie algebra g over R , and K the Killing form of the complexification Lie algebra g C . Then
K 0 ( X , Y ) = K ( X , Y )     X , Y g

My problem:
Understanding the previous equation as K ( X , Y ) K ( X + i 0 , Y + i 0 ) I would get twice the result stated in the book. This is due to the following facts.
1. ad C ( X ) End C ( g C ), acting as
ad C ( X ) [ A + i B ] = [ X + i 0 , A + i B ] C = [ X , A ] + i [ X , B ]
therefore I can write it as a direct sum of the real adjoint map
ad C ( X ) = ad ( X ) ad ( X )
2. Trace [ ad C ( X ) ad C ( Y ) ] = Trace [ ad ( X ) ad ( Y ) ad ( X ) ad ( Y ) ] = 2 Trace [ ad ( X ) ad ( Y ) ]

Background:
1. Helgason's definition of complexification: g C = g × g g g with the complex structure
J : ( X , Y ) X + i Y ( Y , X ) Y + i X
extending the Lie bracket by C -linearity:
[ X + i Y , Z + i T ] C = [ X , Z ] [ Y , T ] + i [ X , T ] + i [ Y , Z ]
2. Killing form of any Lie algebra over an arbitrary field K :
B ( X , Y ) = Trace [ ad ( X ) ad ( Y ) ]
where ad ( X ) is the K -linear map [ X , ].

Answer & Explanation

enfeinadag0

enfeinadag0

Expert

2022-07-02Added 16 answers

K ( a , b ) = T r ( a d a a d b ) a basis of g induces a basis of g C and a d a a d b have the same matrix in both basis.

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