An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee

Author:Nadir Jeevanjee
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(4.54)

Likewise,

Thus, the Lie algebra of the group of invertible linear operators is simply the space of all linear operators!

Now, consider a finite-dimensional vector space V equipped with a non-degenerate Hermitian form . How can we describe the Lie algebra associated with the isometry group Isom(V )?27 If X is in this Lie algebra, then (4.53) and (4.10) tell us that

Differentiating with respect to t yields

Evaluating at t = 0 and rearranging gives

(4.55)

This is the fundamental description of any Lie algebra of isometries, and will take various concrete forms as we proceed through the examples below. The minus sign, in particular, will play a key role.

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