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Definition/Summary

A Lie algebra (“Lee”) is a set of generators of a Lie group. It is a basis of the tangent space around a Lie group’s identity element, the space of differences between elements close to the identity element and the identity element itself.

Lie algebras include a binary, bi-linear, anti-symmetric operation: commutation. The commutator of two basis vectors is a linear combination of the algebra’s basis vectors (closure).

Lie algebras are valuable as a proxy for Lie groups. They are often more convenient to study than the groups that they generate, and much of what is known about Lie groups has come from studying their algebras, like their representation theory. One has to be careful about global properties, however; groups with isomorphic algebras need not be isomorphic, such as SO(3) and SU(2).

Equations

Commutator:

For matrices: [A,B] = A.B – B.A

For operators: [A,B](X) = A(B(X)) – B(A(X))

The commutator satisfies the Jacobi identity:

[A,[B,C]] + [B,[C,A]] +...

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