nLab
endomorphism Lie algebra

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Definition

Given a vector space VV, then its endomorphism Lie algebra 𝔢𝔫𝔡(V)\mathfrak{end}(V) is the Lie algebra whose elements are the linear maps ϕ:VV\phi \colon V \to V and whose Lie bracket is their commutator

[ϕ 1,ϕ 2]=ϕ 1ϕ 2ϕ 2ϕ 1. [\phi_1, \phi_2] = \phi_1 \circ \phi_2 - \phi_2 \circ \phi_1 \,.

A Lie algebra homomorphisms 𝔤𝔢𝔫𝔡(V)\mathfrak{g} \to \mathfrak{end}(V) is a Lie action or Lie algebra representation of 𝔤\mathfrak{g} on VV.

Created on January 5, 2017 at 02:58:40. See the history of this page for a list of all contributions to it.