We have said that a Lie group is an analytic manifold; and so it makes sense to talk about the tangent space to that manifold, and in particular the tangent space at the identity of the group. That tangent space is called the Lie algebra. In the case of a linear group it can be computed explicitly by differentiating curves through the identity.
KeywordsTangent Space Adjoint Representation Heisenberg Algebra Trace Zero Nilpotent Algebra
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