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New entry (!) tangent Lie algebra. Significant changes at invariant differential form with redirect invariant vector field reflecting the vector fields and other tensor cases. Many more related entries listed at and the whole entry reworked extensively at Lie theory. Some changes at Lie’s three theorems and local Lie group. New stubs Chevalley group and Sigurdur Helgason.
By the way, when writing tangent Lie algebra, I had the problem with finding the correct font for the standard symbol of Lie algebra of vector fields on a manifold. Usually one has varchi symbol which looks like Greek chi but with dash through middle. The varchi symbol is not recognized and I put mathcal X which is slanted and script, just alike, but without dash through middle.
By the way, on a real Lie group $G$ of dimension $n$, if one expresses the right invariant vector field in terms of left invariant vector fields then at each point there is a $\mathbb{R}$-linear operator which sends any frame of left invariant vector fields to the corresponding frame in right invariant vector fields; this gives a $GL_n(\mathbb{R})$-valued real analytic function on $G$ (or, in local coordinates, on a neighborhood of the unit element). In other words, if I take a frame in a Lie algebra and interpret it in two ways, as a frame of left invariant vector fields and a frame in right invariant vector fields, then I can take a matrix of real analytic functions on a Lie group and multiply the frame of left invariant vector fields with this matrix to get the correspoding frame of right invariant vector fields. I use in my current research some computations involving this matrix function. Does anybody know of any reference in literature which does any computations involving this matrix valued function on $G$ ?
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