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Hi,

I am a theoretical physicist with no formal "pure math" education, so please calibrate my questions accordingly.

Consider a finite-dimensional Lie algebra, A, spanned by its d generators, X_1,...,X_d. If it matters, I do not assume that A is simple or semisimple, but for my (practical) purposes, one may assume the following structure constants: [X_k, X_p] = i f_{kp}^q X_q, where all structure constants f's are real. Here and below summation over repeating indices is assumed.

My questions are:

  1. Does it make sense to define a stand-alone exponential g=exp(i B^k X_k) WITHOUT a reference to a particular representation and if so, how? [here and below one may assume that all B^k's are arbitrary REAL numbers (i.e., B=B^k X_k is a real form), but does not have to do so].

  2. Either way, take two arbitrary elements in the algebra B = B^k X_k and C = C^k X_k and define P= P^k X_k ==BCH(B,C) as exp(iP)=exp(iB)exp(iC) via the Baker-Campbell-Hausdorff (BCH) relation. Is this a mathematically sensible definition of a map A*A->A? Is convergence of the BCH series ever an issue for a finite dimensional Lie algebra?

  3. My main question: Is it correct to say that the universal covering group, G[A] (i.e., a simply-connected Lie group to which A is the Lie algebra] is generated by the exponential: G[A]=exp(iA)? By "generated," I mean that every element g in G[A] is represented by at least one element in the algebra via the exponential.

  4. More specifically, take the minimal-dimensional faithful matrix representation of the algebra, T[A]. Is it correct that exp(i T[A]) maps the algebra GLOBALLY into the ENTIRE covering group, G[A], via standard matrix multiplication?

Any (professional) advice would be much appreciated. Thank you, Victor
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Does a finite-dimensional Lie algebra always exponentiate into a universal covering group

Hi,

I am a theoretical physicist with no formal "pure math" education, so please calibrate my questions accordingly.

Consider a finite-dimensional Lie algebra, A, spanned by its d generators, X_1,...,X_d. If it matters, I do not assume that A is simple or semisimple, but for my (practical) purposes, one may assume the following structure constants: [X_k, X_p] = i f_{kp}^q X_q, where all structure constants f's are real. Here and below summation over repeating indices is assumed.

My questions are:

  1. Does it make sense to define a stand-alone exponential g=exp(i B^k X_k) WITHOUT a reference to a particular representation and if so, how? [here and below one may assume that all B^k's are arbitrary REAL numbers (i.e., B=B^k X_k is a real form), but does not have to do so].

  2. Either way, take two arbitrary elements in the algebra B = B^k X_k and C = C^k X_k and define P= P^k X_k ==BCH(B,C) as exp(iP)=exp(iB)exp(iC) via the Baker-Campbell-Hausdorff (BCH) relation. Is this a mathematically sensible definition of a map A*A->A? Is convergence of the BCH series ever an issue for a finite dimensional Lie algebra?

  3. My main question: Is it correct to say that the universal covering group, G[A] (i.e., a simply-connected Lie group to which A is the Lie algebra] is generated by the exponential: G[A]=exp(iA)?

  4. More specifically, take the minimal-dimensional faithful matrix representation of the algebra, T[A]. Is it correct that exp(i T[A]) maps the algebra GLOBALLY into the ENTIRE group via standard matrix multiplication?

Any (professional) advice would be much appreciated. Thank you, Victor