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1. • CommentRowNumber1.
   • CommentAuthorelif
   • CommentTimeSep 20th 2015
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   Author: elif
   Format: TextIs there any definition of finite-dimensional Lie algebra crossed modules? Is the category of simply connected Lie algebra crossed modules and the category of finite-dimensional Lie algebra crossed modules equivalent?

   Is there any definition of finite-dimensional Lie algebra crossed modules? Is the category of simply connected Lie algebra crossed modules and the category of finite-dimensional Lie algebra crossed modules equivalent?
2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeSep 21st 2015
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   Author: DavidRoberts
   Format: MarkdownItexDo you mean crossed modules internal to fd Lie algebras? Or something invariant under (weak) equivalence? One could talk of "finite type" Lie algebra crossed modules, for instance $\widehat{\Omega\mathfrak{g}} \to P\mathfrak{g}$ is equivalent to the $L_\infty$-algebra $i\mathbb{R} \to \mathfrak{g}$, and so as an $L_\infty$-algebra it has finite-dimensional homology groups (of the underlying chain complex). The question of simple-connectedness is a bit more tricky. If you have a Lie algebra crossed module that integrates to a smooth group stack, then one could ask whether the underlying shape of that group stack is simply-connected.

   Do you mean crossed modules internal to fd Lie algebras? Or something invariant under (weak) equivalence? One could talk of “finite type” Lie algebra crossed modules, for instance $\widehat{\Omega\mathfrak{g}} \to P\mathfrak{g}$ is equivalent to the $L_\infty$-algebra $i\mathbb{R} \to \mathfrak{g}$, and so as an $L_\infty$-algebra it has finite-dimensional homology groups (of the underlying chain complex).

   The question of simple-connectedness is a bit more tricky. If you have a Lie algebra crossed module that integrates to a smooth group stack, then one could ask whether the underlying shape of that group stack is simply-connected.