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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 is equivalent to the -algebra , and so as an -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.