Timeline for Lie algebra semisimple if and only if perfect?
Current License: CC BY-SA 4.0
9 events
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| May 2, 2022 at 8:01 | history | edited | YCor | CC BY-SA 4.0 |
fixed error ($V$ shouldn't be trivial) and added notational remark
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| May 2, 2022 at 7:59 | comment | added | YCor | A remark on this answer, is that every non-semisimple perfect Lie algebra (fin. dim., in char 0) has a quotient of the form you're describing. So, in a sense, these are the minimal counterexamples (well, it's minimal if the representation is faithful, in the sense that in this case $L\ltimes V$ is not semisimple but all its proper quotients are semisimple). | |
| Feb 15, 2017 at 6:15 | comment | added | Patrick Da Silva | Note that the one-dimensional vector space $V$ for which $\rho : L \to \mathfrak{gl}(V)$ is zero is also a simple $L$-module, and in this case $[L \times V, L \times V] = L \subsetneq L \times V$. You need to exclude that case, otherwise it works fine since $[L,V]$ contains a non-zero vector, hence is equal to $V$. | |
| Jun 11, 2015 at 21:36 | comment | added | LSpice | @JimHumphreys, thanks! I didn't mean to ask you to do the search for me; I just thought you might have some pointers to good entrées to this literature. (My approach is that of someone who knows Lie algebras only as things coming from (algebraic) Lie groups, so basically I am interested in anything about them that sheds light on structure theory in that setting (but, unfortunately for the history you mention, particularly in positive characteristic).) | |
| Jun 11, 2015 at 21:13 | comment | added | Jim Humphreys | @LSpice: Besides the paper I mentioned, there are many others listed on MathSciNet under "perfect Lie algebra" though I'm not sure what would interest you. I guess my point was that the purely algebraic theory of Lie algebras (often in characteristic 0) has been studied by many people over the past century; thus, much is known. A random example is the paper by Baranov and Zalesskii: Plain representations of Lie algebras, J. London Math. Soc. (2) 63 (2001), no. 3, 571–591. You can also find some related posts here by searching MO for "perfect Lie algebra". | |
| Jun 11, 2015 at 18:49 | comment | added | LSpice | @JimHumphreys, you mention the substantial literature on this subject—where else would one look? | |
| Apr 5, 2011 at 13:08 | comment | added | Jim Humphreys | Maybe it's also helpful to mention the substantial though often isolated literature on perfect Lie algebras and related structure theory? There are some interesting connections with other questions, as in the paper by Benkart and Zelmanov in Invent. Math. 126 (1996), 1-45. | |
| Apr 4, 2011 at 8:23 | vote | accept | mark | ||
| Apr 3, 2011 at 22:13 | history | answered | Faisal | CC BY-SA 2.5 |