# LIE ALGEBRA coboundary

There seems to be a problem in the literature about the definition of the 'standard' coboundary on the 'Cartan-Chevalley-Eilenberg' algebra - the problem is the signs!

Where/when did things go wrong? And what's the best way to reference so the next generation learns only the correct signs?

with more precision: the usual C-E coboundary has two summations - for the module structure and for the algebra each is separately correct in all references but some have the sum squaring to zero and others not -i.e. wrong relative sign for the two summations

Here's the original reference: Chevalley, Claude; Eilenberg, Samuel (1948), "Cohomology Theory of Lie Groups and Lie Algebras", Transactions of the American Mathematical Society (Providence, R.I.: American Mathematical Society) 63 (1): 85–124,

and some others that may or may not copy the first Hilton, P. J.; Stammbach, U. (1997), A course in homological algebra, Graduate Texts in Mathematics, 4 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-94823-2, MR 1438546 Knapp, Anthony W. (1988), Lie groups, Lie algebras, and cohomology, Mathematical Notes, 34, Princeton University Press, ISBN 978-0-691-08498-5, MR 938524

The signs are correct (I believe) in

http://www.scholarpedia.org/article/User:Jan_A._Sanders/An_introduction_to_Lie_algebra_cohomology/Lecture_1#the_coboundary_operator

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Hi Jim, I edited the tags to fit the recommended style, and fixed some quote marks (we don't use the TeX convention here). Also, for your information, you have currently one vote to close for being 'not a real question'. – David Roberts Aug 2 '12 at 13:17
Could you add some details about where, specifically, these sign problems are? Is the problem that different sources have made the opposite arbitrary choice of sign convention, or do certain references use signs that are clearly wrong (as opposed to non-standard)? – Kevin Walker Aug 2 '12 at 13:20
That means you may need to provide some more information than just 'the signs are wrong in the literature', for example (off the top of my head) some references to where they are wrong, what that reference says and what the 'right' convention should be. This will shoo away the impending closing votes. – David Roberts Aug 2 '12 at 13:20
Snap. (I wrote my comment without seeing Kevin's - not being condescending!) – David Roberts Aug 2 '12 at 13:21
I've no idea how the signs went wrong, but the easiest way to recover signs when in doubt that I choose for myself is to remember the Koszul resolution $\Lambda(\mathfrak{g})\otimes U(\mathfrak{g})$ of the trivial module by free modules; then all signs are recovered in a jiffy via the interpretation of (co)homology as Tor/Ext. The Koszul complex is pleasant enough to be learned by the next generation, which later on can observe that this puts this story in the context of general Koszul duality between quadratic associative dg algebras and associative algebras with quadratic-linear relations! – Vladimir Dotsenko Aug 11 '12 at 15:41