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The Pfaffian and the Lie algebra homology of skew-symmetric matrices. (English) Zbl 1060.17011
Let \(k\) be a field of characteristic zero and \(A\) a commutative \(k\)-algebra. The orthogonal group \(O(n,k)\), which is not connected, acts on the Lie algebra \({\mathfrak {so}}(n,A)\) and hence on its homology; the action on the homology has eigenvalues \(+1\) and \(-1\). The main result shows that many of the eigenspaces vanish or, if they don’t vanish, can be identified to Kähler differentials, via the Pfaffian. Moreover, stability results are shown, identifying the homology of \({\mathfrak {so}}(n,A)\) with that of \({\mathfrak {so}}(n+1,A)\) if \(n\) is large relative to the homology degree.

17B55 Homological methods in Lie (super)algebras
20G10 Cohomology theory for linear algebraic groups
20J05 Homological methods in group theory
18G99 Homological algebra in category theory, derived categories and functors
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