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Homology of symplectic and orthogonal algebras. (English) Zbl 0716.17019
Let A be an associative algebra with unity over a field k of characteristic zero. Let \({\mathfrak gl}_ n(A)\) denote the Lie algebra of matrices of type (n,n) over A, and let \({\mathfrak gl}(A)=\lim_{n\to \infty}{\mathfrak gl}_ n(A)\). J.-L. Loday and D. Quillen [Comment. Math. Helv. 59, 565-591 (1984; Zbl 0565.17006)] characterized the space of primitive elements of the Hopf algebra \(H_*({\mathfrak gl}(A);k)\). They proved that Prim \(H_*({\mathfrak gl}(A);k)=HC_{*-1}(A)\), where \(HC_*\) is the cyclic homology.
Let us suppose in addition that A is endowed with an anti-involution \(a\mapsto \bar a\) (i.e. \(\overline{ab}=\bar b \bar a)\). Then the Lie algebras \({\mathfrak o}(A)\) (resp. \({\mathfrak sp}(A))\) of orthogonal (resp. symplectic) matrices can be defined in the usual way. The authors prove a result analogous to the above one, namely that Prim \(H_*({\mathfrak o}(A);k)=_{-1}HD_{*-1}(A)\), Prim \(H_*({\mathfrak sp}(A);k)=_{- 1}HD_{*-1}(A)\), where \(_{-1}HD_*\) is the dihedral homology [see J.-L. Loday, Adv. Math. 66, 119-148 (1987; Zbl 0627.18006)]. As a consequence of this, we can see that \({\mathfrak o}(A)\) and \({\mathfrak sp}(A)\) have the same homology. Further the authors study the stabilization homomorphisms \(H_ p({\mathfrak o}_{n-1}(A);k)\to H_ p({\mathfrak o}_ n(A);k)\) and \(H_ p({\mathfrak sp}_{2n-2}(A);k)\to H_ p({\mathfrak sp}_{2n}(A);k)\). They determine the stable range and the first obstruction to stability.
The technique of the paper depends substantially on the invariant theory for orthogonal and symplectic matrices. This theory is reviewed in detail in the first part of the paper.
Reviewer: J.Vanžura

17B56 Cohomology of Lie (super)algebras
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
19D55 \(K\)-theory and homology; cyclic homology and cohomology
18G60 Other (co)homology theories (MSC2010)
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
Full Text: DOI
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