Homology of symplectic and orthogonal algebras
type: Item
instance of: scholarly article
title: Homology of symplectic and orthogonal algebras
zbMATH Open document ID: 0716.17019
DOI: 10.1016/0001-8708(88)90061-8
author: J.-L. Loday
author: Claudio Procesi
published in: Advances in Mathematics
publication date: 1988-01-01T00:00:00Z
review text: 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 \({\mathfrak gl}(A)=\lim_{n\to \infty}{\mathfrak gl}_ n(A)\). \textit{J.-L. Loday} and \textit{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 \textit{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.
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 19D55
Mathematics Subject Classification ID: 18G60
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 4179610
zbMATH Keywords: homology of Lie algebra
zbMATH Keywords: orthogonal Lie algebra
zbMATH Keywords: symplectic Lie algebra
zbMATH Keywords: primitive elements
zbMATH Keywords: dihedral homology
zbMATH Keywords: stable range
zbMATH Keywords: obstruction
zbMATH Keywords: invariant theory
reviewed by: Jiří Vanžura
MaRDI profile type: Publication
full work available at URL: 0001-8708(88)90061-8
OpenAlex ID: W2031975988
cites work: Q3774051
cites work: Homologie et cohomologie des algèbres de Lie
cites work: Homologies diédrale et quaternionique. (Dihedral and quaternionic homology)
cites work: Cyclic homology and the Lie algebra homology of matrices
cites work: The invariant theory of \(n\times n\) matrices
cites work: Q3869479
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description: scientific article; zbMATH DE number 4179610