Cosocle
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In mathematics, the term cosocle (socle meaning pedestal in French) has several related meanings. In group theory, a cosocle of a group G, denoted by Cosoc(G), is the intersection of all maximal normal subgroups of G.[1] If G is a quasisimple group, then Cosoc(G) = Z(G).[1]
In the context of Lie algebras, a cosocle of a symmetric Lie algebra is the eigenspace of its structural automorphism that corresponds to the eigenvalue +1. (A symmetric Lie algebra decomposes into the direct sum of its socle and cosocle.)[2]
In the context of module theory, the cosocle of a module over a ring R is defined to be the maximal semisimple quotient of the module.[3]
See also
References
- ↑ 1.0 1.1 Adolfo Ballester-Bolinches, Luis M. Ezquerro, Classes of Finite Groups, 2006, ISBN:1402047185, p. 97
- ↑ Mikhail Postnikov, Geometry VI: Riemannian Geometry, 2001, ISBN:3540411089,p. 98
- ↑ Braden, Tom; Licata, Anthony; Phan, Christopher; Proudfoot, Nicholas; Webster, Ben (2011). "Localization algebras and deformations of Koszul algebras". Selecta Math. 17 (3): 533–572. doi:10.1007/s00029-011-0058-y. "Lemma 3.8".
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