Subdirect sums of Lie algebras. (English) Zbl 1441.17019
Results of G. Baumslag and J. Roseblade on subdirect products of free groups [J. Lond. Math. Soc., II. Ser. 30, 44–52 (1984; Zbl 0559.20018)] have been generalised by M. R. Bridson et al. to subdirect products of limit groups [Geom. Dedicata 92, 95–103 (2002; Zbl 1048.20009); Ann. Math. (2) 170, No. 3, 1447–1467 (2009; Zbl 1196.20047)].
The authors develop a similar theory for Lie algebras. This requires substantial changes in the approach, in particular because the homotopical methods of group theory are not generally available. The paper under review appeals to homological methods to study subdirect sums of Lie algebras of homological type \(F P_{s}\). Among the result one obtains there is a Lie algebra version of results of the above cited papers: Theorem B states that if \(L\) is a Lie algebra of type \(FP_{k}\) which is a subdirect product of the finitely generated Lie algebras \(F_{1}, \dots, F_{k}\), that is, \(L \le F_{1} \oplus \dots \oplus F_{k}\), with all projections of \(L\) on the \(F_{i}\) being surjective, then \(L\) is a direct sum of at most \(k\) free Lie algebras.
The paper also consider fibre sums of Lie algebras, and tackles the Lie algebra analogue of the \(n\)-\((n+1)\)-\((n+2)\) conjecture of B. Kuckuck [Q. J. Math. 65, No. 4, 1293–1318 (2014; Zbl 1353.20037)], which generalises the 1-2-3 Theorem of G. Baumslag et al. [Comment. Math. Helv. 75, No. 3, 457–477 (2000; Zbl 0973.20034)].
The authors develop a similar theory for Lie algebras. This requires substantial changes in the approach, in particular because the homotopical methods of group theory are not generally available. The paper under review appeals to homological methods to study subdirect sums of Lie algebras of homological type \(F P_{s}\). Among the result one obtains there is a Lie algebra version of results of the above cited papers: Theorem B states that if \(L\) is a Lie algebra of type \(FP_{k}\) which is a subdirect product of the finitely generated Lie algebras \(F_{1}, \dots, F_{k}\), that is, \(L \le F_{1} \oplus \dots \oplus F_{k}\), with all projections of \(L\) on the \(F_{i}\) being surjective, then \(L\) is a direct sum of at most \(k\) free Lie algebras.
The paper also consider fibre sums of Lie algebras, and tackles the Lie algebra analogue of the \(n\)-\((n+1)\)-\((n+2)\) conjecture of B. Kuckuck [Q. J. Math. 65, No. 4, 1293–1318 (2014; Zbl 1353.20037)], which generalises the 1-2-3 Theorem of G. Baumslag et al. [Comment. Math. Helv. 75, No. 3, 457–477 (2000; Zbl 0973.20034)].
Reviewer: Andrea Caranti (Trento)
MSC:
| 17B55 | Homological methods in Lie (super)algebras |
| 17B05 | Structure theory for Lie algebras and superalgebras |
| 17B30 | Solvable, nilpotent (super)algebras |
Keywords:
subdirect products; finitely generated Lie algebras; free Lie algebras; homological methods; fibre sumsReferences:
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