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Multipliers of the terms in the lower central series of the Lie algebra of strictly upper triangular matrices. (English) Zbl 1259.17009

Summary: A Lie algebra multiplier parallels the idea of group theory’s Schur multiplier. This paper classifies the Lie algebra multipliers for all Lie algebras in the lower central series of strictly upper triangular matrices. Multipliers are central, so the classification is focused on computing their dimensions. The calculations are lengthy because balancing various matrix positions plays an important role in determining these dimensions \(\dim M(L^k)\). The result divides into six cases and the dimensions are given as polynomials in the size of the matrices and the position in the lower central series.
Results for \(\dim M(L)\) have been obtained in [P. Batten and E. Stitzinger, On covers of Lie algebras, Commun. Algebra 24, No. 14, 4301–4317 (1996; Zbl 0893.17004)] and for \(\dim M(L^2)\) in the author’s paper [Int. Electron. J. Algebra 9, 69-77 (2011; Zbl 1258.17015)].

MSC:

17B30 Solvable, nilpotent (super)algebras
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