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 Mat. Sb., 1996, Volume 187, Number 2, Pages 59–80 (Mi msb109)

A free Lie algebra as a module over the full linear group

V. M. Zhuravlev

M. V. Lomonosov Moscow State University

Abstract: In this paper we consider a tree Lie algebra over a field of characteristic zero. This algebra is a module over the full linear group, and the spaces of homogeneous elements are invariant under this action. We study the decomposition of the homogeneous spaces into irreducible components and calculate their multiplicities. One method for calculating these multiplicities involves their connection with the values of the irreducible characters of the symmetric group on conjugacy classes of elements corresponding to a product of independent cycles of the same length. In the second section we give an explicit formula for calculating such character values. This formula is analogous to the hook formula for the dimension of the irreducible modules of the symmetric group. In the second method for calculating multiplicities we make use of Witt's formula for the dimensions of the polyhomogeneous components of a free Lie algebra. The rest of this paper deal with relations between the Hilbert series of a free two-generator Lie algebra and the generating series of the multiplicities of the irreducible modules in this algebra.

DOI: https://doi.org/10.4213/sm109

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English version:
Sbornik: Mathematics, 1996, 187:2, 215–236

Bibliographic databases:

UDC: 512.5
MSC: 17B01, 17B05

Citation: V. M. Zhuravlev, “A free Lie algebra as a module over the full linear group”, Mat. Sb., 187:2 (1996), 59–80; Sb. Math., 187:2 (1996), 215–236

Citation in format AMSBIB
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This publication is cited in the following articles:
1. Hong, J, “Decomposition of free Lie algebras into irreducible components”, Journal of Algebra, 197:1 (1997), 127
2. Zhuravlev, VM, “Polynilpotent components in the free Lie algebra and the wreath products of the cyclic groups”, Communications in Algebra, 25:10 (1997), 3189
3. Klyachko, A, “Some identities and asymptotics for characters of the symmetric group”, Journal of Algebra, 206:2 (1998), 413
4. Kovacs, LG, “Lie powers of the natural module for GL(2)”, Journal of Algebra, 229:2 (2000), 435
5. Tirao, P, “On the homology of free nilpotent Lie algebras”, Journal of Lie Theory, 12:2 (2002), 309
6. Ames, G, “The GL-module structure of the Hochschild homology of truncated tensor algebras”, Journal of Pure and Applied Algebra, 193:1–3 (2004), 11
7. Kovacs, LG, “A combinatorial proof of Klyachko's Theorem on Lie representations”, Journal of Algebraic Combinatorics, 23:3 (2006), 225
8. Satoh, T, “THE COKERNEL OF THE JOHNSON HOMOMORPHISMS OF THE AUTOMORPHISM GROUP OF A FREE METABELIAN GROUP”, Transactions of the American Mathematical Society, 361:4 (2009), 2085
9. Satoh T., “A Reduction of the Target of the Johnson Homomorphisms of the Automorphism Group of a Free Group”, Trans Amer Math Soc, 363:3 (2011), 1631–1664
10. Enomoto H., Enomoto N., “Sp-Irreducible Components in the Johnson Cokernels of the Mapping Class Groups of Surfaces, i”, J. Lie Theory, 24:3 (2014), 687–704
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