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

This article is cited in 10 scientific papers (total in 10 papers)

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.


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

Bibliographic databases:

UDC: 512.5
MSC: 17B01, 17B05
Received: 12.07.1995

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
\by V.~M.~Zhuravlev
\paper A free Lie algebra as a~module over the~full linear group
\jour Mat. Sb.
\yr 1996
\vol 187
\issue 2
\pages 59--80
\jour Sb. Math.
\yr 1996
\vol 187
\issue 2
\pages 215--236

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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  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    3. Klyachko, A, “Some identities and asymptotics for characters of the symmetric group”, Journal of Algebra, 206:2 (1998), 413  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    4. Kovacs, LG, “Lie powers of the natural module for GL(2)”, Journal of Algebra, 229:2 (2000), 435  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    5. Tirao, P, “On the homology of free nilpotent Lie algebras”, Journal of Lie Theory, 12:2 (2002), 309  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    7. Kovacs, LG, “A combinatorial proof of Klyachko's Theorem on Lie representations”, Journal of Algebraic Combinatorics, 23:3 (2006), 225  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    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  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    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  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    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  mathscinet  zmath  isi
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