Derivation Lie algebra;
Free Lie algebra;
Symplectic representation;
Young diagram;
Johnson homomorphism;
MAPPING CLASS-GROUPS;
CENTRAL SERIES;
HOMOLOGY;
SURFACES;
REPRESENTATIONS;
QUOTIENT;
GRAPHS;
MODULI;
D O I:
10.1016/j.aim.2015.06.017
中图分类号:
O1 [数学];
学科分类号:
0701 ;
070101 ;
摘要:
We study the structure of the symplectic invariant part h(g,1)(Sp) of the Lie algebra h(g),(1) consisting of symplectic derivations of the free Lie algebra generated by the rational homology group of a closed oriented surface Sigma(g) of genus g. First we describe the orthogonal direct sum decomposition of this space which is induced by the canonical metric on it and compute it explicitly up to degree 20. In this framework, we give a general constraint which is imposed on the Sp-invariant component of the bracket of two elements in h(g),(1). Second we clarify the relations among h(g),(1) and the other two related Lie algebras h(g),* and h(g) which correspond to the cases of a closed surface Sigma(g) with and without base point * is an element of Sigma(g). In particular, based on a theorem of Labute, we formulate a method of determining these differences and describe them explicitly up to degree 20. Third, by giving a general method of constructing elements of h(g,1)(Sp), we reveal a considerable difference between two particular submodules of it, one is the Sp-invariant part of a certain ideal j(g,1) and the other is that of the Johnson image. Finally we combine these results to determine the structure of h(g,1). completely up to degree 6 including the unstable cases where the genus 1 case has an independent meaning. In particular, we see a glimpse of the Galois obstructions explicitly from our point of view. (C) 2015 Elsevier Inc. All rights reserved.
机构:
Univ Insubria, Dipartimento Sci & Alta Tecnol, Via Valleggio 11, I-22100 Como, ItalyUniv Insubria, Dipartimento Sci & Alta Tecnol, Via Valleggio 11, I-22100 Como, Italy
Bazzoni, Giovanni
Freibert, Marco
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h-index: 0
机构:
Christian Albrechts Univ Kiel, Math Seminar, D-24118 Kiel, GermanyUniv Insubria, Dipartimento Sci & Alta Tecnol, Via Valleggio 11, I-22100 Como, Italy
Freibert, Marco
Latorre, Adela
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h-index: 0
机构:
Univ Politecn Madrid, Dept Matemat Aplicada, Avda Juan de Herrera 4, Madrid 28040, SpainUniv Insubria, Dipartimento Sci & Alta Tecnol, Via Valleggio 11, I-22100 Como, Italy
Latorre, Adela
Tardini, Nicoletta
论文数: 0引用数: 0
h-index: 0
机构:
Univ Parma, Dipartimento Sci Matemat Fis & Informat, Unita Matemat & Informat, Parco Area Sci 53-A, I-43124 Parma, ItalyUniv Insubria, Dipartimento Sci & Alta Tecnol, Via Valleggio 11, I-22100 Como, Italy
机构:
CUNY, Grad Ctr, PhD Program Math, New York, NY 10016 USA
CUNY, Grad Ctr, PhD Program Phys, New York, NY 10016 USA
CUNY, Dept Math, New York City Coll Technol, Brooklyn, NY 11201 USA
Univ Toronto, Dept Math, Toronto, ON M5S 2E4, CanadaCUNY, Grad Ctr, PhD Program Math, New York, NY 10016 USA
Douglas, Andrew
Repka, Joe
论文数: 0引用数: 0
h-index: 0
机构:
Univ Toronto, Dept Math, Toronto, ON M5S 2E4, CanadaCUNY, Grad Ctr, PhD Program Math, New York, NY 10016 USA