Total cohomology of solvable lie algebras and linear deformations

TY - JOUR

T1 - Total cohomology of solvable lie algebras and linear deformations

AU - Cagliero, Leandro

AU - Tirao, Paulo

N1 - Publisher Copyright: © 2015 American Mathematical Society.

PY - 2016/5

Y1 - 2016/5

N2 - Given a finite-dimensional Lie algebra g, let Γo(g) be the set of irreducible g-modules with non-vanishing cohomology. We prove that a gmodule V belongs to Γo(g) only if V is contained in the exterior algebra of the solvable radical s of g, showing in particular that Γo(g) is a finite set and we deduce that H∗(g, V) is an L-module, where L is a fixed subgroup of the connected component of Aut(g) which contains a Levi factor. We describe Γo in some basic examples, including the Borel subalgebras, and we also determine Γo(sn) for an extension sn of the 2-dimensional abelian Lie algebra by the standard filiform Lie algebra fn. To this end, we described the cohomology of fn. We introduce the total cohomology of a Lie algebra g, as (formula presented) and we develop further the theory of linear deformations in order to prove that the total cohomology of a solvable Lie algebra is the cohomology of its nilpotent shadow. Actually we prove that s lies, in the variety of Lie algebras, in a linear subspace of dimension at least dim(s/n)2, n being the nilradical of s, that contains the nilshadow of s and such that all its points have the same total cohomology.

AB - Given a finite-dimensional Lie algebra g, let Γo(g) be the set of irreducible g-modules with non-vanishing cohomology. We prove that a gmodule V belongs to Γo(g) only if V is contained in the exterior algebra of the solvable radical s of g, showing in particular that Γo(g) is a finite set and we deduce that H∗(g, V) is an L-module, where L is a fixed subgroup of the connected component of Aut(g) which contains a Levi factor. We describe Γo in some basic examples, including the Borel subalgebras, and we also determine Γo(sn) for an extension sn of the 2-dimensional abelian Lie algebra by the standard filiform Lie algebra fn. To this end, we described the cohomology of fn. We introduce the total cohomology of a Lie algebra g, as (formula presented) and we develop further the theory of linear deformations in order to prove that the total cohomology of a solvable Lie algebra is the cohomology of its nilpotent shadow. Actually we prove that s lies, in the variety of Lie algebras, in a linear subspace of dimension at least dim(s/n)2, n being the nilradical of s, that contains the nilshadow of s and such that all its points have the same total cohomology.

KW - Lie algebra vanishing cohomology

KW - Linear deformations

KW - Nilshadow

KW - Total cohomology

UR - http://www.scopus.com/inward/record.url?scp=84955288123&partnerID=8YFLogxK

U2 - 10.1090/tran/6424

DO - 10.1090/tran/6424

M3 - 文章

AN - SCOPUS:84955288123

SN - 0002-9947

VL - 368

SP - 3341

EP - 3358

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 5

ER -