Total cohomology of solvable Lie algebras and linear deformations
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- by Leandro Cagliero and Paulo Tirao PDF
- Trans. Amer. Math. Soc. 368 (2016), 3341-3358 Request permission
Abstract:
Given a finite-dimensional Lie algebra $\mathfrak {g}$, let $\Gamma _\circ (\mathfrak {g})$ be the set of irreducible $\mathfrak {g}$-modules with non-vanishing cohomology. We prove that a $\mathfrak {g}$-module $V$ belongs to $\Gamma _\circ (\mathfrak {g})$ only if $V$ is contained in the exterior algebra of the solvable radical $\mathfrak {s}$ of $\mathfrak {g}$, showing in particular that $\Gamma _\circ (\mathfrak {g})$ is a finite set and we deduce that $H^*(\mathfrak {g},V)$ is an $L$-module, where $L$ is a fixed subgroup of the connected component of $\operatorname {Aut}(\mathfrak {g})$ which contains a Levi factor.
We describe $\Gamma _\circ$ in some basic examples, including the Borel subalgebras, and we also determine $\Gamma _\circ (\mathfrak {s}_n)$ for an extension $\mathfrak {s}_n$ of the 2-dimensional abelian Lie algebra by the standard filiform Lie algebra $\mathfrak {f}_n$. To this end, we described the cohomology of $\mathfrak {f}_n$.
We introduce the total cohomology of a Lie algebra $\mathfrak {g}$, as $TH^*(\mathfrak {g})=$ $\bigoplus _{V\in \Gamma _\circ (\mathfrak {g})} H^*(\mathfrak {g},V)$ 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 $\mathfrak {s}$ lies, in the variety of Lie algebras, in a linear subspace of dimension at least $\dim (\mathfrak {s}/\mathfrak {n})^2$, $\mathfrak {n}$ being the nilradical of $\mathfrak {s}$, that contains the nilshadow of $\mathfrak {s}$ and such that all its points have the same total cohomology.
References
- L. Auslander and L. W. Green, $G$-induced flows, Amer. J. Math. 88 (1966), 43–60. MR 199308, DOI 10.2307/2373046
- Emmanuel Breuillard, Geometry of locally compact groups of polynomial growth and shape of large balls, Groups Geom. Dyn. 8 (2014), no. 3, 669–732. MR 3267520, DOI 10.4171/GGD/244
- Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, AMS Chelsea Publishing, Providence, RI, 2006. Reprint of the 1962 original. MR 2215618, DOI 10.1090/chel/356
- J. Dixmier, Cohomologie des algèbres de Lie nilpotentes, Acta Sci. Math. (Szeged) 16 (1955), 246–250 (French). MR 74780
- Nick Dungey, A. F. M. ter Elst, and Derek W. Robinson, Analysis on Lie groups with polynomial growth, Progress in Mathematics, vol. 214, Birkhäuser Boston, Inc., Boston, MA, 2003. MR 2000440, DOI 10.1007/978-1-4612-2062-6
- Fritz Grunewald and Joyce O’Halloran, Deformations of Lie algebras, J. Algebra 162 (1993), no. 1, 210–224. MR 1250536, DOI 10.1006/jabr.1993.1250
- G. Hochschild and J.-P. Serre, Cohomology of Lie algebras, Ann. of Math. (2) 57 (1953), 591–603. MR 54581, DOI 10.2307/1969740
- Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0143793
- G. Leger and E. Luks, Cohomology theorems for Borel-like solvable Lie algebras in arbitrary characteristic, Canadian J. Math. 24 (1972), 1019–1026. MR 320104, DOI 10.4153/CJM-1972-103-1
- D. V. Millionshchikov, Cohomology of solvable Lie algebras, and solvmanifolds, Mat. Zametki 77 (2005), no. 1, 67–79 (Russian, with Russian summary); English transl., Math. Notes 77 (2005), no. 1-2, 61–71. MR 2158698, DOI 10.1007/s11006-005-0006-2
- Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282, DOI 10.1017/CBO9780511609589
Additional Information
- Leandro Cagliero
- Affiliation: CIEM-FaMAF, Universidad Nacional de Córdoba, Argentina
- Paulo Tirao
- Affiliation: CIEM-FaMAF, Universidad Nacional de Córdoba, Argentina
- Received by editor(s): December 2, 2012
- Received by editor(s) in revised form: March 11, 2014
- Published electronically: September 15, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 3341-3358
- MSC (2010): Primary 17B56; Secondary 17B30, 16S80
- DOI: https://doi.org/10.1090/tran/6424
- MathSciNet review: 3451879