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Non-Existence of Complex Structures on Filiform Lie Algebras [article]

M. Goze, E. Remm
2003 arXiv   pre-print
The aim of this work is to prove the nonexistence of complex structures over nilpotent Lie algebras of maximal class (also called filiform).  ...  Then no 2n−dimensional complex filiform Lie algebra is a vectorial direct sum of two n−dimension filiform sub-algebras. Proof.  ...  Thus g 1 or σ(g 1 ) is a n−dimensional complex filiform Lie algebra.  ... 
arXiv:math/0103035v2 fatcat:plmqd6jbcne3vhoi2vwyuie26q

Links among Characteristically Nilpotent, $C$-Graded and Derived Filiform Lie Algebras

J.C. Benjumea, F.J. Echarte, M.C. Márquez, J. Núñez
2005 Rocky Mountain Journal of Mathematics  
At the same time we offer an algorithm that will allow us to know, given a complex filiform Lie algebra, which of those properties hold.  ...  The aim of this paper is to study the interconnectivity among Lie algebras that are characteristically nilpotent, derived and c-graded filiform, establishing some characterization theorems involving them  ...  It was used by Ancochea and Goze to classify complex nilpotent Lie algebras of dimension 7 and complex filiform Lie algebras of dimension 8, since it is an invariant of these algebras, in the sense of  ... 
doi:10.1216/rmjm/1181069676 fatcat:jg2ujpxm5jakporp6qjlwbdhla

Page 1861 of Mathematical Reviews Vol. , Issue 2004c [page]

2004 Mathematical Reviews  
We find that there exist 188 families of complex filiform Lie algebras of dimension 11.” Athanassios I.  ...  Summary: “In this paper we give the explicit classification of complex filiform Lie algebras of dimension 11.  ... 

An algorithm to obtain laws of families of filiform Lie algebras

J.R. Gómez, A. Jiménez-Merchán, J. Núñez-Valdés
1998 Linear Algebra and its Applications  
By this recursive algorithm, we obtain parametrizations of the affine algebraic sets of filiform Lie algebras of dimensions 11 and 12. 0 1998 Elsevier Science Inc. All rights reserved.  ...  We present a polynomial time algorithm which generates families of filiform Lie algebras of dimension n.  ...  It allows to deal with the variety in a easier way. 0 The laws of filiform Lie algebras of dimensions 11 and 12 In this section we consider the set of laws of filiform Lie algebras of dimensions 11 and  ... 
doi:10.1016/s0024-3795(97)10077-5 fatcat:hyy3r65ndjhirpvng6isbnw22u

Symbolic and Iterative Computation of Quasi-Filiform Nilpotent Lie Algebras of Dimension Nine

Mercedes Pérez, Francisco Pérez, Emilio Jiménez
2015 Symmetry  
This paper addresses the problem of computing the family of two-filiform Lie algebra laws of dimension nine using three Lie algebra properties converted into matrix form properties: Jacobi identity, nilpotence  ...  This structure theorem permits the exhaustive classification of the quasi-filiform nilpotent Lie algebras of dimension nine with current computational methodologies.  ...  López from University of Sevilla for their support and orientation on the knowledge of Lie Algebras.  ... 
doi:10.3390/sym7041788 fatcat:4gqgq5w4ojhltcfrupjdcgsn3a

Invariant functions and contractions of certain types of Lie algebras of lower dimensions

J.M. Escobar, J. Núñez, P. Pérez-Fernández
2018 Journal of Nonlinear Mathematical Physics  
We focus the study of these functions in the frame of the filiform Lie algebras, trying to extend to these algebras some of the properties of such functions over semi-simple Lie algebras.  ...  In this paper, we deal with contractions of Lie algebras.  ...  Indeed, although by using a different procedure, we have confirmed some results by those authors and have also obtained new results in the case of filiform Lie algebras of dimension 5.  ... 
doi:10.1080/14029251.2018.1494705 fatcat:zvfvlwipx5ebldax6unqcmeoay

Low-dimensional filiform Lie algebras

J.R. Gómez, A. Jimenéz-Merchán, Y. Khakimdjanov
1998 Journal of Pure and Applied Algebra  
We give a complete classification up to isomorphisms of complex filiform Lie algebras of dimension m with M 5 11.  ...  IYk"r(Xk,Xk+l) =x" &,u" E ~2('%,-bz)~ Let 5% be the variety of filiform Lie algebras laws in a m-dimensional vectorial space.  ...  Filiform Lie algebras of dimension m 5 11 In this section we present the classes of filiform Lie algebras laws of dimension 11 or less which are classified in the sections below.  ... 
doi:10.1016/s0022-4049(97)00096-0 fatcat:kgb6hgywbfeh7gkd2x2eg2x7tu

Characteristically nilpotent Lie algebras and symplectic structures [article]

Dietrich Burde
2004 arXiv   pre-print
Among these Lie algebras are filiform CNLAs of dimension n< 14. It turns out that there are many examples of CNLAs which admit a symplectic structure.  ...  A generalization of a sympletic structure is an affine structure on a Lie algebra.  ...  All complex Lie algebras of dimension n ≤ 4 are affine except for sl 2 (C), and all complex nilpotent Lie algebras of dimension n ≤ 7 are affine.  ... 
arXiv:math/0409251v1 fatcat:ay56ubzyi5fudj6zq7gwged2ca

A class of nilpotent lie algebras

J.M. Cabezas, L.M. Camacho, J.R. Gómez, R.M. Navarro
2000 Communications in Algebra  
A pfiliform Lie algebra g is a nilpotent Lie algebra for which Goze's invariant is (np, 1,. . . , I ) . These Lie algebras are well known for p 2 n -4, n = dirn(g).  ...  In this paper we describe the pfiliform Lie algebras, for p = n -5 and we give their classification when the derived subalgebra is maximal.  ...  ACKNOWLEDGEMENTS This work was partly supported by the PAICYT, FQM143 of the Junta de Andalucia (Spain) and project UPV 066.163-EA032/98 of Univ. Pals Vasco (Spain).  ... 
doi:10.1080/00927870008827102 fatcat:ky3ddyjprbgadf2zwwqxxk4m4q

Characteristically nilpotent Lie algebras and symplectic structures

Dietrich Burde
2006 Forum mathematicum  
Among these Lie algebras are filiform CNLAs of dimension n ≤ 14. It turns out that there are many examples of CNLAs which admit a symplectic structure.  ...  A generalization of a sympletic structure is an affine structure on a Lie algebra.  ...  Every 10-dimensional complex symplectic filiform Lie algebra is isomorphic to one of the following laws: µ 9 10 (α), µ 10 10 , µ 11 10 , µ 13 10 (α, β), µ 16 10 (α, β), µ 26 10 (0, β), µ 27 10 , µ 29 10  ... 
doi:10.1515/forum.2006.038 fatcat:sf4s5hgivbdjdkw2xyeg6kiebu

On filiform Lie algebras. Geometric and algebraic studies [article]

Elisabeth Remm
2017 arXiv   pre-print
A finite dimensional filiform K-Lie algebra is a nilpotent Lie algebra g whose nil index is maximal, that is equal to dim g -1.  ...  If we fix a Vergne's basis, the set of filiform n-dimensional Lie algebras is a closed Zariski subset of an affine space generated by the structure constants associated with this fixed basis.  ...  We do the same thing for the dimensions 10 and 11, giving the description of the contact 11-dimensional filiform Lie algebras and consequently the description of the symplectic 10-dimensional filiform  ... 
arXiv:1712.00318v1 fatcat:3ohecpyaljesnlkuxdt47p6j7y

Graded filiform Lie algebras and symplectic nilmanifolds [article]

Dmitri V. Millionschikov
2002 arXiv   pre-print
It is proved that a symplectic filiform Lie algebra is a filtered deformation of some N-graded symplectic filiform Lie algebra. But this condition is not sufficient.  ...  In particular we describe the spaces of symplectic cohomology classes for all even-dimensional algebras of the list.  ...  Acknowledgments This work started from discussions on results of [19] , [20] , [24] with S. Salamon during the visit to the Torino University in December 2000. The author is very grateful to A.  ... 
arXiv:math/0205042v1 fatcat:wsdfdb2ltnhf5hsemnhu2sebem

Filiform Lie algebras of dimension 8 as degenerations [article]

Joan Felipe Herrera-Granada, Paulo Tirao
2013 arXiv   pre-print
For each complex 8-dimensional filiform Lie algebra we find another non isomorphic Lie algebra that degenerates to it.  ...  Since this is already known for nilpotent Lie algebras of rank > 1, only the caracteristically nilpotent ones should be considered.  ...  This paper is part of the PhD. thesis of the first author. He thanks CONICET for the Ph.D. fellowship awarded that made this possible.  ... 
arXiv:1308.4580v1 fatcat:vk5jg7kv6ng4fboivsiy24r2hm

On nilpotent filiform Lie algebras of dimension eight

P. Barbari, A. Kobotis
2003 International Journal of Mathematics and Mathematical Sciences  
The aim of this paper is to determine both the Zariski constructible set of characteristically nilpotent filiform Lie algebrasgof dimension8and that of the set of nilpotent filiform Lie algebras whose  ...  group of automorphisms consists of unipotent automorphisms, in the variety of filiform Lie algebras of dimension8overC.  ...  Consider the set of complex filiform Lie algebras. Consider C 8 with  ... 
doi:10.1155/s016117120311201x fatcat:j4yvn7t3zzdedkaxdc7w3aldx4

There are no rigid filiform Lie algebras of low dimension [article]

Paulo Tirao, Sonia Vera
2017 arXiv   pre-print
We prove that there are no rigid complex filiform Lie algebras in the variety of (filiform) Lie algebras of dimension less than or equal to 11.  ...  This follows by constructing non trivial linear deformations in a Zariski open dense set of the variety of filiform Lie algebras of dimension 9, 10 and 11. (In lower dimensions this is well known.)  ...  This paper is part of the PhD. thesis of the second author. She thanks CONICET for the Ph.D. fellowship awarded that made this possible.  ... 
arXiv:1709.04793v1 fatcat:yjajr4j5bbhfnkplwc6prgf6r4
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