For each of these algebras, a multiplication table was found. In the 40s and 50s, morozov and dixmier began with the. Nilpotent lie algebras in dimension six were studied as far back as. For complex nilpotent lie algebras with a small second derived quotient, it was found that u is the direct sum of a generalized heisenberg algebra and an abelian lie algebra, and that k is one of fourteen algebras. For each class of algebra a matrix representation and a system of vector fields representation is given. To achieve the classification we use the action of the automorphism group on the second cohomology space, as isomorphism types of nilpotent lie algebras correspond to orbits of subspaces under this action.
Table i classification of nilpotent lie algebras of dimension 5 le,e31e4 ce. Symplectic forms on six dimensional real solvable lie algebras. Complex structures on indecomposable 6dimensional nilpotent. In 5 we have classi ed which lie algebras of the form g g, for a three dimensional lie algebra g, admit an integrable complex structure. Jan 01, 2005 the moduli space of six dimensional twostep nilpotent lie algebras this space is homeomorphic to a cone over a four dimensional contractible simplicial complex. Classification of complex structures on 6dimensional nilpotent lie. We give a classification of five and sixdimensional complex onegenerated nilpotent novikov algebras. We study complex product structures on nilpotent lie algebras, establishing some of their main properties, and then we restrict ourselves to 6 dimensions, obtaining the classification of 6 dimensional nilpotent lie algebras admitting such structures. Introduction the rst important research about nilpotent lie algebras is due to k. Pdf onegenerated nilpotent novikov algebras semantic scholar. Pdf pseudokahler metrics on sixdimensional nilpotent. Integrable complex structures on some 6dimensional lie algebras.
Limiting processes contractions of lie algebras appear in different areas of physics and mathematics, where lie algebras arise, e. The moduli space of sixdimensional twostep nilpotent lie. Varieties of nilpotent lie algebras of dimension less than six core. Sixdimensional nilpotent lie algebras with center of dimension one. The proof of this classification is essentially constructive. The recent results and references on sevendimensional nilpotent lie algebras can be found in 55.
Complex product structures on 6dimensional nilpotent lie. Degenerations of nilpotent lie algebras dimension less than or equal to 5. Research article classification of ordered type soliton. Integrable complex structures on some 6dimensional lie. As regards the classification of the fivedimensional nilradical algebras they were studied by mubarakzyanov 1, who. Request pdf classification of 6dimensional nilpotent lie algebras over fields of characteristic not 2 nilpotent 6dimensional lie algebras. The classification of sixdimensional lie algebras with one nilpotent basis element, izv. Case six comprises sixdimensional nilpotent lie algebras that were classified by morozov 8 and in the latter reference representations were supplied 12. We summarize and synthesize these various accounts. The inverse problem for sixdimensional codimension two. Nilpotent lie algebras are very important in the classi cat ion theory of lie algebras, where they play a central role as a consequence of t he levi theorem and the works of malcev. This space is homeomorphic to a cone over a four dimensional contractible simplicial comple x. Minimal matrix representations for sixdimensional nilpotent lie algebras, ryad ghanam and gerard thompson.
We also give a minimal faithful representation and. Classification of 6dimensional nilpotent lie algebras. The moduli space of sixdimensional t w ostep nilpotent. Niu, classification of 8dimensional nilpotent lie algebras with 4dimensional center, j. There is just one isomorphism type of nilpotent lie algebras with dimension 2, two isomorphism types in dimension 3, three isomorphism type in dimension 4, and 9. Vcuqatar publications vcu scholars compass virginia. In the proof a method due to skjelbred and sund is used. Representations of sixdimensional mubarakazyanov lie algebras. W e determine the moduli space of metric tw ostep nilpotent lie algebras of dimension up to 6.
Favre discovered the lowest dimensional cnla known until then 35, which additionally was of the same nature as dyers example. To achieve the classification we use the action of the automorphism group on the second cohomology space, as isomorphism types of nilpotent lie algebras. Sixdimensional nilpotent lie algebras sciencedirect. Classification of the quasifiliform nilpotent lie algebras of dimension 9 perez, mercedes, perez, francisco p. An upper bound has been given in 6 for the dimension of x g. Summary and details of some other classifications can be found in. A corrected list of the mubarakzyanov algebras, that is, six dimensional indecomposable solvable lie algebras for which the nilradical is five dimensional, is presented. We give a full classification of 6 dimensional nilpotent lie algebras over an arbitrary field, including fields that are not algebraically closed and fields of characteristic2. Camina lie algebras asianeuropean journal of mathematics. Pdf pseudokahler metrics on sixdimensional nilpotent lie.
Pseudokahler metrics on sixdimensional nilpotent lie algebras. Computing with nilpotent orbits in simple lie algebras of exceptional type, lms jcm, 11, 280297, 2008. Moreo ver, we exhibit standard metric representati ves of the seven isomorphism types of six dimensional tw ostep nilpotent lie algebras within our picture. The author constructs the symplectic structures on real, solvable, nonnilpotent lie algebras of dimension six. It develops and generalizes the result due to heineken for the. The set of all nilpotent ide algebra multiplications on a fixed vector space of dimension n over a field k forms an affine variety n. We give a full classification of sixdimensional nilpotent lie algebras over an arbitrary field, including fields that are not algebraically closed and. The complete set of the lie algebra couples that do not admit contraction is obtained.
This paper is devoted to characterizing all capable nilpotent lie algebras of class two with derived subalgebra of dimension. This space is homeomorphic to a cone over a four dimensional contractible simplicial complex. In 1958 morozov obtained a classi cation of the six dimensional nilpotent lie algebras over r 10. The classification of six dimensional lie algebras with one nilpotent basis element, izv. The same problem was considered in 26 for the sixdimensional real lie algebras with fourdimensional nilradicals. The multiplication tables for these lie algebras are given in table i. In the literature, sixdimensional nilpotent lie algebras have been classified by algorithmic approaches. Minimal matrix representations for sixdimensional nilpotent. A corrected list of the mubarakzyanov algebras, that is, six dimensional indecomposable solvable lie algebras.
Therefore, when constructing nilpotent lie algebras as central extensions we want to avoid constructing those with central components. Popovych wolfgang pauli institute, vienna, austria tuesday, september 20, 2016 2. Varieties of nilpotent lie algebras of dimension less than six. Nilpotent, solvable and levi decomposable lie algebras 87 100. Abstractwe give a full classification of sixdimensional nilpotent lie algebras over an arbitrary field, including fields that are not algebraically closed and fields of characteristic 2.
Pdf onegenerated nilpotent novikov algebras semantic. In 1958 morozov obtained a classi cation of the sixdimensional nilpotent lie algebras over r 10. Its classical, but i would not suggest reading the original paper even if you read italian. As an application, the behaviour of the associated. The moduli space of sixdimensional t w ostep nilpotent lie. We determine the moduli space of metric twostep nilpotent lie algebras of dimension up to 6. Computation of invariants of lie algebras by means of. Identification of the radical and levi factor 63 76. The famous feiginfrenkel duality ff1 of principal w algebras asserts that for all noncritical levels kin cone has see acl2, thm. Pdf we classify complex structures on 6dimensional nilpotent lie algebras up to equivalence. Moreover, we exhibit standard metric representatives of the seven isomorphism types of sixdimensional twostep nilpotent lie algebras within our picture. Classification of 6dimensional nilpotent lie algebras over. This article is concerned with finding representations for a certain class of sixdimensional lie algebras, specifically. Ados theorem asserts that every real lie algebra g of dimension n has a faithful representation as a subalgebra of glp,r for some p.
The structure of lie algebras and the classification problem. The four and sevendimensional nilradical cases were classi ed. This is a step toward a classi cation in dimension six at leas for decomposable algebras. It remains to consider the indecomposable case when the nilradical has dimension five.
The rst signi cant research about n ilpotent lie algebras is due to umlauf in the 19th century. We give a full classification of 6dimensional nilpotent lie algebras over an arbitrary field, including fields that are not algebraically closed and fields of characteristic2. The structure of lie algebras and the classification. The moduli space of sixdimensional twostep nilpotent lie algebras article pdf available in annals of global analysis and geometry 271. Moreo ver, we exhibit standard metric representati ves of the seven isomorphism types of sixdimensional tw ostep nilpotent lie algebras within our picture. Minimal matrix representations of fivedimensional lie algebras. In particular, it is known that a complex structure can be. Introduction let g be a simplyconnected, connected and rational sstep nilpotent lie group of real dimension 2n. In this paper we prove this statement for filiform lie algebras, quasifiliform lie algebras, nilpotent lie algebras with nilindex 2 and 2generated nilpotent metabelian lie algebras. This space is homeomorphic to a cone over a fourdimensional contractible simplicial complex. A lie algebra l is said to be metabelian if l 2 is abelian.
Moreover, we exhibit standard metric representatives of the seven isomorphism types of six dimensional twostep nilpotent lie algebras within our picture. Abeliancomplexstructures on6dimensionalcompactnilmanifolds. In fact for the convenience of the reader we provide also an smatrix for all indecomposable lie algebras of dimension and less. Matrix representation for sevendimensional nilpotent lie algebras r, ghanam, mustafa b, basim, mt, mustafa, and g, thompson, journal of physical mathematics, 2016.
This completes mubarakzyanovs classification of the real six. All possible pairs of complex sixdimensional nilpotent lie algebras are considered and necessary contraction conditions are verified. In dimension seven and lower, nilsoliton metric lie algebras have been classified 29. The work falls into two cases, when the algebra is decomposable into two lower dimensional ideals and when it is indecomposable with four dimensional nilradical. Minimal representations for 6dimensional nilpotent lie algebra. Contraction admissible pairs of complex sixdimensional. Classification of 6dimensional nilpotent lie algebras over fields of. Abstractwe give a full classification of six dimensional nilpotent lie algebras over an arbitrary field, including fields that are not algebraically closed and fields of characteristic 2. Recently, the authors in a joint paper obtained the structure of all capable nilpotent lie algebras with derived subalgebra of dimension at most.
Dec 11, 2016 all possible pairs of complex six dimensional nilpotent lie algebras are considered and necessary contraction conditions are verified. Nilpotent 6dimensional lie algebras over any field of characteristic not 2 are classified. In the 40s and 50s, morozov and dixmier began with the systematical study of this class of algebras. Lg where l gis the langlands dual lie algebra of and the dual level. But googling bianchi classification might help you find references. Classification of ordered type soliton metric lie algebras. The theorem offers no practical information about the size of p in relation to n and in principle p may be very large compared to n. Ams proceedings of the american mathematical society. Pdf the moduli space of sixdimensional twostep nilpotent. Limiting processes contractions of lie algebras appear in different areas of.