Let v be a nite dimensional vector space over an algebraically closed eld of characteristic 0. The generalised nilradical of a lie algebra nasaads. If 1 is a nilpotent element of l, it will be represented by a nilpotent matrix in any finitedimensional representation. We give the classification of all solvable leibniz superalgebras such that even part is a lie algebra and nilradical is a lie superalgebra with a maximal. Identification of the radical and levi factor 63 76. From the classical theory of finitedimensional lie algebras it is known that an arbitrary lie algebra is a semidirect sum of the solvable radical and. A solvable lie algebra l has the property that its nilradical n contains its own centraliser.
A lie algebra is semisimple if and only if its radical is. All solvable, indecomposable, finitedimensional, complex lie superalgebras. In this example we calculate the nilradical of a certain 4dimensional lie algebra. Sevendimensionalliealgebraswithnilradical isomorphicto a r. Here we consider several possible generalisations of the nilradical for which this property holds in any lie algebra. Pdf a class of solvable lie algebras and their casimir invariants. Creative commons attributionnoncommercialnoderivatives 4. Abstractwe construct all solvable lie algebras with a specific ndimensional nilradical nn,3 which contains the previously studied filiform n2dimensional nilpotent algebra nn2,1 as a subalgebra but not as an ideal. Some features of rank one real solvable cohomologically. In this paper we show that the method for describing solvable lie algebras with given nilradical by means of nonnilpotent outer derivations of the nilradical is also applicable to the case of leibniz algebras. In mathematics, a lie algebra is solvable if its derived series terminates in the zero subalgebra.
Towers department of mathematics, lancaster university lancaster la1 4yf, england email. We observe that a similar assertion holds for characteristic sequences n 1. For an ideal, i, in a ring, r, the set of all elements, a, in r for which a n is a member of i for some positive integer n. We study solvable lie groups which admit a leftinvariant metric of strictly negative ricci curvature. Pdf solvable leibniz algebras with filiform nilradical. Pdf solvable lie algebras with abelian nilradicals researchgate. A nitedimensional leibniz algebra is a semisimple lie algebra adjoined by a semidirect sum with a solvable radical 5. Any such group is essentially determined by the nilradical of its lie algebra, which is then called an einstein nilradical. Then the dimension of q is not greater than the maximal number of nilindependent derivations of n. Solvable lie algebras with abelian nilradicals 407 and of the restriction of ad l to the nilradical of l. Any solvable lie algebra has a descending chain of ideals g g 0. Can someone provide two concrete examples the lie algebra which is solvable, but not nilpotent. Solvable leibniz algebras with filiform nilradical.
Citeseerx document details isaac councill, lee giles, pradeep teregowda. U, where u is the nilradical in gs, called the unipotent hull of g 10, 11 or the nilshadow of g 4. This approach is based on the nilindependent deriva tions of nilradical. Using this method, we obtain the classification of three and fourdimensional solvable lie algebras over fields of any characteristic. Solvable can be divided into two parts, nilpotent and nonnilpotent. The generalised nilradical of a lie algebra lancaster. Pdf solvable leibniz algebras with heisenberg nilradical. Einstein solvable lie algebras with a free nilradical y. It turns out that such a metric always exists, except for in the two cases, when is one of the algebras of rank two, or, and is a one. Is there a condition for the nilradical to equal the derived subalgebra. Right and left solvable extensions of an associative.
The generalized casimir invariants for the ngraded nilradicals and their associated solvable extensions are computed by the method of moving frames. It is shown that for a finitedimensional solvable rigid lie algebra r, its rank is upper bounded by the length of the characteristic sequence cn of its nilradical n. In this paper, we continue the description of solvable leibniz algebras whose nilradical is a filiform algebra. We extend this result to a classification of solvable leibniz algebras with heisenberg nilradical. A lie algebra is reductive if and only if its radical equals its center. Nilpotent and solvable algebras math 649, 20 dan barbasch february 14 dan barbasch nilpotent and solvable algebras math 649, 20. Cohomologically rigid solvable lie algebras with a.
In this paper, we investigate solvable leibniz superalgebras whose nilradical is a lie superalgebra with maximal nilindex. Solvable leibniz algebras with triangular nilradical core. All solvable lie algebras whose nilradical is the naturally graded. And further explain the subtle differences between the solvable lie algebra and the nilpotent lie algebra. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. On solvable lie groups of negative ricci curvature. Some features of rank one real solvable cohomologically rigid. Rigidity problem for lattices in solvable lie groups. Let l be a solvable leibniz algebra and n be its nilradical. In this paper we describe a simple method for obtaining a classification of smalldimensional solvable lie algebras. The purpose of this article is to form the nilpotent frame 2 of a solvable lie algebra i. It should be noted that lie superalgebra with a maximal nilindex only exists in the variety of lie 2,m when m is odd. Sixdimensional solvable lie multiplication groups with vedimensional nilradical using necessary conditions we found in 5, theorems 3.
All finitedimensional indecomposable solvable lie algebras. In algebra, the nilradical of a lie algebra is a nilpotent ideal, which is as large as possible the nilradical of a finitedimensional lie algebra is its maximal nilpotent ideal, which exists because the sum of any two nilpotent ideals is nilpotent. We give necessary and sufficient conditions of the existence of a left. All solvable extensions of a class of nilpotent lie algebras. It is well known that the nilradical of a finitedimensional lie algebra over a field of characteristic p 0 need not be characteristic that is, invariant under all derivations of the algebra, but is there an example of a solvable lie algebra with noncharacteristic nilradical. Representations of codimension one nonabelian nilradical. Since solvable leibniz algebras whose nilradical is quasi. It should be noted that lie superalgebra with a maximal nilindex only exists in the variety of lie2,m when m is odd.
Lie algebras, generalised nilradical, quasi nilpotent radical, quasiminimal, quasisimple, socle, centraliser. Solvable lie groups and hermitian geometry anna fino dipartimento di matematica universit a di torino. Show that if h is solvable and gh is solvable, then g is solvable too. On indecomposable solvable lie superalgebras having a. A nilpotent lie algebra has an outer automorphism, that is, an automorphism that is not in the image of ad. Solvable leibniz algebras with triangular nilradical by lindsey boskodunbar, matthew burke, jonathan dunbar, j. Solvable lie algebras with heisenberg nilradical 1 144. The class of solvable lie algebras with an ngraded nilradical of maximal nilpotency index is classified. Solvable lie algebras with abelian nilradicals 107 120. We obtain necessary and sufficient conditions of the existence of such a metric for lie groups the nilradical of whose lie algebra is either abelian or heisenberg or standard filiform and discuss some open questions. Solvable leibniz algebras with naturally graded nonlie p.
In fact, solvable leibniz algebras whose nilradical is a naturally graded filiform leibniz algebra are known. Identification of lie algebras through the use of invariants 39 52. Pdf nilpotency degree of the nilradical of a solvable. Cohomologically rigid solvable lie algebras with a nilradical. Using this method we extend the classification of solvable lie algebras with naturally graded filiform lie algebra to the case of leibniz algebras. Classification of solvable leibniz algebras with naturally. We classify solvable lie groups with a free nilradical admitting an einstein leftinvariant metric. Solvable lie algebras and their nilradicals 99 112. In algebra, the nilradical of a l ie algebra is a nilpotent ideal, which is as large as possible the nilradical of a finitedimensional lie algebra is its maximal nilpotent ideal, which exists because the sum of any two nilpotent ideals is nilpotent. We show that among the free lie algebras, there are very few einstein nilradicals. This restriction zinr 1 is realized by matrices a e krxr where k is the ground field assumed to be of characteristic zero. Classification of solvable algebras with the given.
Solvable leibniz algebras with an abelian nilradical. Classification of solvable lie algebras with a given nilradical by. Jul 31, 20 all solvable lie algebras with heisenberg nilradical have already been classified. We give the classification of all solvable leibniz superalgebras such that even part is a lie algebra and nilradical is a lie. A left invariant riemannian metric on a lie group g will be identi. Jan 11, 2015 we study solvable lie groups which admit a leftinvariant metric of strictly negative ricci curvature. Solvable leibniz algebras with filiform nilradical springerlink. Indeed, his lifelong interest was in rings and nonassociative algebras, with emphasis on their structure theory. Solvable leibniz algebras with quasifiliform lie algebras.
Moreover, we establish that solvable leibniz algebras with filiform lie. Dan barbasch nilpotent and solvable algebras math 649, 20. Solvable lie algebras with an ngraded nilradical of. Solvable leibniz algebras with quasifiliform lie algebras of. Classification of solvable algebras with the given nilradical. Rather surprisingly it turns out that the classification of such solvable algebras can be deduced from the classification of solvable algebras with the nilradical. We construct all solvable lie algebras with a specific n dimensional nilradical n n, 3 which contains the previously studied filiform n2 dimensional nilpotent algebra n n2, 1 as a subalgebra but not as an ideal. Nilradical article about nilradical by the free dictionary.
Einstein solvable lie algebras with a free nilradical 2006. Solvable lie algebra with noncharacteristic nilradical. It is shown that such solvable extensions are unique up to isomorphism. Solvable lie algebras from now on we assume that chark 0 and that the dimension of all lie algebras is.
Jacobson does mention the notion of reductive lie algebra in later exercises, but early in his book he focuses mostly on the ideal structure including his versions of solvable and nil radicals. The rigidity of such leibniz algebras with twodimensional complemented. Pdf a procedure is presented for classifying solvable lie algebras with abelian nilradicals. The quotient of a lie algebra by its nilradical is a reductive lie algebra. In the mathematical field of lie theory, the radical of a lie algebra is the largest solvable ideal of the radical, denoted by, fits into the exact sequence. Classification of solvable lie algebras new approaches and. Twostep nilpotent lie algebras attached to graphs hamidreza fana department of mathematical sciences sharif university of technology p. With a value of 2, an outline of the main steps is displayed while a value of 3 provides all the intermediate steps. Lies theorem states that if is a finitedimensional vector space over an algebraically closed field of characteristic zero, and is a solvable lie algebra, and if is a representation of over, then there exists a simultaneous eigenvector. Solvable lie algebras with abelian nilradicals iopscience. Pdf nilpotency degree of the nilradical of a solvable lie. The most natural example of a solvable lie algebra is the set of all upper triangular n nmatrices over an algebraically closed eld of characteristic zero. Recognition of a lie algebra given by its structure constants 37 50. A definitive version was subsequently published in journal of algebra, 470, 2016 doi.
The computations can be examined by setting i nfolevel nilradical to 2 or 3. The derived subalgebra of a finite dimensional solvable lie algebra over a field of characteristic 0 is nilpotent. When the ground field has characteristic zero and has finite dimension, then levis theorem states that this exact sequence splits. It should be noted that the concept of the unipotent hull u of a solvable lie group. We give the classification of all solvable leibniz superalgebras such that even part is a lie algebra and nilradical is a lie superalgebra with a. Right and left solvable extensions of an associative leibniz. In fact, for a given solvable leibniz algebra with a xed nilradical the complemented space. Construction of solvable lie algebras with the nilradical n n,3 firstly, we recall how the knowledge of automorphisms and derivations of a given nilpotent lie algebra n can be employed in the construction of all solvable lie algebras s with the nilradical n. Classification of solvable lie algebras new approaches. The generalised nilradical of a lie algebra lancaster eprints. The generalised nilradical of a lie algebra research portal. In this article, solvable leibniz algebras, whose nilradical is quasifiliform lie algebra of maximum length, are classified. Solvable extensions of negative ricci curvature of. Here, we extend the description to solvable leibniz algebras whose nilradical is a filiform algebra.
In this article we concentrate on, the case when the solvable lie algebra l has an. Jul 30, 2015 in this paper, we continue the description of solvable leibniz algebras whose nilradical is a filiform algebra. On solvable lie groups of negative ricci curvature springerlink. Classification of solvable lie algebras with a given. This is interesting because gives a representation of l as a subalgebra of the derivation algebra of its nilradical with kernel equal to the centre of n.
Solvable lie algebras with quasifiliform nilradicals. Classification of solvable leibniz algebras with nullfiliform nilradical. Further solvable lie algebras whose nilradical is the naturally graded. Nilpotent, solvable and levi decomposable lie algebras 87 100. Also they found 22 non isomorphic classes of solvable leibniz. All solvable lie algebras with heisenberg nilradical have already been classified. They showed that a v edimensional solvable leibniz algebra with a threedimensional heisenberg nilradical is a lie algebra. Throughout the paper vector spaces and algebras are. The former is not nilpotent its simple, and the latter is. Every solvable lie algebra l has a uniquely defined upto equivalence maximal 2. Although the theory of lie algebras doesnt require an algebraic closed scalar field and many real lie algebras are important it is more than convenient as soon as a lie algebra is a matrix algebra, i. Solvable leibniz superalgebras whose nilradical is a lie.
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