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