Dimension of the c nilpotent multiplier of lie algebras. There is no ndimensional nilpotent lie algebra of nilpotency class c. The schur multiplier of the quaternion group is trivial, but the schur multiplier of dihedral 2groups has order 2. The lie algebras tm is shown to have m2 functionally independent invariants. N a pair of lie algebras where n is an ideal of the lie algebra l. This is to say that a central extension of a nilpotent lie algebra by a nilpotent lie algebra is nilpotent. In particular, if all sylow psubgroups of g are cyclic, then. On the multiplier of nilpotent nlie algebras sciencedirect. A lie algebra is nilpotent if and only if all elements of are ad nilpotent. The following result provides an upper bound for dimension of multiplier of nilpotent nlie algebras which is less than the bound in corollary 3.
It was proved by the second author the dimension of the schur multiplier is equal to 1 2n 1n 2. Representations of nilpotent lie algebras mathematics stack. We further obtain an inequality between dimensions of c nilpotent multiplier of lie algebra l and tensor product of a central ideal by its abelianized factor lie algebra. We give a direct proof of an upper bound for the dimension of the schur multiplier as a function of class and the minimum number of generators of the algebra. A nilpotent lie algebra has an outer automorphism, that is, an automorphism that is not in the.
Journal of lie theory volume 2003 16 c 2003 heldermann verlag on the multiplier of a lie algebra bill yankosky communicated by j. Introduction the rst important research about nilpotent lie algebras is due to k. Favre discovered the lowest dimensional cnla known until then 35, which additionally was of the same nature as dyers example. These results are applied to derive a new bound on the dimension of the schur multiplier of a nilpotent lie algebra.
Some upper bounds on the dimension of the schur multiplier. On characterizing pairs of nonabelian nilpotent and. The schur liemultiplier of leibniz algebras quaestiones. Abstractwe give a bound on the dimension of the schur multiplier of a finite dimensional nilpotent lie algebra which sharpens the earlier. Recently, rai proved that the dimension of schur multiplier of l is bounded by. We consider the multipliers of lie algebras of maximal class, classifying these algebras with a certain additional property. Invariants of the nilpotent and solvable triangular lie algebras. A remark on the schur multiplier of nilpotent lie algebras. Abstract this thesis is concerned with the dassitication of 7dimensional nilpotent lie alge bras. Skjelbred and sund 1978 36 reduce the classincation of nilpotent algebras in a given dimension.
If the algebra has dimension greater than one, then the schur multiplier is nonzero. The covering groups of the alternating and symmetric groups are of considerable recent interest. Characterization of nilpotent lie algebras pair by their. On the dimension of the schur multiplier of nilpotent lie algebras. Pdf a note on the schur multiplier of a nilpotent lie. Jun 22, 2016 in this paper, we present an explicit formula for the 2 nilpotent multiplier of a direct product of two lie algebras. The schur multiplier is very useful as a tool to classify pgroups and nilpotent lie algebras, thanks to. In her dissertation on the lie algebra version of the schur multiplier, peggy. The lie algebra analogue to the schur multiplier has been investigated in a number of recent articles. The rst step is to translate the problem, which concerns lattices in a simply connected nilpotent lie group, into a problem about discrete cocompact subrings in its lie algebra. Finally we compute the bogomolov multipliers of some simple complex lie algebras. One may check that mcl is independent of the choice of the free presentation of l. Skjdbred and sund have published in 1977 their method of constructing all nilpotent lie algebras of dimension rr given those algebras of dimension nilpotent schur lie multiplier is useful in characterizing lie nilpotency and c lie stem covers of leibniz algebras, and in proving the existence of c lie stem covers for nite dimensional leibniz algebras and the non existence of ccovering on certain lie nilpotent leibniz.
Because of its length it is omitted from this paper. Request pdf the c nilpotent shur liemultiplier of leibniz algebras we introduce the notion of c nilpotent schur liemultiplier of leibniz algebras. The classification of finite dimensional nilpotent lie algebras. Characterization of finite dimensional nilpotent lie algebras. The schur multipliers of lie algebras of maximal class. L dim l 2 and then we show that all of them are capable. In this paper, we classify all nilpotent lie algebras of maximal class for s l. We notice that the results relative to the liezation functor in the paper are analogue to absolute properties of the cnilpotent multiplier of lie algebras.
Request pdf characterization of nilpotent lie algebras pair by their schur multipliers let n, l be a pair of finite dimensional nilpotent lie algebras, in which n is an ideal in l and. We then derive some consequences of the construction. A lie algebra with a regular automorphism that is, one having no fixed point except zero of prime period is nilpotent. Then its adjoint representation will consist of elements which are nilpotent matrices over rationals. Nilpotent subalgebras that coincide with their normalizer cartan subalgebras play an essential role in the classification of simple lie algebras of finite dimension. An nbyn matrix a with entries from a field is nilpotent if and only if its characteristic polynomial is t n. Nilpotent lie algebras having the schur multiplier of maximum. In particular we provide characterizations of capability of leibniz algebras by means of both their characteristic ideal and nilpotent schur multiplier. The cnilpotent shur liemultiplier of leibniz algebras. Then g is nilpotent i for each a2g, ad an 0 for some n0. On dimension of the schur multiplier of nilpotent lie algebras. A nonabelian tensor product of lie algebras glasgow.
All nonzero nilpotent elements are zero divisors an nbyn matrix a with entries from a field is nilpotent if and only if its characteristic polynomial is t n if x is nilpotent, then 1. Invariants of the coadjoint representation of two classes of lie algebras are calculated. N be a pair of nite dimensional nilpotent lie algebras and k be the complement of n in l. In this paper, we present an explicit formula for the 2nilpotent multiplier of a direct product of two lie algebras. Let f be a free lie algebra of 2 generators and l ff4. Pdf some results on the schur multiplier of nilpotent. In this paper, we obtain the structure of all nilpotent lie algebras that attain this bound. Characterization of finite dimensional nilpotent lie.
The schur multiplier of lie algebra l 6, 26 is an abelian lie algebra of dimension 8. We consider the schur multipliers of finite dimensional nilpotent lie algebras. On the capability and schur multiplier of nilpotent lie algebra of class two authors. Classification of pair of nilpotent lie algebras by their schur multipliers elaheh khamseh and somaieh alizadeh niri communicated by vasile br nzanescu let lbe a nite dimensional nilpotent lie algebra and n, k be ideals of l such that l n k, with dimn n and dimk m.
Some results on the schur multipliers of nilpotent lie algebra. Lie algebra of nilpotent lie group mathematics stack. Under fairly general conditions, we extend the 5sequence of cohomology for nilpotent lie algebras a step further. On dimension of schur multiplier of nilpotent lie algebras ii. Here, we give the structure of all nilpotent lie algebras of maximal class l when dim. Mar 09, 2011 we consider the schur multipliers of finite dimensional nilpotent lie algebras. The dimension of schur multiplier of such lie algebras is also bounded by dim l 2. Representations of nilpotent lie algebras mathematics. Let l be a nonabelian nilpotent lie algebra of dimension n and ml be its schur multiplier. Then l is a lie algebra of 2 generators and class 3. Every non zero lie algebra has a nilpotent element. Usually, h 2 l, z is called schur multiplier of l and denoted by m l. For a twosided ideal of a leibniz algebra, we construct a fourterm exact sequence relating the schur lie multipliers of and, which is applied to study and characterize lie nilpotency, lie stem covers and lie capability of leibniz algebras.
Ams proceedings of the american mathematical society. Some properties of the cnilpotent multiplier of lie algebras. Condition 5 corresponds to the property of connected nilpotent algebraic groups being the direct product of a unipotent group by a torus. List of 7dimensional nilpotent lie algebras a multiplication table for each algebra is given below nonzero brackets only. If a sylow psubgroup of g is cyclic for some p, then the order of. The purpose of this paper is to derive some inequalities for dimension of the c nilpotent multiplier of finite dimensional lie algebras and their factor lie algebras. For a twosided ideal of a leibniz algebra, we construct a fourterm exact sequence relating the schur liemultipliers of and, which is applied to study and characterize lienilpotency, liestem covers and lie. The schur multipliers of the finite simple groups are given at the list of finite simple groups. On the capability and schur multiplier of nilpotent lie algebra of class two. For a nilpotent lie algebra l of dimension n and dim l 2 m. Schur multipliers of nilpotent lie algebras 4 example 4. Lie algebras by their schur multipliers arabyani, et al.
Let us outline the method used to obtained the estimates of x1. For a twosided ideal n of a leibniz algebra g, we construct a fourterm exact sequence relating the schur lie multipliers of g and gn, which is applied to study and characterize lie nilpotency, lie stem covers and lie capability of leibniz algebras. Condition 1 shows that they have similar module theoretic properties. Some upper bounds on the dimension of the schur multiplier of. Given a nilpotent lie algebra l of dimension dim l n, it is wellknown that the second homology lie algebra h 2 l, z of l with coefficients in z is again a finite dimensional lie algebra. Pdf a note on the schur multiplier of a nilpotent lie algebra. The cnilpotent schur liemultiplier of leibniz algebras. Later santharoubane 1979 2829 fiirther generaiizes this idea and establishes a link between nilpotent lie algebras and kacmoody lie algebras. For a twosided ideal n of a leibniz algebra g, we construct a fourterm exact sequence relating the schur liemultipliers of g and gn, which is applied to study and characterize lienilpotency, liestem covers and liecapability of leibniz algebras. Pair of lie algebras, schur multiplier, nilpotent lie algebra 1. Classification of pair of nilpotent lie algebras by their schur multipliers.
Schur multiplier, nilpotent lie algebras, derived subalgebra, nonabelian tensor product. The theory of the schur multiplier and covers for finite dimensional lie algebras has been developed analogously to that for finite groups, and proved to be a powerful tool in the study of certain types of problems in lie algebra theory, such as the classification of finite nilpotent lie algebras. Then we compute the bogomolov multipliers for abelian, heisenberg and nilpotent lie algebras of class at most 6. A lie algebra is nilpotent if and only if all elements of are adnilpotent. No nilpotent element can be a unit except in the trivial ring 0, which has only a single element 0 1. Possible use of the rank of a nilpotent lie algebra to construct a maximal dimensional solvable lie algebra 3 proof or counter example. The following example shows not only there exists a nilpotent lie algebra that attains the bound of the main theorem, but also the new upper bound is given in the main theorem is better than the bound 1. Request pdf on dimension of schur multiplier of nilpotent lie algebras ii let formula presented. Dec 31, 2020 in this paper, we extend the notion of the bogomolov multipliers and the cpextensions to lie algebras. Invariants of the nilpotent and solvable triangular lie. Introduction throughout this paper, we denote by l. Assume n and k are of dimension n and m respectively. Introduction the rst important research about nilpotent lie.
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