Nonabelian tensor product and homology of lie superalgebras. Some properties of the non abelian tensor product and the non abelian homology of lie algebras are established. Pdf completions of lie algebras mohammad shahryari. Leibniz algebra, capable leibniz algebra, non abelian tensor and exterior prod. For every finite pgroup g of order p n with derived subgroup of order p m, rocco n. A lie algebra is called semisimple if it is isomorphic to a direct sum of simple algebras.
An example of solvable lie algebra is given by the algebra of uppertriangular matrix of size n. This bibliography can be used with the brief introduction pdf file or. A non abelian tensor product of lie algebras by graham j. A non abelian lie algebra g is called simple if it has no non trivial ideals. Q dim e l, entangled idealsin this section, we introduce the concept of entangled ideals in a tensor product e. We also give a definition of a non abelian tensor product in lie rinehart algebras based on the construction of ellis of non abelian tensor product of lie algebras. Universal central extensions of lierinehart algebras. Empg1209 hwm1206 dmusmp1205 non abelian tensor multiplet equations from twistor space. A lie algebra is simple if it has no non trivial ideals and is not abelian. In this section we introduce a non abelian tensor product of lie superalgebras, which generalizes the non abelian tensor product of lie algebras, and study its properties. Since the nonabelian tensor product of groups and lie algebras plays an important. For every finite pgroup g of order pn with derived subgroup of order pm, rocco in 10 proved that the order of tensor square of g is at most pnn.
Schur multiplier, nilpotent lie algebras, derived subalgebra, nonabelian tensor product. Abstract hom lie algebras can be considered as a deformation of lie algebras. In 3 ellis introduced and investigated the non abelian tensor product of lie algebras. Feb 02, 2015 decomposition of the nonabelian tensor product of lie algebras via the diagonal ideal february 2015 the bulletin of the malaysian mathematical society series 2 424. Later this non abelian exterior product is applied to the construction of an eight term exact sequence in leibniz homology. Given a commutative ring r and any ring a the tensor product r. The non abelian tensor product is then defined as a piece of this internal crossed square. Nonabelian tensor products and nonabelian homology of groups. It is clear that one should be able to develop an analogous theory of tensor products for other algebraic structures such as lie algebras or commutative algebras. Pdf on the nonabelian tensor product of lie algebras.
Oct 28, 1996 the non abelian tensor product of groups introduced by brown and loday is generalized. Meromorphic tensor equivalence for yangians and quantum loop. For more information about the non abelian tensor and exterior product of a pair of lie algebras see. Guin see 5 has shown that, given crossed lie galgebras. If a and b are commutative, then the tensor product is commutative as well. The tensor product turns the category of r algebras into a symmetric monoidal category. Graded modules over simple lie algebras with a group grading the cartan decomposition of a semisimple lie algebra with respect to a cartan subalgebra can be regarded as a grading by a free abelian group.
Now simmilar theory for lie algebras will be provided. Capability and schur multiplier of a pair of lie algebras. Rocco, on a construction related to the nonabelian tensor square of a group, bol. In the present paper we study finitedimensional graded simple algebras over an. Ellis, a non abelian tensor product of lie algebras, glasg. A nonabelian exterior product and homology of leibniz algebras. Non abelian tensor product of hom lie algebras is constructed and studied. Nonabelian tensor products and nonabelian homology of. Sep 05, 2014 which generalizes the nonabelian tensor product of l ie algebras 6, and study its properties. Homleibniz algebra, nonabelian tensor product, universal.
Since a and b may both be regarded as rmodules, their tensor product. Also its relationship to the universal quadratic functor is established, which is applied to the comparison of the. Then we study the lifting of automorphisms and derivations to central extensions. Sep 05, 2014 nonabelian tensor product of hom lie algebras is constructed and studied. We find when a finitedimensional semisimple lie algebra admits non trivial hom lie algebra structures and the isomorphic classes of non trivial hom lie algebras are determined. We shall assume throughout that the deformation parameters and q are related. Pdf nonabelian tensor multiplet equations from twistor.
Pdf on the tensor square of nonabelian nilpotent finite. We shall use the term lie algebra to mean a lie algebra over. Feb 01, 2018 similar to the work of miller, in the group case, batten et al. Applying this tensor product, in 5 guin defined the lowdimensional h0 and h1 non abelian homologies of lie algebras with coefficients in crossed modules and gave applications to cyclic homology and milnors additive ktheory for non commutative.
In this paper, we introduce the non abelian tensor square of precrossed modules in lie algebras and investigate some of its properties. Recall the definition of the nonabelian tensor product g. On the triple tensor product of nilpotent lie algebras. Schur multipliers and non abelian tensor product of a pair. Q l and we show that they have close connections with the completions of l. A nilpotent algebra is the algebra of strictly uppertriangular matrix of size n, u nc. Nonabelian minimal closed ideals of transitive lie. Now similarly, one can prove that the class f l is not elementary for any abelian lie algebra l. As a simple case, f l is the trivial lie algebra, of dimension 1 and abelian. Nonabelian exterior products of lie algebras and an exact. In particular, the goal of the present paper is to consider the relative notions of central extension and commutator when the semi abelian category is leib, the category of leibniz algebras, and its birkho. Ar salemkar, h tavallaee, h mohammadzadeh, b edalatzadeh. The kunneth formula was originally obtained by schur in 1904 see 8.
Of ourse,c abelian lie algebras are nilpotent algebras too k1. Available formats pdf please select a format to send. A nonabelian tensor product of homlie algebras nasaads. Pdf decomposition of the nonabelian tensor product of. Pdf a note on the schur multiplier of a nilpotent lie. Citeseerx nonabelian tensor product of lie algebras and. Oct 15, 2015 non abelian tensor product of lie superalgebras. The homology groups of groups are constructed with coefficients in any group, as the left derived functors of the non abelian tensor product, which generalize the classical theory of homology of groups. Z a can be given the structure of an r algebra by defining r s.
Nonabelian tensor product of hom lie algebras is constructed and studied. That is to say, a onedimensionalnecessarily abelian lie algebra is by definition not simple, even though it has no nontrivial ideals. In the present paper we study finitedimensional graded simple algebras. Gradings on lie algebras by various abelian groups arise in the theory of symmetric spaces, kacmoody algebras, and color lie superalgebras. Sorry, we are unable to provide the full text but you may find it at the following locations. Jul 22, 2017 we introduce a non abelian exterior product of two crossed modules of leibniz algebras and investigate its relation to the lowdimensional leibniz homology. Keywords hom lie algebra homaction semidirect product derivation. Let m and n be two lie superalgebras with actions on. This tensor product is used to describe universal \alpha central. Roughly speaking, given the nonabelian tensor product. Decomposition of the nonabelian tensor product of lie algebras via the diagonal ideal february 2015 the bulletin of the malaysian mathematical society series 2 424.
In this note, we prove that the hom lie algebra structures on finitedimensional simple lie algebras are trivial. Nonabelian minimal closed ideals of transitive lie algebras. A non abelian tensor product of lie algebras by graha jm. Let gbe a complex semisimple lie algebra, and y gand u qlgthe yangian and quantum loop algebra of g. Pdf decomposition of the nonabelian tensor product of lie. A nonabelian tensor product of homlie algebras core. A nonabelian tensor product of homlie algebras springerlink. Knapp, basic algebra, digital second edition east setauket, ny. Mar 27, 2014 then we study the lifting of automorphisms and derivations to central extensions.
Decomposition of the nonabelian tensor product of lie. This tensor product is used to describe universal central extensions of hom lie algebras and to establish a relation between cyclic and milnor cyclic homologies of homassociative algebras satisfying certain additional condition. The centroid of a lie triple algebra liu, xiaohong and chen, liangyun, abstract and applied analysis, 20. One of the main problems, concerning the study of nonabelian tensorproductsofliealgebras,isrelatedtothesizeofthesocalleddiagonalidealsee the following section for details, because this gives a measure of how far is our lie algebra from being abelian. Let abe an algebra over fa vector space with an associa. The nonabelian tensor and exterior products of crossed modules of lie algebras hajar ravanbod faculty of mathematical sciences, shahid beheshti. Ocarroll, some localizations which are hubert rings, j. Ellis, a nonabelian tensor product of lie algebras, glasgow math. In 7, 6 using the non abelian tensor product of groups see 1 and its non abelian derived functors the non abelian homology of groups is constructed and studied, generalizing the classical eilenbergmaclane homology of groups and extending the non abelian homology introduced in 4. First we need to introduce the notion of first we need to introduce the notion of an action of multiplicative lie ring s.
A nonabelian tensor product of lie algebras cambridge university. Ellis, non abelian exterior products of lie algebras and an exact sequence in the homology of lie algebras, j. We relate this non abelian tensor product to the universal central extension. Nonabelian tensor product of lie algebras and its derived. At 6 dec 20 abstract we investigate the resonance varieties attached to a commutative differential graded algebra and to a representation of a lie algebra, with emphasis on how these varieties behave under finite products and coproducts. Ladra, non abelian tensor product of lie algebras and its derived functors, extracta mathematicae 17 2002 281288. A nonabelian tensor product of leibniz algebra numdam. Recall that the latter are deformations of the enveloping algebras of the current lie algebra gsof gand its loop algebra gz,z. Universal central extensions and nonabelian tensor. A nonabelian exterior product and homology of leibniz. A nonabelian tensor product of lie algebras glasgow. This tensor product is used to describe universal \\alpha \central extensions of hom lie algebras and to establish a relation between cyclic and milnor cyclic homologies of homassociative algebras satisfying certain additional condition. In this articl we e have chosen to work with lie algebras. May 22, 2015 non abelian tensor products of groups 3, 4 and lie alg ebras 11.
Ellis, a non abelian tensor product of lie algebras, glasgow math. The nilpotent multipliers of the direct sum of lie algebras. Nonabelian tensor and exterior products modulo q and. Tensor products the tensor product of two r algebras is also an r algebra in a natural way. This non abelian tensor product is the homversion of the non abelian tensor product of lie algebras developed by ellis 6 and used in 7, 9 for the construction of the non abelian homology of. Some partial results concerning graded simple algebras with non abelian torsion free grading were also obtained in 11. The tensor product can be given the structure of a ring by defining the product on elements of the form a. The nonabelian tensor square g g of a group g was introduced by r. Here we state some theorems and definitions on schur multipliers and non abelian tensor products of a pair of lie algebras. There is the abelian lie algebra, which we denote l ab, and there is the non abelian lie algebra. Apr 06, 1992 journal of pure and applied algebra 78 1992 9160 9 northholland non abelian tensor and exterior products modulo q and universal qcentral relative extension daniel conduch u. Pdf nonabelian tensor product of lie algebras and its. We show that if a is the category of groups or the category of lie algebras, this general construction coincides with the specific notions of non abelian tensor products already known in these settings. A nonabelian tensor product of precrossed modules in lie algebras.
This is a general fact of homology, which is known as the kunneth formula see 7, and it is true also for two finite dimensional lie algebras h and k. An example of simple lie algebra is sl2 and sln in general with n 2 2. Let r be a commutative ring and let a and b be r algebras. Finally, the relation between ellis non abelian tensor product and the non abelian tensor product modulo qof two lie algebras with compatible actions on each other is given theorem 1. There are exactly two lie k algebras of dimension two, up to isomorphism.
1065 985 573 532 512 1418 1405 1038 807 1409 211 1642 548 660 1370 511 1594 1050 125 1104 337 1452 169 795