In this paper, we obtain the structure of all nilpotent lie algebras that attain this bound. Also, we present a criterion for nilpotent lie algebras lacking any covers with respect to the variety of nilpotent lie algebras of class at most c. Brauer algebra and lie superalgebra and double centralizer and mixed. 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. A schurtrivial group is defined as a group whose schur multiplier is trivial.
Lecture 7 complete reducibility of representations of. The schur multipliers of lie algebras of maximal class. Multiplicative lie algebras and schur multiplier sciencedirect. Characterization of finite dimensional nilpotent lie. Throughout the paper, we may assume that l is a lie algebra over an algebraically closed. Schurweyl duality for the brauer algebra and the orthosymplectic lie superalgebra michael ehrig and catharina stroppel abstract. On the capability and schur multiplier of nilpotent lie algebra of class two. It turns out that cyclic implies schurtrivial, free implies schurtrivial, and any finite group generated by schurtrivial subgroups of relatively prime indices is schurtrivial. Under fairly general conditions, we extend the 5sequence of cohomology for nilpotent lie algebras a step further. Please see 3 for a collection of schurs contributions to this area. Contents preface pagexi 1 introduction 1 2 liegroups. For a twosided ideal n of a leibniz algebra g, we construct a fourterm exact sequence relating the schur liemultiplier of gand gn, which is applied to study and characterize. We consider the multipliers of lie algebras of maximal class, classifying these algebras with a certain additional property.
A stronger statement would be that the schur multiplier ofri 2 is not finite dimensional. In the present paper, we will characterize all finite dimensional nilpotent lie algebras with sl 1. Introduction the lie algebra analogue of the schur multiplier was investigated in the dissertations of kay moneyhun and peggy batten see 8 and 2. For instance, 7, chapters 3 and 4 is a classic reference, which describes the aspects. Available formats pdf please select a format to send. A note on the schur multiplier of a nilpotent lie algebra. The lie algebra analogue to the schur multiplier has been investigated in a number of recent articles.
Introduction the rst important research about nilpotent lie algebras is due to k. Schur multipliers for group extensions and for lie groups also where are they written for lie algebras. Suppose there is a presentation ofa 0ira0 wherer is a finitely generated freekalgebra andi is nonzero. On the multiplier of a lie algebra bill yankosky communicated by j. Then schurs lemma says that the endomorphism ring of the module m is a division algebra over the field k. Full text views reflects the number of pdf downloads, pdfs. But it only discuss the lie group, so whats about the projective representation of discrete group like finite group or infinite discrete group. Abstractwe give a bound on the dimension of the schur multiplier of a finite dimensional nilpotent lie algebra which sharpens the earlier. A schur trivial group is defined as a group whose schur multiplier is trivial. We then derive some consequences of the construction. Multiplicative lie algebras and schur multiplier article in journal of pure and applied algebra 2239 december 2018 with 54 reads how we measure reads.
The schur liemultiplier of leibniz algebras quaestiones. Nilpotent lie algebras having the schur multiplier of maximum. Russo submitted on 4 jan 2010 v1, revised 27 may 2010 this version, v3, latest version 19 jun 2012. The concept of the schur multiplier of a group originated from a work by schur 33 on projective representations in 1904. This is the second in a series of three volumes dealing with important topics in algebra. We give a proof of a schurweyl duality statement between the brauer algebra and the orthosymplectic lie superalgebra ospv. Homological aspects of lie algebra crossed modules. Let l, i be a pair of lie algebras, then the schur multiplier of the pair l, i is defined to be the abelian lie algebra m l, i. This cited by count includes citations to the following articles in scholar.
Linear algebra, galois theory, representation theory, group extensions and schur multiplier ramji lal auth. A remark on the schur multiplier of nilpotent lie algebras. Some upper bounds on the dimension of the schur multiplier of. Let l be a nonabelian nilpotent lie algebra of dimension n and ml be its schur multiplier. The schur hopf formula is established for multiplicative lie algebras.
In this paper, we classify all nilpotent lie algebras of. The schur multiplier of the quaternion group is trivial, but the schur multiplier of dihedral 2groups has order 2. Let l, n be a pair of finitedimensional nilpotent lie algebras, in which n is an ideal in l. The structure, capability and the schur multiplier of generalized heisenberg lie algebras. Phd students batten and moneyhun introduced the schur multiplier of a lie algebra in 3, 4, 23. On dimension of the schur multiplier of nilpotent lie. We give a bound on the dimension of the schur multiplier of a. Some upper bounds on the dimension of the schur multiplier of l. We also introduce the group of nontrivial relations satisfied by the lie product in a multiplicative lie algebra, and study it as a functor arising from the presentations of. Lecture 7 complete reducibility of representations of semisimple algebras september 27, 2012 1 new modules from old a few preliminaries are necessary before jumping into the representation theory of semisimple algebras. In this article, we indicate that the schur multiplier of every nilpotent lie algebra of finite dimension at least 2 is nonzero. Pdf the structure, capability and the schur multiplier of. Pdf a note on the schur multiplier of a nilpotent lie. Then any multiplicative lie algebra structure on g is uniquely determine by a gequivariant homomorphism from g, g to g as described in the previous proposition.
Recently, nayak 24, 25 has generalized this notion to lie superalgebras. Russo universita degli studi di palermo, palermo, italy. I have made them public in the hope that they might be useful to others, but these are not o cial notes in any way. We examine su cient conditions for making the schur multiplier of a lie algebra nontrivial proving an elegant theorem involving the dimension of the lie algebra. Russo submitted on 4 jan 2010 v1, revised 27 may 2010 this version, v3, latest version 19 jun 2012 v4. Ams proceedings of the american mathematical society. On the dimension of the cnilpotent schur liemultiplier of leibniz algebras.
On the dimension of the cnilpotent schur lie multiplier of leibniz algebras. Characterization of finite dimensional nilpotent lie algebras. The homology theory of multiplicative lie algebras, schur multiplier in terms of homology, multiplicative steinberg lie algebra of a. 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. In addition, we deal with some properties of covers of lie algebras whose schur multipliers are. 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. Irreducible representations of the symmetric group 8 2. Schurs lemma is frequently applied in the following particular case.
Some upper bounds on the dimension of the schur multiplier. Francesco g 2011, a restriction on the schur multiplier of nilpotent lie algebras, electronic journal of linear algebra, volume 22. Suppose l is a nite dimensional lie algebra over a eld with characteristic not equal to two. If is such a representation, then can be interpreted as a mapping such that. On the schur multiplier of nlie algebras request pdf. We consider the schur multipliers of finite dimensional nilpotent lie algebras. Russo, a note on the schur multiplier of a nilpotent lie algebra, comm.
A note on the schur multiplier of a nilpotent lie algebra authors. It turns out that cyclic implies schur trivial, free implies schur trivial, and any finite group generated by schur trivial subgroups of relatively prime indices is schur trivial. The cohomology group, where is the multiplicative group of complex numbers with trivial action. Niroomand, peyman and russo, francesco g 2011, a restriction on the schur multiplier of nilpotent lie algebras, electronic journal of linear algebra, volume 22. Ifa is infinite dimensional overk, lewin proved thatri 2 is not finitely presented. Then schur s lemma says that the endomorphism ring of the module m is a division algebra over the field k. The latest versions of adobe reader do not support viewing pdf files within. Schur multiplier, nilpotent lie algebras, derived subalgebra, nonabelian tensor.
It was proved by the second author the dimension of the schur multiplier is equal to 1 2n. Reference recommendation for projective representation. The classification of finite dimensional nilpotent lie algebras. She gives some results on the structure of the schur multiplier of a lie superalgebra l, denoted by ml, and provides several upper bounds for the dimension of ml, as well. In this paper we derive some inequalities for the dimension of the schur multiplier of the pair l, n in terms of the dimension of the commutator subalgebra l, n. Recently, rai proved that the dimension of schur multiplier of l is bounded by. Lie f,r lie is called the schur multiplier of grelative to the liezation functor or schur liemultiplier. Let l be a nonabelian nilpotent lie algebra of dimension n and. In her dissertation on the lie algebra version of the schur multiplier, peggy. Schur multiplier definition of schur multiplier by the free. A multiplier is the lie algebra analogue of the schur multiplier from group theory. Capability and schur multiplier of a pair of lie algebras. Schur s lemma is frequently applied in the following particular case.
Schur multiplier synonyms, schur multiplier pronunciation, schur multiplier translation, english dictionary definition of schur multiplier. Introduction and preliminaries let lbe a lie algebra over a eld with characteristic di erent from 2 and a free presentation 0. A nonabelian tensor product of lie algebras glasgow. Suppose that r is an algebra over a field k and the vector space m n is a simple module of r. 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. Schur in his work on finitedimensional complex projective representations of a group cf. Some restrictions on the betti numbers of a nilpotent lie algebra. In this article, we give the structure of all covers of nlie algebras with finite dimensional schur multipliers, which generalizes an earlier work of salemkar et. Nearly a century later, stitzinger and his phd students batten and moneyhun introduced the schur multiplier of a lie algebra in 3, 4, 23. So for simple lie group, the projective representation is the representation of universal covering group. On dimension of the schur multiplier of nilpotent lie algebras, cent.
For a twosided ideal n of a leibniz algebra g, we construct a fourterm exact sequence relating the schur lie multiplier of gand gn, which is applied to study and characterize lie nilpotency, lie stem covers and lie capability of. The ones marked may be different from the article in the profile. Schur multiplier definition of schur multiplier by the. Letk be a field, anda a finitely generatedkalgebra, with augmentation. Pair of lie algebras, schur multiplier, nilpotent lie algebra 1. N are obtained without considering the existence of a complement for n. On one hand, several authors investigated the deep connections among land the structure of a. If the algebra has dimension greater than one, then the schur multiplier is nonzero. Lie f,r lie is called the schur multiplier of grelative to the liezation functor or schur lie multiplier.
Peyman niroomand damghan university, damghan, iran, francesco g. Finally, we indicate that if a lie algebra l belongs to some certain classes of lie algebras then so does its cover. Multiplicative lie algebras and schur multiplier request pdf. Linear algebra, galois theory, representation theory, group extensions and schur multiplier infosys science foundation series on free shipping on qualified orders. A restriction on the schur multiplier of nilpotent lie. There are many results concerning the schur multiplier of a group being trivial.
Classification of pair of nilpotent lie algebras by their schur multipliers. On the dimension of the schur multiplier of nilpotent lie algebras. A restriction on the schur multiplier of nilpotent lie algebras. On the schur multiplier of augmented algebras springerlink. Schur duality for the cartan type lie algebrawn kyo nishiyama communicated by g. On the capability and schur multiplier of nilpotent lie algebra of class two p niroomand, f johari, m parvizi proceedings of the american mathematical society 144 10, 41574168, 2016. A note on the schur multiplier of a nilpotent lie algebra by peyman niroomand and francesco g. It was proved by the second author the dimension of the schur multiplier is equal to 1 2n 1n 2. These results are applied to derive a new bound on the dimension of the schur multiplier of a nilpotent lie algebra. Brookes part iii course on lie algebras, given at cam bridge university in michaelmas term, 2012.
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