In this paper, we obtain the structure of all nilpotent lie algebras that attain this bound. Nathan jacobson, presently henry ford ii professor of mathematics at yale university, is a wellknown authority in the field of abstract algebra. The cohomology groups for and 2 have, in a number of cases, simple interpretations. Linear algebraic group reductive group abelian variety elliptic curve v t e.
You wont get quite far with this book it covers the main definitions and gives the structure theorem for semisimple lie algebras, but if you do the exercises, you will have a good foundation. Then i moved to humphreys introduction to lie algebras and representation theory which has already been mentioned and is the absolute best. Schur algebras and representation theory cambridge tracts in. Though it presupposes knowledge of linear algebra, it is not overly theoretical and can be readily used for selfstudy. It is aimed at researchers working in group theory, in particular, graduate students in algebra. Lecture 7 complete reducibility of representations of. A for a nilpotent stem lie algebra h of class two with derived subalgebra of dimension two and an abelian lie algebra a. This book is intended for a oneyear graduate course on lie groups and lie algebras. The designation e8 comes from the cartankilling classification of the complex simple lie algebras, which fall into four infinite series labeled an, bn, cn, dn, and five. Now we can derive a new bound for the dimension of the schur multiplier of a nilpotent lie algebra. It is also described the structure of all nilpotent lie algebras such that the bound is. C 1 nc 2 which associates to each object x pc 1 an object fpxqpc 2, associates to each morphism f. In mathematical group theory, the schur multiplier or schur multiplicator is the second.
In fact, we show that the schur multiplier of each nilpotent lie algebra of finite dimension at least 2, is nonzero. N appearing in the following natural exact sequence of lie algebras h 3l. Multiplicative lie algebras and schur multiplier request pdf. Also, we consider some of the features of central extension of an arbitrary lie algebra. Reference recommendation for projective representation. The schur multiplier of the quaternion group q 8 is trivial.
Then g g possesses a schur cover, whose central subgroup is the schur multiplier a uni h 2 g. In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of lie type, or one of 26 sporadic groups the list below gives all finite simple groups, together with their order, the size of the schur multiplier, the size of the outer automorphism group. Schur multiplier synonyms, schur multiplier pronunciation, schur multiplier translation, english dictionary definition of schur multiplier. We consider the schur multipliers of finite dimensional nilpotent lie algebras. In mathematics, e8 is any of several closely related exceptional simple lie groups, linear algebraic groups or lie algebras of dimension 248. Edalatzadehturk j math proof if l is an abelian lie algebra.
The analogous theory for the schur multiplier of a lie algebra was developed in the dessertation of batten and it has been further investigated in many recent papers1,2,3,4,5,6,8,9,11, 12,14. An improvement of a bound of yankosky 2003 is presented in this paper, thanks to a restriction which has been recently obtained by the authors on the schur multiplier ml of a finite dimensional nilpotent lie algebra l. Lie algebra of nilpotency class c with the derived subalgebra of dimension m. F2r,f, which is the usual schur multiplier of the lie algebra l see 1,2,5,8,12, for more information on the schur multiplier of a. This result does not hold, in general, for nilpotent groups. The aim of this paper is to introduce the concept of the schur multiplier of a pair of lie algebras and to obtain some inequalities for the dimension of. The schur multiplier of the quaternion group is trivial, but the schur multiplier of dihedral 2groups has order 2. Prove that the lie algebra from example 2 is isomorphic to o3 by comparing the structure constants. If a m nk, then the space of n nmatrices has the structure of a lie algebra with lie bracket x.
In the present paper, we will characterize all finite dimensional nilpotent lie algebras with s l 1. According to,,, we may define the schur multiplier, m l, for a lie algebra l. Some restrictions on the betti numbers of a nilpotent lie. By the way, i just found out that this is discussed and proved in the book lie algebras with triangular decompositions by moody and pianzola, page 33, in case someone reads this in the future. Suppose l is a nite dimensional lie algebra over a eld with characteristic not equal to two. The schur multiplier of groups is appeared in the works of schur in 1904. Here, we give the structure of all nilpotent lie algebras of maximal class l when dim. A restriction on the schur multiplier of nilpotent lie. The cnilpotent schur liemultiplier of leibniz algebras. The material it treats has relevance well beyond the theory of lie groups and algebras, ranging from the geometry of regular polytopes and paving problems to current work on finite simple groups having a b,npair structure, or tits systems.
His book, lie algebras, is a classic handbook both for researchers and students. Every semisimple lie algebra over an algebraically closed field of characteristic 0 is a direct sum of simple lie algebras by definition, and the finitedimensional simple lie algebras fall in four families a n, b n, c n, and d n with five exceptions e 6, e 7, e 8, f 4, and g 2. The reason why you want to study lie algebras can have a great impact on what books one would recommend. Keywords automorphism central automorphism coclass tree divisibility property group cohomology schur multiplier uniform lie algebra wells exact sequence group extension. N be a pair of lie algebras, where n is an ideal in l. 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. I heard its related to group cohomology, schur s multiplier and group extension. A 1 9 the schur multiplier has an extra z3z, so the schur multiplier of the simple group has order 6 instead of 2. Nilpotent lie algebras having the schur multiplier of maximum. This is the second in a series of three volumes dealing with important topics in algebra. The schur multiplier of the elementary abelian group of order 16 is an elementary abelian group of order 64, showing that the multiplier can be strictly larger than the group itself. Analogous to the schur multiplier of a group, the schur multiplier of a lie algebra, ml, can be defined as ml. The book describes developments on some wellknown problems regarding the relationship between orders of finite groups and that of their automorphism groups. Pdf a note on the schur multiplier of a nilpotent lie algebra.
On dimension of the schur multiplier of nilpotent lie algebras. The lie algebra of spn is given by the quaternionic skewhermitian matrices, the set of nbyn quaternionic matrices that satisfy. The free lie algebra on x is the lie subalgebra in tw x generated by x. The book then shows the characters of irreducible representations of semi simple lie algebras. L dim l 2 and then we show that all of them are capable. The dimension of schur multiplier of such lie algebras is also bounded by dim l 2.
The book goes beyond the representation theory of compact lie groups, which is the basis of many texts, and provides a carefully chosen range of material to give the student the bigger picture. If the algebra has dimension greater than one, then the schur multiplier is nonzero. Schur in his work on finitedimensional complex projective representations of a group cf. These tools are used to characterize lie nilpotency and c lie stem covers of leibniz algebras. This book provides an introduction to lie groups, lie algebras, and repre sentation theory, aimed at graduate students in mathematics and physics.
Symplectic group project gutenberg selfpublishing ebooks. Some upper bounds on the dimension of the schur multiplier. Let l, i be a pair of lie algebras, then the schur multiplier of the pair l, i is. Some properties of the schur multiplier and covers of lie. As a second introduction to representation theory after finite groups. This schur multiplier kept popping up at each loop iteration, making a finite presentation impossible. Some cases where the group is perfect but has a schur multiplier that is larger than expected include. Characterization of finite dimensional nilpotent lie. Cohomology of algebras encyclopedia of mathematics. So for simple lie group, the projective representation is the representation of universal covering group.
An important class of infinitedimensional real lie algebras arises in differential topology. 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. In order to reach our goals, the content is organized as follows. We also introduce the group of nontrivial relations satisfied by the lie product in a multiplicative lie algebra. The cohomology group, where is the multiplicative group of complex numbers with trivial action. The homology theory of multiplicative lie algebras, schur multiplier in terms of. A note on the schur multiplier of a nilpotent lie algebra, comm. A 1 4 the schur multiplier has an extra z2z, so the schur multiplier of the simple group has order 2 instead of 1.
The lie algebra analogue of the schur multiplier was investigated in the dissertations of kay moneyhun and peggy batten see and. This book can be useful for students in advance algebra or who have a background in linear algebra. Alg algebras, algebra homomorphisms lie lie algebras, lie algebra homomorphisms a functor is a map between categories f. Araskhan, on dimension and homological methods of the higher schur multiplier of a pair of lie algebras, comm. For v v a vector space, the general linear lie algebra or endomorphism lie algebra. The book addresses the group of automorphisms and the derivation algebra of a lie algebra and schur s lemma. Fpxqnfpyqpc 2, and preserves both identity morphisms and composition of morphisms. Sedighi, some results on the integer valued polynomials over matrix rings, comm. There are many results on the schur multiplier of finite pgroups and the reader can see for instance. The canonical action of the schur algebra on a tensor power of the standard representation. As explained below, the lie algebra of spn is a real form of the complex symplectic lie algebra sp2n, c. For instance, 7, chapters 3 and 4 is a classic reference, which describes the aspects.
Multiplicative lie algebras and schur multiplier sciencedirect. We consider the multipliers of lie algebras of maximal. A note on the schur multiplier of a nilpotent lie algebra. However, if k is a finite multiplicative lie algebra, then h m l 2 k, c. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Martin gives a selfcontained account of this algebra and those links, covering the basic ideas and their quantum analogues. If is such a representation, then can be interpreted as a mapping such that. 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. One of the key ideas in the theory of lie groups is to replace the global object, the group, with its local or linearized version, which lie himself called its infinitesimal group and which has since become known as its lie algebra. The schur algebra is an algebraic system providing a link between the representation theory of the symmetric and general linear groups both finite and infinite.
The lie algebra analogue to the schur multiplier has been investigated in a number of recent articles. The schur multipliers of lie algebras of maximal class. The schur multipliers of lie algebras of maximal class international. Schur algebras and representation theory cambridge tracts.
This lie algebra is unitarily equivalent to the usual spin physics angularmomentum component operators for spin1 particles in quantum mechanics. A remark on the schur multiplier of nilpotent lie algebras. Russo submitted on 4 jan 2010 v1, revised 27 may 2010 this version, v3, latest version 19 jun 2012 v4. V \mathfrakglv of v v is the lie algebra whose elements are linear endomorphisms v v v \to v and whose lie bracket is given by the commutator of endomorphisms. Peyman niroomand damghan university, damghan, iran, francesco g. Schur algebras are an algebraic system that provide a link between the representation theory of the symmetric and general linear groups. Capability and schur multiplier of a pair of lie algebras. The structure, capability and the schur multiplier of. Edalatzadeh, the multiplier and the cover of direct sums of lie algebras, asianeuropean journal of mathematics, 52 2012. A note on the schur multiplier of a nilpotent lie algebra authors. A schur trivial group is defined as a group whose schur multiplier is trivial. Schur multiplier, nilpotent lie algebras, settore mat02 algebra, settore mat03 geometria.
Schur multiplier definition of schur multiplier by the free. So my moral from this little story is that you study schur multipliers because you cannot avoid them. Brauer algebra and lie superalgebra and double centralizer and mixed tensor space and invariant theory 1. The complete collection of groups of lie type is beautifully exposed in carters book using the simple complex lie algebras as a starting point. That c acts by constant multiplication by some 2f follows from schurs lemma. Although there are already several excellent books that cover many of the same topics, this book has two distinctive features that i hope will make it a useful addition to the literature. Schur multipliers for group extensions and for lie groups also where are they written for lie algebras. Leibniz n algebra, lie cover, schur lie multiplier. For example, the schur multiplier of the quaternion group q n.
Russo universita degli studi di palermo, palermo, italy submitted on 4 jan 2010, last revised 19 jun 2012 this version, v4. Nilpotent lie algebras having the schur multiplier of. Let l be a nonabelian nilpotent lie algebra of dimension n and. We obtain exact sequences and formulas of the dimensions of the underlying vector spaces relating the cnilpotent schur liemultiplier of a leibniz algebra q and its quotient by a twosided ideal. The basic example of a lie algebra arises from using the commutator in an associative algebra, so x. The schur hopf formula is established for multiplicative lie algebras.
On dimension of the schur multiplier of nilpotent lie. Lie groups are smooth differentiable manifolds and as such can be studied using differential calculus, in contrast with the case of more general topological groups. On one hand, several authors investigated the deep connections among land the structure of a. Russo universita degli studi di palermo, palermo, italy. But it only discuss the lie group, so whats about the projective representation of discrete group like finite group or infinite discrete group. Linear algebra, galois theory, representation theory, group extensions and schur multiplier algebra 2 and trigonometry algebra. The lie algebras sl2, o3, t2, n3 all have dimension 3. In the text dr martin gives a full, selfcontained account of this algebra and these links, covering both the basic theory of schur algebras and related areas. The schur multiplier of the pair l, n, where l is a lie algebra with ideal, is the abelian lie algebra ml, n which appears in the following natural exact sequence of mayervietoris type mathematical expression not reproducible in ascii. And the reason was that the finitedimensional lie algebra i was starting from had a nontrivial schur multiplier.
Lie algebra g, then the casimir endomorphism c acts by constant multiplication, with the constant equal to dimgdimv. Edalatzadeh, some notes on the schur multiplier of a pair of lie algebras, journal of lie theory, 2320, 483492. Aug 07, 2003 this book provides an introduction to lie groups, lie algebras, and repre sentation theory, aimed at graduate students in mathematics and physics. Linear algebra, galois theory, representation theory, group extensions and schur multiplier ramji lal auth. Lie algebra cohomology, nonabelian lie algebra cohomology, lie algebra extensions.
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