Pdf lie superalgebras whose enveloping algebras satisfy. In this paper we study deformations of some classes of metric lie superalgebras over the complex field. On the freudenthals construction of exceptional lie algebras yamaguti, kiyosi and asano, hiroshi, proceedings of the japan academy, 1975. Moreover we know when the enveloping algebra ug is a domain, when it has. We refer the reader to 1 for basic background about lie superalgebras and their enveloping algebras and to 2, 15, 16 for notation. The dictionary extensively covers finite dimensional algebras and superalgebras, which moreover are indispensable to the study of more elaborate.
Engel condition on enveloping algebras of lie superalgebras salvatore siciliano and hamid usefi abstract. In so doing, instead of lie algebras, we consider color lie superalgebras, and instead of a primitively generated hopf algebra, we take a hopf algebra h whose semigroup. Introduction to finite and infinite dimensional lie super. Zolotykh crc press boca raton new york london tokyo.
Several authors have tried to construct the enveloping algebras of homlie algebras. For lie superalgebras of type i, this was done in l2 using a bijection. Lie algebras, lie superalgebras, vertex algebras and. Identities in the enveloping algebras for modular lie. The reader may skip these chapters if he or she feels su ciently versed in that topic. Lie superalgebras and enveloping algebras about this title. We describe a type of lie colour algebra, which we call generic, whose universal enveloping algebra is a domain with finite global dimension. Petrogradski department amechanics and mathematics algebra, moscow state university, moscow 117234, ussr communicated by susan montgomery received november 1, 1989 introduction the universal enveloping algebra of a lie algebra in characteristic zero satisfies a nontrivial identity. Request pdf engel condition on enveloping algebras of lie superalgebras let l be a lie superalgebra over a fled of characteristic p eq 2 with enveloping algebra ul or let l be a restricted. Let g be a nite abelian group and let l be a, possibly restricted, ggraded lie color algebra.
M87 2012 510dc23 2011044064 copying and reprinting. Let g g0 g1, be a basic classical simple lie superalgebra, and h a cartan subalgebra. Let l be a lie superalgebra with its enveloping algebra ul over a field f. Let dbe a subset of an associative algebra uover a. The first part, chapters 15, introduces the reader to the concept of lie superalgebras. Then the enveloping algebra ul is also ggraded, and we. In this work we want to study the cohomology of lie algebras and lie superalgebras and apply the results to several examples. We define an analogue of shapovalov forms for qtype lie superalgebras and factorize the corresponding shapovalov determinants which are responsible for simplicity of the highest weight modules. Request pdf engel condition on enveloping algebras of lie superalgebras let l be a lie superalgebra over a fled of characteristic p\neq 2 with enveloping algebra ul or let l be a restricted. Quantized enveloping algebras associated with simple lie. We also characterize l when ul is lie solvable, lie nilpotent, or lie supernilpotent. Lie superalgebras and enveloping algebras american.
Lie algebras, lie superalgebras, vertex algebras and related topics share this page. These conditions are not equivalent for lie superalgebras. Infinite dimensional lie algebras an introduction progress in. Infinite dimensional lie algebras an introduction progress.
Also, we show that a classical finite wsuperalgebra can be obtained by a zhu algebra of a classical affine wsuperalgebra. Shirshov basis for a lie superalgebra if and only if. Classical affine walgebras associated to lie superalgebras. It is reasonable to try to find similar techniques in the lie superalgebra context. Introduction to finite and infinite dimensional lie algebras and superalgebras introduces the theory of lie superalgebras, their algebras, and their representations.
Let l be a restricted lie superalgebra with its restricted enveloping algebra u l over a field f of characteristic p 2. Identities in the enveloping algebras for modular lie superalgebras. Lie superalgebras and enveloping algebras musson, ian. In this paper, we prove classical affine walgebras associated to lie superalgebras wsuperalgebras, which can be constructed in two different ways. Lie superalgebras are generalizations of lie algebras, useful for depicting supersymmetry the symmetry relating fermions and bosons. Petrogradski department amechanics and mathematics algebra, moscow state university, moscow 117234, ussr communicated by susan montgomery received november 1, 1989 introduction the universal enveloping algebra of a lie algebra in characteristic zero satisfies. It turned out that the orbit method had much wider applications. It is intended to be useful to mathematical and theoretical physicists, from the level of the graduate student upwards. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. A nilpotent element e 2 gis regular nilpotent if e.
The universal enveloping algebra of a finitedimensional lie algebra is a filtered quadratic algebra. Lie superalgebras and enveloping algebras, by ian musson. In section 2, after introducing the basic facts such as superlyndonshirsho words monomials. Pdf lie superalgebras whose enveloping algebras satisfy a non. Download lie superalgebras and enveloping algebras by ian. A polynomial identity is called nonmatrix if it is not. A kalgebra is a kvector space with two linear maps m. Universal enveloping algebras recall that, for an associative algebra a with unity 1, a lie algebra structure on a is given by the lie bracket abab. Rt 24 sep 2004 on some representations of nilpotent lie algebras and superalgebras shantala mukherjee. On antistructurable algebras and extended dynkin diagrams kamiya, noriaki and mondoc, daniel, journal of generalized lie theory and applications, 2009. In most of these theories, the even elements of the superalgebra correspond to bosons and odd elements to fermions but this is not always true. Centers of universal enveloping algebras of lie superalgebras in prime characteristic. Bergen, jeffrey, pacific journal of mathematics, 1994. Automorphic forms and lie superalgebras urmie ray auth.
The enveloping algebra of the lie superalgebra osp1. We give a generalization of the wellknown theorem stating that the category of primitively generated hopf algebras is equivalent to the category of restricted lie algebras. Recall that a lie superalgebra is just a z2graded vector. In the study of group algebras and enveloping algebras of ordinary lie algebras, many questions have been solved using amethods. Semisimplicity of restricted enveloping algebras of lie superalgebras.
Lie superalgebras are a ordinary generalization of lie algebras, having functions in geometry, quantity thought, gauge box idea, and string idea. Universal enveloping algebras 24 free products 26 4. The basic features of the theory of representations of simple lie superalgebras make them kindred to lie algebras in characteristic p, while if p 2 there is almost no differ. In a previous paper, we studied the homogenized enveloping algebra of the lie algebra s. Lie superalgebras whose enveloping algebras satisfy a nonmatrix polynomial identity article pdf available in israel journal of mathematics 1961 august 20 with 52 reads how we measure reads. An associative unitary respectively, lie algebra is said to be proper in case the intersection of all of its cofinite twosided respectively, lie ideals is zero.
First we observe that every lie supernilpotent superalgebra is indeed lie solvable. Click download or read online button to get infinite dimensional lie algebras an introduction progress in mathematics book now. Combinatorial aspects of lie superalgebras alexander a. Musson, university of wisconsin, milwaukee, milwaukee, wi. We construct quantum deformations of enveloping algebras of borcherds superalgebras, their verma modules, and their irreducible highest weight modules. This booklet develops the speculation of lie superalgebras, their enveloping algebras, and their representations. The orbit method was created by kirillov in the attempt to describe the unitary dual n. Lie superalgebras of types a, q and recently for the orthosymplectic lie algebras bru03, bru04, clw09, gs09, ps97a, ps97b, ser96. Grobnershirshov bases for lie superalgebras and their. Generic lie colour algebras bulletin of the australian. A construction of lie algebras and lie superalgebras by freudenthalkantor triple systems, i kakiichi, yoshiaki, proceedings of the japan academy, series a, mathematical sciences, 1985. Contents introduction v 1 preliminaries 1 1 lie superalgebras 1 algebras 1 lie algebras 3 ggraded algebras 4 lie superalgebras 5 2 color lie superalgebras 7.
We characterize l when ul satisfies a nonmatrix polynomial identity. The universal enveloping algebra of the lie algebra g is. These are later used in a discussion of the enveloping algebra ul of a lie superalgebra l from the point of view of noncommutative ring theory. Graduate students and research mathematicians interested in lie algebras, lie superalgebras, and vertex algebras. Using the hopf algebra structure of ul, it is shown that over. We provide an application and show grobner basis methods can be used to study universal enveloping algebras of factors of generic lie colour algebras. For lie superalgebras of type i, this was done in l2. L\ dimkl, lie algebras, the universal enveloping algebra of the lie superalgebra can be given a hopf algebra structure. In mathematics, a lie superalgebra is a generalisation of a lie algebra to include a z 2grading. Dictionary on lie algebras and superalgebras luc frappat. It is a borcherdskacmoody lie algebra with lorentzian root lattice. Then l is a functor which converts associative algebras into lie algebras. Lie superalgebras and enveloping algebras university of wisconsin. We characterize l when u l is lie solvable, lie nilpotent, or lie supernilpotent.
This book is a detailed reference on lie algebras and lie superalgebras presented in the form of a dictionary. Lie algebras 3 ggraded algebras 4 lie superalgebras 5 2 color lie superalgebras 7 commutation factor 7 color tensor product 10 free ecommutative associative algebra 10 color lie superalgebras 12 skew envelope of a color lie superalgebra 19 differential lie superalgebras 20 3 universal constructions 21 free associative algebras 21 the free. The construction of the group algebra for a given group is in many ways analogous to constructing the universal enveloping algebra for a given lie algebra. Journal of algebra 145, 121 1992 identities in the enveloping algebras for modular lie superalgebras v. Mar 26, 2017 lie superalgebras are a ordinary generalization of lie algebras, having functions in geometry, quantity thought, gauge box idea, and string idea. A polynomial identity is called nonmatrix if it is not satisfied by the algebra of 2. Chapters 1 and 2 give an introduction to the theory of lie algebras and lie superalgebras as well as to their representations. Lie identities on enveloping algebras of restricted lie superalgebras article in journal of algebra 393.
This book develops the theory of lie superalgebras, their enveloping algebras, and their representations. Passman depaul university university of wisconsinmadison abstract. Mondragon in our work on groebner basis algebras, to give the. Lie superalgebras are a natural generalization of lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. Lie identities on enveloping algebras of restricted lie. The first example is the lie superalgebra glmn realized as the algebra of. This is in complete contrast with the enveloping algebras of ordinary lie superalgebras satisfying a nonmatrix pi, see theorem 1. Oct 19, 20 let l be a lie superalgebra with its enveloping algebra ul over a field f. Dictionary on lie algebras and superalgebras presents a detailed description of the structure of lie algebras and lie superalgebras and an extensive study of their finite dimensional representation theory. As one can talk about associative and lie algebras there, i can imagine the definition in terms of the universal property but i am interested in its existence, a construction, if you may.
Centers of universal enveloping algebras of lie superalgebras. Kerade attains the minimal dimension, which is equal to the rank of g. We apply the factorization to obtain a description of. Then u l can be viewed both as a lie algebra and a lie superalgebra. Shapovalov determinants of qtype lie superalgebras.
Color lie superalgebras and hopf algebras springerlink. Engel condition on enveloping algebras of lie superalgebras. Lie superalgebra analog of the lie delta ideal, used by the authors in the study of enveloping rings, and also of the finite conjugate center of a group, used in the study of group algebras and. Nov 19, 2007 on the freudenthals construction of exceptional lie algebras yamaguti, kiyosi and asano, hiroshi, proceedings of the japan academy, 1975. Finite dimensional simple lie superalgebras over algebraically closed fields of characteristic zero were classified by v. These superalgebras include abelian lie superalgebras, simple lie superalgebras with the nondegenerate killing form, trivial extensions of the semisimple lie algebras, and the metric lie superalgebras with dimension. One can see that g0 is a lie algebra and g1 is a g0module. Download lie superalgebras and enveloping algebras by ian m. Lie superalgebras whose enveloping algebras satisfy a non. On some representations of nilpotent lie algebras and.