Bergen, jeffrey, pacific journal of mathematics, 1994. Moreover we know when the enveloping algebra ug is a domain, when it has. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. The reader may skip these chapters if he or she feels su ciently versed in that topic. Graduate students and research mathematicians interested in lie algebras, lie superalgebras, and vertex algebras. In this paper, we prove classical affine walgebras associated to lie superalgebras wsuperalgebras, which can be constructed in two different ways.
For lie superalgebras of type i, this was done in l2 using a bijection. We also characterize l when ul is lie solvable, lie nilpotent, or lie supernilpotent. The universal enveloping algebra of the lie algebra g is. 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. Lie superalgebras and enveloping algebras american. Pdf lie superalgebras whose enveloping algebras satisfy a non. These conditions are not equivalent for lie superalgebras. Shirshov basis for a lie superalgebra if and only if. 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. Lie superalgebras are generalizations of lie algebras, useful for depicting supersymmetry the symmetry relating fermions and bosons.
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. Engel condition on enveloping algebras of lie superalgebras salvatore siciliano and hamid usefi abstract. This booklet develops the speculation of lie superalgebras, their enveloping algebras, and their representations. 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. Lie superalgebras and enveloping algebras, by ian musson. First we observe that every lie supernilpotent superalgebra is indeed lie solvable. Let g g0 g1, be a basic classical simple lie superalgebra, and h a cartan subalgebra.
It is a borcherdskacmoody lie algebra with lorentzian root lattice. Pdf lie superalgebras whose enveloping algebras satisfy. 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. 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.
The dictionary extensively covers finite dimensional algebras and superalgebras, which moreover are indispensable to the study of more elaborate. M87 2012 510dc23 2011044064 copying and reprinting. We provide an application and show grobner basis methods can be used to study universal enveloping algebras of factors of generic lie colour algebras. Lie superalgebras and enveloping algebras university of wisconsin. Journal of algebra 145, 121 1992 identities in the enveloping algebras for modular lie superalgebras v. Infinite dimensional lie algebras an introduction progress. Color lie superalgebras and hopf algebras springerlink. Musson, university of wisconsin, milwaukee, milwaukee, wi. 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. Semisimplicity of restricted enveloping algebras of lie superalgebras. We apply the factorization to obtain a description of. Lie superalgebras and enveloping algebras musson, ian. Lie identities on enveloping algebras of restricted lie.
In a previous paper, we studied the homogenized enveloping algebra of the lie algebra s. Lie algebras, lie superalgebras, vertex algebras and related topics share this page. Automorphic forms and lie superalgebras urmie ray auth. Centers of universal enveloping algebras of lie superalgebras in prime characteristic.
In the study of group algebras and enveloping algebras of ordinary lie algebras, many questions have been solved using amethods. 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. Using the hopf algebra structure of ul, it is shown that over. Lie superalgebras are a natural generalization of lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. Introduction to finite and infinite dimensional lie algebras and superalgebras introduces the theory of lie superalgebras, their algebras, and their representations. Let dbe a subset of an associative algebra uover a. We refer the reader to 1 for basic background about lie superalgebras and their enveloping algebras and to 2, 15, 16 for notation. The first part, chapters 15, introduces the reader to the concept of lie superalgebras. It is intended to be useful to mathematical and theoretical physicists, from the level of the graduate student upwards. We construct quantum deformations of enveloping algebras of borcherds superalgebras, their verma modules, and their irreducible highest weight modules. 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.
In this paper we study deformations of some classes of metric lie superalgebras over the complex field. 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. Kerade attains the minimal dimension, which is equal to the rank of g. 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. Download lie superalgebras and enveloping algebras by ian. Identities in the enveloping algebras for modular lie. Then the enveloping algebra ul is also ggraded, and we. A polynomial identity is called nonmatrix if it is not. 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. In mathematics, a lie superalgebra is a generalisation of a lie algebra to include a z 2grading.
It is reasonable to try to find similar techniques in the lie superalgebra context. This book is a detailed reference on lie algebras and lie superalgebras presented in the form of a dictionary. We characterize l when u l is lie solvable, lie nilpotent, or lie supernilpotent. 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. Rt 24 sep 2004 on some representations of nilpotent lie algebras and superalgebras shantala mukherjee. Let l be a restricted lie superalgebra with its restricted enveloping algebra u l over a field f of characteristic p 2. Lie superalgebras and enveloping algebras about this title. Universal enveloping algebras 24 free products 26 4.
Lie superalgebras are a ordinary generalization of lie algebras, having functions in geometry, quantity thought, gauge box idea, and string idea. Centers of universal enveloping algebras of lie superalgebras. Shapovalov determinants of qtype lie superalgebras. Grobnershirshov bases for lie superalgebras and their. 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. Also, we show that a classical finite wsuperalgebra can be obtained by a zhu algebra of a classical affine wsuperalgebra. 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.
Identities in the enveloping algebras for modular 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. Finite dimensional simple lie superalgebras over algebraically closed fields of characteristic zero were classified by 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. Passman depaul university university of wisconsinmadison abstract. Several authors have tried to construct the enveloping algebras of homlie algebras. Click download or read online button to get infinite dimensional lie algebras an introduction progress in mathematics book now. Then l is a functor which converts associative algebras into lie algebras. Lie superalgebras of types a, q and recently for the orthosymplectic lie algebras bru03, bru04, clw09, gs09, ps97a, ps97b, ser96.
Zolotykh crc press boca raton new york london tokyo. 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. 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. 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. Lie superalgebras whose enveloping algebras satisfy a non.
The universal enveloping algebra of a finitedimensional lie algebra is a filtered quadratic algebra. Combinatorial aspects of lie superalgebras alexander a. This is in complete contrast with the enveloping algebras of ordinary lie superalgebras satisfying a nonmatrix pi, see theorem 1. Nov 19, 2007 on the freudenthals construction of exceptional lie algebras yamaguti, kiyosi and asano, hiroshi, proceedings of the japan academy, 1975. The first example is the lie superalgebra glmn realized as the algebra of. We characterize l when ul satisfies a nonmatrix polynomial identity. Then u l can be viewed both as a lie algebra and a lie superalgebra. Engel condition on enveloping algebras of lie superalgebras. L\ dimkl, lie algebras, the universal enveloping algebra of the lie superalgebra can be given a hopf algebra structure. One can see that g0 is a lie algebra and g1 is a g0module.
This book develops the theory of lie superalgebras, their enveloping algebras, and their representations. Let l be a lie superalgebra with its enveloping algebra ul over a field f. On antistructurable algebras and extended dynkin diagrams kamiya, noriaki and mondoc, daniel, journal of generalized lie theory and applications, 2009. On some representations of nilpotent lie algebras and. 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. Recall that a lie superalgebra is just a z2graded vector. The enveloping algebra of the lie superalgebra osp1. Download lie superalgebras and enveloping algebras by ian m. The orbit method was created by kirillov in the attempt to describe the unitary dual n. Lie identities on enveloping algebras of restricted lie superalgebras article in journal of algebra 393. 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. Infinite dimensional lie algebras an introduction progress in. Let g be a nite abelian group and let l be a, possibly restricted, ggraded lie color algebra. Lie algebras, lie superalgebras, vertex algebras and.
A polynomial identity is called nonmatrix if it is not satisfied by the algebra of 2. Classical affine walgebras associated to lie superalgebras. For lie superalgebras of type i, this was done in l2. A nilpotent element e 2 gis regular nilpotent if e. Generic lie colour algebras bulletin of the australian. Mondragon in our work on groebner basis algebras, to give the. Dictionary on lie algebras and superalgebras luc frappat. Quantized enveloping algebras associated with simple lie. Oct 19, 20 let l be a lie superalgebra with its enveloping algebra ul over a field f. A kalgebra is a kvector space with two linear maps m.
On the freudenthals construction of exceptional lie algebras yamaguti, kiyosi and asano, hiroshi, proceedings of the japan academy, 1975. Introduction to finite and infinite dimensional lie super. In this work we want to study the cohomology of lie algebras and lie superalgebras and apply the results to several examples. Chapters 1 and 2 give an introduction to the theory of lie algebras and lie superalgebras as well as to their representations. It turned out that the orbit method had much wider applications.
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