This book develops the theory of lie superalgebras, their enveloping algebras, and their representations. On the distribution of arcomponents of restricted lie algebras. The universal enveloping algebra of a finitedimensional lie algebra is a filtered quadratic algebra. Lie superalgebras and enveloping algebras, by ian musson. Classical simple lie superalgebras borel subalgebras and dynkinkac diagrams contragredient lie superalgebras the pbw theorem and filtrations on enveloping algebras methods from ring theory enveloping algebras of classical simple lie superalgebras verma modules schur. Our main result is that any graded primitive ideal is the annihilator of a graded simple quotient of a verma module.
Introduction to finite and infinite dimensional lie algebras and superalgebras introduces the theory of lie superalgebras, their algebras, and their representations. Dec 08, 2004 semisimplicity of restricted enveloping algebras of lie superalgebras. Lie superalgebras and enveloping algebras ebook, 2012. We characterize l when ul satisfies a nonmatrix polynomial identity. Lie superalgebras and lie supergroups alan huckleberry november 10, 2009. Universal enveloping algebras of lie antialgebras springerlink. In this talk, i will discuss how essentially the same ideas, with some modifcations, can be used to construct canonical bases for basic type lie superalgebras, in the. Engel condition on enveloping algebras of lie superalgebras salvatore siciliano and hamid usefi abstract. If you are working over a field of positive characteristic, then the result is also true if you replace the universal enveloping algebra by the restricted enveloping algebra see theorem 6. On the theory of frobenius extensions and its application to lie superalgebras.
Introduction in 2007, ovsienko 11 introduced a class of nonassociative superalgebras called lie antialgebras. Mixed cohomology of lie superalgebras pdf free download. Lie identities on enveloping algebras of restricted lie superalgebras article in journal of algebra 393. Lie algebras of order three or more generally lie algebras of order f were introduced in 8 as a possible generalisation of lie superalgebras, in order to implement nontrivial extensions of the poincar. A serre type theorem for affine lie superalgebras and their quantized enveloping superalgebras yamane, hiroyuki, proceedings of the japan academy, series a, mathematical sciences, 1994. Musson, university of wisconsin, milwaukee, milwaukee, wi. Lie superalgebras and enveloping algebras musson, ian. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. 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 elements form an abelian group g gh, and h is generated by its relatively primitive elements which supercommute with the.
We also characterize l when ul is lie solvable, lie nilpotent, or lie supernilpotent. We describe a type of lie colour algebra, which we call generic, whose universal enveloping algebra is a domain with finite global dimension. A classification of primitive ideals in the enveloping. We compare a number of different definitions of structure algebras and tkk constructions for jordan super algebras appearing in the literature. Because casimir operators commute with all elements of a lie algebra. Kinematical superalgebras and lie algebras of order. 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.
A cohomology of lie colour algebras were introduced and investigated in 16,17 and representations of lie colour algebras were explicitly described in 8. Another application of this method is the chevalley type restriction theorem for simple finitedimensional lie superalgebras. 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. We show that every lie antialgebra is canonically related to a lie superalgebra and prove that its enveloping algebra is a quotient of the enveloping algebra of the corresponding lie superalgebra. Note on frobenius extensions and restricted lie superalgebras. Primitive ideals, twisting functors and star actions for. Moreover, one can be obtained as the lie superalgebra of superderivations of the other. During the preparation of the following pages, it has appeared to us necessary to update the lie algebra part. Classification of classical simple lie superalgebras. In mathematics, a universal enveloping algebra is the most general algebra that contains all representations of a lie algebra. Mikhaleva composition lemma and the equality problem for colored lie superalgebras.
Let g g0 g1, be a basic classical simple lie superalgebra, and. Neu, training manual on transport and fluids, 2010. The tensor product of two superalgebras is defined by. Primitive ideals, twisting functors and star actions for classical lie superalgebras kevin coulembier and volodymyr mazorchuky abstract we study three related topics in representation theory of classical lie superalgebras. Universal enveloping algebras of lie antialgebras, algebras. The full universal enveloping algebra ejh of the lie superalgebra. In both references the rank 2 classical simple lie superalgebras just appear at. The main topics deal with the structure of simple lie superalgebras and their. Lie identities on enveloping algebras of restricted lie.
In this work we will study the cohomology of lie algebras and lie superalgebras. Simple lie algebras in characteristic 2 recovered from. In mathematics, a lie superalgebra is a generalisation of a lie algebra to include a z2grading. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. Finite dimensional simple lie superalgebras over algebraically closed fields of characteristic zero were classified by v. We provide an application and show grobner basis methods can be used to study universal enveloping algebras of factors of generic lie colour algebras.
Automorphic forms and lie superalgebras urmie ray auth. Formally, a lie superalgebra is a nonassociative z 2graded algebra, or superalgebra, over a commutative ring typically r or c whose product, called the lie superbracket or supercommutator, satisfies the two conditions analogs of the usual lie algebra axioms, with grading. Laplace operators of infinitedimensional lie algebras and. Most known examples of lie superalgebras with a related automorphic form such as the fake monster lie algebra whose reflection group is given by the leech lattice arise from superstring theory and can be. This together with other recent papers yields theorem. Generalizations of lie algebras some other generalizations of lie algebras are i color lie algebras and superalgebras 1978 i lie. Centers of universal enveloping algebras of lie superalgebras in prime characteristic. The explicit expressions for indecomposable representations of nine squareroot lie algebras of vector type, are obtained on the space of universal enveloping algebra of twostate heisenberg. Representation theory of lie superalgebras and related topics. The universal enveloping algebra of the lie algebra g is. I tangent algebras to analytic loops a loop is a nonassociative counterpart of a group exp map to loop. Bergen, jeffrey, pacific journal of mathematics, 1994. The material covered ranges from basic definitions of lie groups to the classification of finitedimensional representations of semisimple lie algebras.
Dictionary on lie superalgebras cern document server. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. 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. Jul 22, 2010 universal enveloping algebras of lie antialgebras leidwanger, severine. Lie superalgebras and enveloping algebras about this title. Im reading mussons book on lie superalgebras and enveloping algebras as well kac advances paper on this subject. Generic lie colour algebras bulletin of the australian. Certain structural features of these algebras account for various phenomena in lie theory that. It is a borcherdskacmoody lie algebra with lorentzian root lattice. An isomorphism psi from the supersymmetric algebra superl l of pll to ukappapll. Infinite dimensional lie algebras an introduction progress.
Lie superalgebras whose enveloping algebras satisfy a non. Browse other questions tagged lie algebras or ask your own question. The first example is the lie superalgebra glmn realized as the algebra of endomorphisms of a z2graded vector space v with dim v0 m, dim. Introduction to finite and infinite dimensional lie super. Lie antialgebras is a class of supercommutative algebras recently appeared in symplectic geometry. While discussing all classes of finite and infinite dimensional lie algebras and lie superalgebras in terms of their different classes of root systems, the book focuses on kacmoody algebras.
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. This is an analogue of the wellknown theorem of duflo d on primitive ideals in the enveloping algebra of a semisimple lie algebra. The straightening laws for the enveloping algebra ukappapll of the general linear lie superalgebra of a finite dimensional z2graded vector space are investigated. The universal enveloping algebras of color lie superalgebras are koszul. Lie superalgebras appear in 4 as lie algebras of certain generalized groups, nowadays called lie supergroups, whose function algebras are algebras with commuting and anticommuting variables. We define the notion of enveloping algebra of a lie antialgebra and study its properties. The success of this publication as well as the encouragements of many of our collegues convinced us to repeat the same exercise for superalgebras. Let j be the jordan lie superalgebra with basis as above. Primeness of the enveloping algebra of the special lie. Representations of lie colour algebras request pdf. Integral bases for the universal enveloping algebras of map. Note on frobenius extensions and restricted lie superalgebras note on frobenius extensions and restricted lie superalgebras farnsteiner, rolf 199605 00. Infinite dimensional lie algebras an introduction progress in.
Consideruas a gmodule with respect to the adjoint action. The main ideas will be illustrated with examples in. A polynomial identity is called nonmatrix if it is not satisfied by the algebra of. Let l be a finitedimensional simple lie superalgebra over an algebraically closed field of. With numerous exercises and worked examples, it is ideal for graduate courses on lie groups and lie algebras.
We construct quantum deformations of enveloping algebras of borcherds superalgebras, their verma modules, and their irreducible highest weight modules. Recently, a satisfactory theory, similar to lie s theory, has been developed on the connection between lie supergroups and lie superalgebras 5. Nowadays enveloping algebras and their deformations are studied in a number of interrelated fields such as quantum groups, lie superalgebras or restricted lie algebras. Download infinite dimensional lie algebras an introduction progress in mathematics or read online books in pdf, epub, tuebl, and mobi format. Universal enveloping algebras are the lie theoretic analogues of group algebras. In this paper, we prove classical affine w algebras associated to lie superalgebras w superalgebras, which can be constructed in two different ways. The first part, chapters 15, introduces the reader to the concept of lie superalgebras. Some of the articles by experts in the field survey recent developments while others include new results in representations of lie algebras, and quantum groups, vertex operator algebras and lie superalgebras. Click download or read online button to get infinite dimensional lie algebras an introduction progress in mathematics book now. Color lie superalgebras and hopf algebras springerlink. Newest liesuperalgebras questions mathematics stack exchange. Classical affine walgebras associated to lie superalgebras.
Citeseerx document details isaac councill, lee giles, pradeep teregowda. On representations of lie superalgebras furutsu, hirotoshi and hirai, takeshi, proceedings of the japan academy, series a, mathematical sciences, 1987. Full text get a printable copy pdf file of the complete article 611k, or click on a page image below to browse page by page. The rst one is classi cation of primitive ideals, i. Auslanderreiten theory for restricted lie algebras. Lie superalgebras are generalizations of lie algebras, useful for depicting supersymmetry the symmetry relating fermions and bosons. Lie algebras for physicists written fteen years ago see ref. This is in complete contrast with the enveloping algebras of ordinary lie superalgebras satisfying a nonmatrix pi, see theorem 1. Let g be a complex semisimple lie algebra, let ube its universal enveloping algebra, and let zg be the centre of u. Using the hopf algebra structure of ul, it is shown that over.
M87 2012 510dc23 2011044064 copying and reprinting. Centers of universal enveloping algebras of lie superalgebras in prime characteristic article pdf available october 2012 with 161 reads how we measure reads. Modular lie algebras universal enveloping algebras. We demonstrate that, for unital superalgebras, all the definitions of the structure algebra and the tkk constructions reduce to one of two cases. The enveloping algebra of the lie superalgebra osp1. Engel condition on enveloping algebras of lie superalgebras. Universal enveloping algebras are used in the representation theory of lie groups and lie algebras. Ohio state mathematical research institute publications 7 1998, 165186. 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. Hompseudo superalgebras of associative and lie types in this section we will introduce the concept and construction theorems of homhpseudo superalgebras of associative and lie types, and show some examples of hom lie hpseudo superalgebras that are neither hom lie superalgebras nor hom lie pseudo algebras. Centers of universal enveloping algebras of lie superalgebras. Lowdimensional z2z2 lie algebras and lie superalgebras. The book begins with five chapters on the basic properties of lie superalgebras, including explicit constructions for all the classical simple lie superalgebras.
In mathematics, a lie superalgebra is a generalisation of a lie algebra to include a z 2grading. Lie superalgebras are a natural generalization of lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. In most of these theories, the even elements of the superalgebra correspond to bosons and odd elements to fermions. Geometric differences between the use of lie algebras and. First we observe that every lie supernilpotent superalgebra is indeed lie solvable. Lie superalgebras and enveloping algebras book, 2012. Moreover we know when the enveloping algebra ug is a domain, when it has. 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. Universal enveloping algebras arbeitsgruppe algebra. Many more properties and structures of hom lie algebras have been deve l oped see 2023 and references cited therein. Jul 22, 2010 lie antialgebras is a class of supercommutative algebras recently appeared in symplectic geometry. The dictionary extensively covers finite dimensional algebras and superalgebras, which moreover are indispensable to the study of more elaborate. For example, verma modules can be constructed as quotients of the universal enveloping algebra.
Recall that a lie superalgebra is just a z2graded vector. Universal enveloping poisson conformal algebras request pdf. Simple lie algebras in characteristic 2 recovered from superalgebras and on the notion of a simple finite group yuri kochetkov and dimitry leites introduction in what follows, unless mentioned to the contrary, k is an algebraically closed field, char k p. Lie superalgebras and enveloping algebras university of wisconsin. The axioms were established after encountering two unusual algebraic structures in a context of symplectic geometry.
A primeness criterion due to bell is shown to apply to the universal enveloping algebra of the cartan type lie superalgebras sv and e sv. Lie superalgebras of types a, q and recently for the orthosymplectic lie algebras bru03, bru04, clw09, gs09, ps97a, ps97b, ser96. A polynomial identity is called nonmatrix if it is not satisfied by the algebra of 2. 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. The separation theorem of kostant see ko states the existence of a submodule hof usuch that the multiplication map provides. Pdf centers of universal enveloping algebras of lie. Moores 1965 paper on the structure of hopf algebras. On structure and tkk algebras for jordan superalgebras. Lie algebras, lie superalgebras, vertex algebras and related. The universal enveloping algebras considered correspond to a minimal locality function for which there is an injective embedding. In addition, the enveloping algebra gives a precise definition for the casimir operators. Pdf let l be a lie superalgebra with its enveloping algebra ul over a field. Oct 19, 20 let l be a lie superalgebra with its enveloping algebra ul over a field f. Numerous and frequentlyupdated resource results are available from this search.
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