A non abelian lie algebra g is called simple if it has no non trivial ideals. Also, in contrast to the classical case of 2dimensional lie algebras, there is an infinity of 5lie algebra structures on the plane. A polynomial identity is called nonmatrix if it is not satisfied by the algebra of 2. The axioms were established after encountering two unusual algebraic structures in a context of symplectic geometry. This together with other recent papers yields theorem. Questions tagged liesuperalgebras mathematics stack exchange. Lie identities on enveloping algebras of restricted lie. In both references the rank 2 classical simple lie superalgebras just appear at. Lie nalgebras, supersymmetry, and division algebras introduction i everything in this table can be made super. This book develops the theory of lie superalgebras, their enveloping algebras, and their representations.
Algebraic groups and lie algebras atlantic algebra centre. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. 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. Lie groups and lie algebras in robotics 7 u x figure 3.
We now combine these results to obtain a formula for the restriction of. The universal enveloping algebra of a lie algebra is a hopf algebra. Lie algebra, algebraic encyclopedia of mathematics. Lie algebras, lie superalgebras, vertex algebras and related. These algebras form a very special class of jordan superalgebras, they are closely related to lie super algebras and, in some sense, link together commutative and lie algebras.
Pdf centers of universal enveloping algebras of lie. It is found that if nontrivial maps u w and w u are to be included, then the lie algebra structure of endu. Lie superalgebras are a natural generalization of lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. Musson, university of wisconsin, milwaukee, milwaukee, wi. Oct 19, 20 let l be a lie superalgebra with its enveloping algebra ul over a field f. However, if lis the lie algebra of an a ne algebraic group, and mis the lie algebra of a closed subgroup, the following construction is more natural. In 17, by substituting universal enveloping vertex algebras of lcas in the brst complexes with symmetric algebras generated by the lcas, we get classical a. Enveloping algebra of homlie algebras researchgate. Sep 05, 2008 in the complex and quaternionic cases, they constitute generalisations of the baggerlambert hermitian 3 algebras and anti lie triple systems, respectively, which underlie n6 and n5 superconformal chernsimons theories, respectively. Quantum enveloping superalgebras and link invariants core. Both are immediatelyseen to be 1dimensional and smooth. A number of questions about algebraic lie algebras have been solved in the class of nil lie algebras cf.
In both references the rank 2 classical simple lie. An introduction lecture notes in mathematics keywords. Unitary representation theory of lie superalgebras uq espace. Let l be a finitedimensional simple lie superalgebra over an algebraically closed field of characteristic zero. 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 whose enveloping algebras satisfy a non. Lie groups and algebras exponential map if g is a matrix lie group with lie algebra g, then the exponential mapping for g is the map. Given an arbitrary rigid transformation it can always be put in. But since in the universal enveloping x, x 2x2 for any odd x, this difference between lie algebras and lie superalgebras disappears in char 2.
The lie algebra of a group is a complete invariant of the local group structure. The concept of supersymmetry first appeared in particle physics out of a need to merge. The workshop is the continuation of joint activity between the atlantic algebra centre and the network of ontario lie theorists. Finite dimensional simple lie superalgebras over algebraically closed fields. In the process we rederive the relation between certain types of complex 3algebras and metric lie superalgebras. Lie superalgebras and enveloping algebras about this title. The graded lie algebras of an algebra by albert nijenhuis 1 communicated at the meeting of june 24, 1967 l. 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.
Maybe, this is a good point to complement the picture with the lie algebras of gm and ga, since we have looked a lot at those groups before. Recently, a satisfactory theory, similar to lies theory, has been developed on the connection between lie supergroups and lie superalgebras 5. The dictionary extensively covers finite dimensional algebras and superalgebras, which moreover are indispensable to the study of more elaborate. Their universal enveloping algebras are z2 graded associative unital algebras, and we will always. 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. Graduate studies in mathematics publication year 2012. In the finitedimensional case this provides a new proof of a result of harishchandra. Lie groups and algebras for optimisation and motion.
Jul 22, 2010 lie antialgebras is a class of supercommutative algebras recently appeared in symplectic geometry. We define the notion of enveloping algebra of a lie antialgebra and study its properties. American mathematical society 201 charles street providence. For example, verma modules can be constructed as quotients of the universal enveloping algebra. Citeseerx document details isaac councill, lee giles, pradeep teregowda. However we give explicit constructions for eachclassicalsimpleliesuperalgebrag. Given an arbitrary rigid transformation it can always be put in the above form. Lie nalgebras, supersymmetry, and division algebras introduction i yet superstrings and super2branes are exceptional objectsthey only make sense in certain dimensions.
Just as for lie algebras, the universal enveloping algebra of the lie superalgebra can be given a hopf algebra structure. The analogous result for lie p algebras and their restricted universal enveloping algebras holds and is proved. Request pdf elementary lie algebras and lie a algebras a finitedimensional lie algebra l over a field f is called elementary if each of its subalgebras has trivial frattini ideal. Grobnershirshov bases for lie superalgebras and their. We shall do this by constructing explicitly the associated universal matrices. Universal enveloping algebra to any lie algebra l, we assign an associative algebra ul called the universal en. I the theory of lie nalgebravalued connections was. The centers of ugandug0 are denoted by zgandzg0 respectively.
Primeness of the enveloping algebra of the special lie. Let gbe an a ne algebraic group with unit e, and denote the stalk at eof the sheaf of regular functions on open subsets of gby o e. Lie algebra and enveloping algebra mathematics stack. In mathematics, a universal enveloping algebra is the most general unital, associative algebra that contains all representations of a lie algebra universal enveloping algebras are used in the representation theory of lie groups and lie algebras. The centre of the universal enveloping algebra u is denoted z and given by. This is in complete contrast with the enveloping algebras of ordinary lie superalgebras satisfying a nonmatrix pi, see theorem 1. If we combine the two former expressions for u, we get. L in this paper, we introduce a new family of quasitriangular hopf algebras coming from complex simple lie superalgebras. May 07, 2019 the workshop is the continuation of joint activity between the atlantic algebra centre and the network of ontario lie theorists. On the generalized enveloping algebra of a color lie. We also characterize l when ul is lie solvable, lie nilpotent, or lie supernilpotent. Lie algebras, lie superalgebras, vertex algebras and.
Algebraic groups and lie algebras atlantic algebra. I a connection valued in lie nalgebra is a connection on an nbundle, which is like a bundle, but the. I like many exceptional objects in mathematics, they are tied to the division algebras, r, c, h and o. W requires geometries on u and w of the same type that is, both orthogonal, or both symplectic, or both unitary, or both antiunitary whereas the lie superalgebra requires to combine the geometrytypes of u and w in such.
So, for example, the number of paths of length 2 joining i to j is the i. Elementary lie algebras and lie aalgebras request pdf. The enveloping algebra of the lie superalgebra osp1. The universal enveloping algebra of a finitedimensional lie algebra is a filtered quadratic algebra. Pdf file 1608 kb djvu file 349 kb article info and citation. 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. Basic classical simple lie superalgebras are defined in k2.
Simple lie algebras in characteristic 2 recovered from. In the complex and quaternionic cases, they constitute generalisations of the baggerlambert hermitian 3algebras and antilie triple systems, respectively, which underlie n6 and n5 superconformal chernsimons theories, respectively. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Pdf enveloping algebras of color homlie algebras researchgate. Outline in a recent note 11 the author showed the existence of a natural graded lie gla structure on the cohomology hb, ab when a is an associative, commutative or lie algebra and b a subalgebra. In fact, gs is highly nonabelian since it has trivial center and equals its derived algebra. One of the beautiful properties in representation theory of a ne lie algebras is the sl 2pzqinvariance of the space of characters of integrable modules discovered by kacpeterson in. Universal enveloping algebras of lie antialgebras springerlink. This means that two lie groups have isomorphic lie algebras if and only if the groups are locally isomorphic. Quantized enveloping algebras associated with simple lie. Introduction in 2007, ovsienko 11 introduced a class of nonassociative superalgebras called lie antialgebras. In mathematics, a lie superalgebra is a generalisation of a lie algebra to include a z 2grading.
Geometric differences between the use of lie algebras and. Let g be the lie superalgebra osp1, 2r and ug the enveloping algebra 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. In the process we rederive the relation between certain types of complex 3 algebras and metric lie superalgebras. Moores 1965 paper on the structure of hopf algebras. Color lie superalgebras and hopf algebras springerlink. In chapter 6 we define the enveloping algebra uk of a lie superalgebra k. We show that a set of monic polynomials in a free lie superalgebra is a grobnershirshov basis for a lie superalgebra if and only if it is a grobnershirshov basis for its universal enveloping algebra. 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. Geometric differences between the use of lie algebras and lie.
This time we focus on the interrelations and various connections between the lie theory and algebraic groups as well as on its applications to representation theory, graded algebras, superalgebras and quantum groups. The action of zgonmf determines the central character. Lie algebra and enveloping algebra mathematics stack exchange. Translated simple modules for lie algebras and simple. Enveloping algebra of homlie algebras article in journal of generalized lie theory and applications 22 january 2008 with 63 reads how we measure reads. Quantized enveloping algebras associated with simple lie superalgebras and their universal jrmatrices by hiroyuki yamane introduction o. First we observe that every lie supernilpotent superalgebra is indeed lie solvable. Let l be a lie superalgebra with its enveloping algebra ul over a field f. Recently, a satisfactory theory, similar to lie s theory, has been developed on the connection between lie supergroups and lie superalgebras 5.
We construct quantum deformations of enveloping algebras of borcherds superalgebras, their verma modules, and their irreducible highest weight modules. 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 elements of g. An introduction lecture notes in mathematics free ebook. Mock theta functions and representation theory of affine. 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. A subalgebra s of l is stable if 5 remains a subalgebra under small deformations of l. Newest liesuperalgebras questions mathematics stack. Let l be a finitedimensional simple lie superalgebra over an algebraically closed field of. Lie nalgebras, supersymmetry, and division algebras. Im reading mussons book on lie superalgebras and enveloping algebras as well kac advances paper on this subject. Since the pioneering work of baggerlambert and gustavsson, there has been a proliferation of threedimensional superconformal chernsimons theories whose main ingredient is a metric 3algebra. The only property which differentiates lie superalgebras from ll2graded lie algebras is that the former can have a nonzero bracket of an odd element with itself. Borel, linear algebraic groups, benjamin 1969 mr0251042 zbl 0206. Pdf in this paper the universal enveloping algebra of color homlie algebras is studied.
Many lie algebroids but not all of them arise as infinitesimal parts of lie groupoids, just as finitedimensional lie algebras do from lie groups. 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. Mock theta functions and representation theory of affine lie superalgebras and superconformal algebras abstract. I the corresponding lie 2 and lie 3superalgebras are similarly exceptional. Differential algebraic lie algebras 249 in classical lie theory since it is a nonabelian 1parameter lie algebra.
526 694 1240 688 420 695 104 217 1442 519 741 1139 68 141 42 1390 1356 1341 101 1141 1502 218 1240 76 856 565 295 1450 37 1323 1321 1029 1327 888