Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix

Finite dimensional modular Lie superalgebras over algebraically closed fields with indecomposable Cartan matrices are classified under some technical, most probably inessential, hypotheses. If the Cartan matrix is invertible, the corresponding Lie superalgebra is simple otherwise the quotient of the derived Lie superalgebra modulo center is simple (if its rank is greater than 1). Eleven new exceptional simple modular Lie superalgebras are discovered. Several features of classic notions, or notions themselves, are clarified or introduced, e.g., Cartan matrix, several versions of restrictedness in characteristic 2, Dynkin diagram, Chevalley generators, and even the notion of Lie superalgebra if the characteristic is equal to 2. Interesting phenomena in characteristic 2: (1) all simple Lie superalgebras with Cartan matrix are obtained from simple Lie algebras with Cartan matrix by declaring several (any) of its Chevalley generators odd; (2) there exist simple Lie superalgebras whose even parts are solvable. The Lie superalgebras of fixed points of automorphisms corresponding to the symmetries of Dynkin diagrams are also listed and their simple subquotients described

Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix

Finite dimensional modular Lie superalgebras over algebraically closed fields with indecomposable Cartan matrices are classified under some technical, most probably inessential, hypotheses. If the Cartan matrix is invertible, the corresponding Lie superalgebra is simple otherwise the quotient of the derived Lie superalgebra modulo center is simple (if its rank is greater than 1). Eleven new exceptional simple modular Lie superalgebras are discovered. Several features of classic notions, or notions themselves, are clarified or introduced, e.g., Cartan matrix, several versions of restrictedness in characteristic 2, Dynkin diagram, Chevalley generators, and even the notion of Lie superalgebra if the characteristic is equal to 2. Interesting phenomena in characteristic 2: (1) all simple Lie superalgebras with Cartan matrix are obtained from simple Lie algebras with Cartan matrix by declaring several (any) of its Chevalley generators odd; (2) there exist simple Lie superalgebras whose even parts are solvable. The Lie superalgebras of fixed points of automorphisms corresponding to the symmetries of Dynkin diagrams are also listed and their simple subquotients described

Algebry Liego macierzy nieskończonych

The works of S. Lie, W. Killing and E. Cartan were the starting points for systematic development of the theory of finite-dimensional Lie algebras. We mention here the classification of finite-dimensional simple Lie algebras over the algebraically closed fields (for fields of characteristic 0 due to E. Cartan and W. Killing and for characteristic p > 3 given in works of R. E. Block, R. L. Wilson, H. Strade, A. Premet) and the representation theory (the highest weight classification of irreducible modules of general linear Lie algebras). However, at the present time, there is no general theory of the infinitedimensional Lie algebras. There are few classes of infinite-dimensional Lie algebras that were more or less intensively studied from the geometric point of view: the Lie algebras of vector fields, the Lie algebras of smooth mappings of a given manifold into a finite-dimensional Lie algebra, the classical Lie algebras of operators in a Hilbert or Banach space and the Kac-Moody algebras. Algebraic point of view was used in investigations of free Lie algebras and graded Lie algebras. In many papers appear, as examples, the Lie algebra of Z x Z infinite matrices over C which have only finite number of nonzero entries and g j - the Lie algebra of generalized Jacobian matrices, i.e. infinite matrices having nonzero entries in a finite number of diagonals. They play important role in representation theory and physics. We note th at there is no systematic study of Lie algebras of infinite matrices. In this thesis, we consider the Lie algebra of column-finite infinite matrices indexed by positive integers N, describe the lattice of its ideals and describe its derivations. All rings R in the thesis are commutative and with unity. In the chapter 1 we present basic notions used in the thesis. We give descriptions of ideals and derivations of Lie algebras of finite-dimensional matrices. We recall the classification of finite-dimensional simple Lie algebras. In the chapter 2 we survey some directions in study infinite-dimensional Lie algebras and give two fundamental examples of Lie algebras of infinite matrices - and g l j . The third chapter contains results on Lie algebras of infinite matrices. We give the definition of the Lie algebra glcf (N, R) of column-finite matrices over R indexed by positive integers. We prove that glcf (N, R) is isomorphic with the Lie algebra of column-finite matrices indexed by integers. This shows that all results in the thesis are valid for Lie algebras of Z x Z column-finite matrices. We also define fundamental Lie subalgebras of glcf (N, R) and prove some of their properties. The fourth chapter contains results on the Lie algebra slf r (N, R) of infinite matrices having nonzero entries in only finite number of rows and with trace zero. We describe its structure. For any field K , we prove the simplicity of s lfr (N, K ). We note that A. A. Baranov found classification of finitary simple Lie algebras over a field of characteristic 0 and together with H. Strade they gave classification of finitary simple Lie algebras for any algebraically closed field of prime characteristic p > 3 (they use the classification of simple finitedimensional Lie algebras over an algebraically closed field of prime characteristic p > 3). The Lie algebra slcf (N, K ) is a matrix representation of corresponding finitary Lie algebra. In the fifth chapter we prove that every derivation of the Lie algebra of strictly upper triangular infinite matrices over R is a sum of inner and diagonal derivations. We also prove th at every derivation of glcf (N, R) is a sum of inner and central derivations. The last chapter contains description of lattice of ideals of glcf (N, K ). The description does not depend on the characteristic of K. As a corollary, we obtain a new uncountably dimensional simple Lie algebra

The Gelfand-Kirillov dimension of rank 2 Nichols algebras of diagonal type

Most interests in the theory of Nichols algebras emerged from the the theory of pointed Hopf algebras. For their classification it is an essential step to classify finite (Gelfand-Kirillov-) dimensional Nichols-algebras under some finiteness conditions. Nichols algebras have been discussed iby various authors. Especially, those of diagonal type which yielded interesting applications, for example as the positive part of quantized enveloping algebra of a simple finite-dimensional Lie algebras g. Finite-dimensional Nichols algebras of diagonal type have been classified in a series of papers. One important step for this has been the introduction of the root-system and the associated Weyl groupoid.In this context the following implications were observed: (1) If a Nichols algebra is of finite dimension, then the corresponding Weyl grouppoid is finite. (2) If the Weyl grouppoid of a Nichols algebra is finite, the Gelfand-Kirillov dimension of a Nichols algebra is finite. For (1) the converse is true under some circumstances. The converse of (2) has been conjectured to be true. Recently, the topic of finite Gelfand-Kirillov dimensional Nichols algebras has received increased attention. In particular rank 2 Nichols algebras of diagonal type with finite Gelfand-Kirillov dimension over a field of characteristic zero have been classified and were used to also classify finite Gelfand-Kirillov-dimensional Nichols algebras over abelian groups. The goal of this work is to extend this result to any characteristic. Note that there are more braidings yielding a finite root system in positive characteristic, especially there are examples with simple roots a yielding X(a,a) = 1 where X denotes the corresponding bicharacter. Roots of this kind imply infinite Gelfand-Kirillov dimension in characteristic zero. Hence new tools have to be developed generalizing the results for characteristic zero in addition

Classification of finite dimensional simple Lie algebras in prime characteristics

We give a comprehensive survey of the theory of finite dimensional Lie algebras over an algebraically closed field of characteristic p>0 and announce that for p>3 the classification of finite dimensional simple Lie algebras is complete. Any such Lie algebra is up to isomorphism either classical (i.e. comes from characteristic 0) or a filtered Lie algebra of Cartan type or a Melikian algebra of characteristic 5.Comment: Revised version: a list of open problems has been added as suggested by the refere

Towards classification of simple finite dimensional modular Lie superalgebras

A way to construct (conjecturally all) simple finite dimensional modular Lie (super)algebras over algebraically closed fields of characteristic not 2 is offered. In characteristic 2, the method is supposed to give only simple Lie (super)algebras graded by integers and only some of the non-graded ones). The conjecture is backed up with the latest results computationally most difficult of which are obtained with the help of Grozman's software package SuperLie.Comment: 10 page