| Abstract: | 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. |