We would like to know how many essentially different (that is, non-isomorphic) Lie algebras there are and what approaches we can use to classify them. To get some feeling for …
We would like to know how many essentially different (that is, non-isomorphic) Lie algebras there are and what approaches we can use to classify them. To get some feeling for these questions, we shall look at Lie algebras of dimensions 1, 2, and 3. Another reason for looking at these low-dimensional Lie algebras is that they often occur as subalgebras of the larger Lie algebras we shall meet later. Abelian Lie algebras are easily understood: For any natural number n, there is an abelian Lie algebra of dimension n (where for any two elements, the Lie bracket is zero). We saw in Exercise 1.11 that any two abelian Lie algebras of the same dimension over the same field are isomorphic, so we understand them completely, and from now on we shall only consider non-abelian Lie algebras. How can we get going? We know that Lie algebras of different dimensions cannot be isomorphic. Moreover, if L is a non-abelian Lie algebra, then its derived algebra L′ is non-zero and its centre Z(L) is a proper ideal. By Exercise 2.8, derived algebras and centres are preserved under isomorphism, so it seems reasonable to use the dimension of L and properties of L′ and Z(L) as criteria to organise our search.
Abstract It was recently demonstrated that such classical concepts as eigenproblems and Lie algebras are closely related. A family of appropriate eigenproblems generates any finite-dimensional Lie algebra and, conversely, a …
Abstract It was recently demonstrated that such classical concepts as eigenproblems and Lie algebras are closely related. A family of appropriate eigenproblems generates any finite-dimensional Lie algebra and, conversely, a Lie algebra generates a set of pairs ( F, v ), where F is a linear mapping and v its eigenvector. This point of view can be used to describe the invariants of Lie algebras. The application of this procedure is shown in detail on the example of five-dimensional nilpotent Lie algebras.
Nowadays infinite-dimensional Lie theory is a core area of modern mathematics, covering a broad range of branches, such as the structure and classification theory of infinite-dimensional Lie algebras, geometry of …
Nowadays infinite-dimensional Lie theory is a core area of modern mathematics, covering a broad range of branches, such as the structure and classification theory of infinite-dimensional Lie algebras, geometry of infinite-dimensional Lie groups and their homogeneous spaces, and representation theory of infinite-dimensional Lie groups, Lie algebras and Lie-superalgebras. The focus of this workshop was on recent developments in all of these areas with particular emphasis on connections with other branches of mathematics, such as algebraic groups and Galois cohomology. The meeting was attended by 52 participants from many European countries, Canada, the USA, Brazil, Japan and Australia. The meeting was organized around a series of 23 lectures each of 50 minutes duration representing the major recent advances in the area. We feel that the meeting was exciting and highly successful. The quality of the lectures, several of which surveyed recent developments, was outstanding. The exceptional atmosphere of the Oberwolfach Institute provided an optimal environment for bringing people working in algebraically, geometrically or analytically oriented areas of infinite-dimensional Lie theory together, and to create an atmosphere of scientific interaction and cross-fertilization. Without going too much into detail, let us mention some important new developments that were showcased during the workshop. In the structure theory of infinite-dimensional Lie algebras, the classification of extended affine Lie algebras, based on the notion of a Lie torus has now reached a mature state (Neher). In the classification theory of infinite-dimensional Lie algebras, several deep results were obtained recently with Galois cohomology methods exhibiting exciting connections between forms of multiloop algebras and the Galois theory of forms of algebras over rings (Allison, Gille, Chernousov). This branch of structure theory is complemented by the connection between the classification of generalized Kac–Moody algebras and automorphic forms (Scheithauer). In the representation theory of infinite-dimensional Lie algebras, the most exciting new developments concern various kinds of categories of representations of current algebras and Kac–Moody–Lie (super-)algebras (Benkart, Chari, Futorny, Gorelik, Kumar, Littelmann, Serganova). Another interesting, recently very active direction of Kac–Moody theory are Kac–Moody groups over finite fields, which leads to a new class of infinite simple groups (Caprace). On geometric and analytic Lie theory, we had exiting talks on new geometric directions in the representation theory of Banach–Lie groups, related to Banach–Lie–Poisson spaces (Ratiu), and applications of heat kernel measures in the representation theory of loop groups (Pickrell). On the opposite side of the spectrum of Lie group theory, namely direct limit theory, crucial progress has been made on direct limits of unitary representations, as well as in the context of direct limits of infinite-dimensional groups (Wolf, Glöckner). We further had several contributions dealing with geometric aspects such as Chern forms, gerbes and generalized projective geometries (Paycha, Schweigert, Bertram). Finally, we had several exciting talks about several more particular results, dealing with vertex operator algebras, polyzeta values and quantization (Billig, Mathieu, Omori). More specific information is contained in the abstracts which follow in this volume.
The workshop focussed on recent developments in infinite-dimen- sional Lie theory. The talks covered a broad range of topics, such as structure and classification theory of infinite-dimensional Lie algebras, geometry …
The workshop focussed on recent developments in infinite-dimen- sional Lie theory. The talks covered a broad range of topics, such as structure and classification theory of infinite-dimensional Lie algebras, geometry of in- finite-dimensional Lie groups and homogeneous spaces and representation theory of infinite-dimensional Lie groups, Lie algebras and Lie-superalgebras.
Preliminaries on Affine Lie Algebras Characters of Integrable Representations Principal Admissible Weights Residue of Principal Admissible Characters Characters of Affine Orbifolds Operator Calculus Branching Functions W-Algebra Vertex Representations for Affine …
Preliminaries on Affine Lie Algebras Characters of Integrable Representations Principal Admissible Weights Residue of Principal Admissible Characters Characters of Affine Orbifolds Operator Calculus Branching Functions W-Algebra Vertex Representations for Affine Lie Algebras Soliton Equations.
This is the third, substantially revised edition of this important monograph. The book is concerned with Kac–Moody algebras, a particular class of infinite-dimensional Lie algebras, and their representations. It is …
This is the third, substantially revised edition of this important monograph. The book is concerned with Kac–Moody algebras, a particular class of infinite-dimensional Lie algebras, and their representations. It is based on courses given over a number of years at MIT and in Paris, and is sufficiently self-contained and detailed to be used for graduate courses. Each chapter begins with a motivating discussion and ends with a collection of exercises, with hints to the more challenging problems.
The aim of this paper is to demonstrate how a fairly simple nilpotent Lie algebra can be used as a tool to study differential operators on $ℝ^n$ with polynomial coefficients, …
The aim of this paper is to demonstrate how a fairly simple nilpotent Lie algebra can be used as a tool to study differential operators on $ℝ^n$ with polynomial coefficients, especially when the property studied depends only on the degree of the polynomia
An approach for generating Lie algebras is presented by establishing a finite—dimensional Lie algebra.The isospectral Lax problems are given by using the corresponding loop algebra.The compati- bility of the isospectral …
An approach for generating Lie algebras is presented by establishing a finite—dimensional Lie algebra.The isospectral Lax problems are given by using the corresponding loop algebra.The compati- bility of the isospectral problems derives a soliton hierarchy of equations,whose Hamiltonian structure is obtained by use of the quadratic—form identity.
A geometric programme to analyse the structure of Lie algebras is presented with special emphasis on the geometry of linear Poisson tensors. The notion of decomposable Poisson tensors is introduced …
A geometric programme to analyse the structure of Lie algebras is presented with special emphasis on the geometry of linear Poisson tensors. The notion of decomposable Poisson tensors is introduced and an algorithm to construct all solvable Lie algebras is presented. Poisson-Liouville structures are also introduced to discuss a new class of Lie algebras which include, as a subclass, semi-simple Lie algebras. A decomposition theorem for Poisson tensors is proved for a class of Poisson manifolds including linear ones. Simple Lie algebras are also discussed from this viewpoint and lower-dimensional real Lie algebras are analysed.
In this paper we describe a simple method for obtaining a classification of small-dimensional solvable Lie algebras. Using this method, we obtain the classification of three- and fourdimensional solvable Lie …
In this paper we describe a simple method for obtaining a classification of small-dimensional solvable Lie algebras. Using this method, we obtain the classification of three- and fourdimensional solvable Lie algebras (over fields of any characteristic). Precise conditions for isomorphism are given.
The problem of Lie algebras’ classification, in their different varieties, has been dealt with by theory researchers since the early 20th century. This problem has an intrinsically infinite nature since …
The problem of Lie algebras’ classification, in their different varieties, has been dealt with by theory researchers since the early 20th century. This problem has an intrinsically infinite nature since it can be inferred from the results obtained that there are features specific to each field and dimension. Despite the hundreds of attempts published, there are currently fields and dimensions in which only partial classifications of some families of algebras of low dimensions have been obtained. This article intends to bring some order to the achievements of this prolific line of research so far, in order to facilitate future research.
We present a new look at the description of real finite-dimensional Lie algebras. The basic ingredient is a pair (F,v) consisting of a linear mapping F∈End(V) with an eigenvector v. …
We present a new look at the description of real finite-dimensional Lie algebras. The basic ingredient is a pair (F,v) consisting of a linear mapping F∈End(V) with an eigenvector v. This pair allows to build a Lie bracket on a dual space to a linear space V. The Lie algebra obtained in this way is solvable. In particular, when F is nilpotent, the Lie algebra is actually nilpotent. We show that these solvable algebras are the basic bricks of the construction of all other Lie algebras. Using relations between the Lie algebra, the Lie–Poisson structure and the Nambu bracket, we show that the algebra invariants (Casimir functions) are solutions of an equation which has an interesting geometric significance. Several examples illustrate the importance of these constructions.
Let k be a field of any characteristic, V a finite-dimensional vector space over k, and Sd(V*) be the d-th symmetric power of the dual space V*. Given a linear …
Let k be a field of any characteristic, V a finite-dimensional vector space over k, and Sd(V*) be the d-th symmetric power of the dual space V*. Given a linear map φ on V and an eigenvector w of φ, we prove that the pair (φ,w) can be used to construct a new Lie algebra structure on Sd(V*). We prove that this Lie algebra structure is solvable, and in particular, it is nilpotent if φ is a nilpotent map. We also classify the Lie algebras for all possible pairs (φ,w), when k=C and V is two-dimensional.
Abstract Working over an arbitrary field of characteristic different from 2, we extend the Skjelbred-Sund method to compatible Lie algebras and give a full classification of nilpotent compatible Lie algebras …
Abstract Working over an arbitrary field of characteristic different from 2, we extend the Skjelbred-Sund method to compatible Lie algebras and give a full classification of nilpotent compatible Lie algebras up to dimension 4. In case the base field is cubically closed, we find that there are three isomorphism classes and a one-parameter family in dimension 3, and 12 isomorphism classes, 6 one-parameter families and one 2-parameter family in dimension 4.