This chapter is concerned with finite dimensional Lie algebras. A large number of specialised functions are provided for these algebras. We refer to [dG00] for a general introduction to the theory of Lie algebras and their algorithms. For some of the functions described here that rely on a non-trivial algorithm we will indicate a precise reference.
Lie algebras are viewed as free modules over a base ring R with a multiplication satisfying the usual Lie axioms. Some functions require additional conditions on the base ring; for example, many functions require that the base ring be a field.
The main computational machinery in Magma for Lie algebras assumes that they are given either as structure constant algebras or as matrix algebras. Functions are provided which, given a finitely presented finite dimensional Lie algebra, will attempt to construct an isomorphic structure constant Lie algebra. As a structure constant algebra, the Lie algebra L of dimension n over a ring R is defined in Magma by giving the n3 structure constants aijk ∈R (1 ≤i, j, k ≤n) such that, if {e1, e2, ..., en} is the basis of L, ei * ej = ∑k = 1n aijk * ek.
In Magma there is more functionality for Lie algebras defined by structure constants than for matrix Lie algebras. Throughout this chapter the algebra representation appropriate for a given intrinsic will be noted. For information on matrix algebras considered as associative algebras see Chapter MATRIX ALGEBRAS.
In addition, some functions are provided for finitely presented Lie algebras and Lie algebras generated by extremal elements, and databases of solvable Lie algebras, nilpotent Lie algebras, and nilpotent orbits in simple Lie algebras are available.
As mentioned above, the most extensively supported Lie algebras in Magma are structure constant Lie algebras and matrix Lie algebras. The methods for constructing these (by explicitly specifying structure constants or matrices) are described in Sections Constructors for Lie Algebras.
Well known simple finite Lie algebras can be more easily constructed by specifying the type. The classical, reductive, Lie algebras (An, Bn, Cn, Dn, E6, E7, E8, F4, G2) and their twisted variants are described in Section Almost Reductive Lie Algebras, the Witt Lie algebras and its derivatives (of Cartan-Type) in Section Cartan-Type Lie Algebras, and the Melikian Lie algebras in Section Melikian Lie Algebras. Those interested primarily in classical Lie algebras may want to skip to Section Almost Reductive Lie Algebras, which includes a number of examples to get started.
Elementary properties of these Lie algebras (bases, types, Weyl group, etc.) are described in Section Operations on Lie Algebras. This section also contains information about isomorphism testing. Other properties (nilpotency, simplicity) are discussed in Section Properties of Lie Algebras and Ideals. Some further operations that only apply to matrix Lie algebras may be found in Section Operations for Matrix Lie Algebras.
Constructors such as direct sums, subalgebras, centralisers, Cartan subalgebras, derived series, etc. are treated in Sections Construction of Subalgebras, Ideals and Quotients and Operations on Subalgebras and Ideals; construction of homomorphisms can be found in Section Homomorphisms. The construction of elements of Lie algebras is described in Section Construction of Elements, and operations on them in Section Operations on Elements.
In addition to these Lie algebras Magma supports two other types of Lie algebras. Firstly finitely presented Lie algebras, as free Lie algebras modulo a set of relations, described in Section Finitely Presented Lie Algebras. Secondly Lie algebras generated by extremal elements, described in Section Lie Algebras Generated by Extremal Elements.
More specialistic functions for structure constant Lie algebras are described in Sections The Natural Module (the natural module), Automorphisms of Classical- type Reductive Algebras (automorphisms of classical-type Lie algebras), Restrictable Lie Algebras (restrictable Lie algebras), and Universal Enveloping Algebras (universal enveloping algebras).
Magma also provides databases and recognition procedures for small-dimensional solvable and nilpotent Lie algebras (described in Section Solvable and Nilpotent Lie Algebras Classification), a database of semisimple subalgebras of simple Lie algebras (described in Section Semisimple Subalgebras of Simple Lie Algebras), and a database of nilpotent orbits in simple Lie algebras (described in Section Nilpotent Orbits in Simple Lie Algebras).
Other chapters that may be of interest are Chapter KAC-MOODY LIE ALGEBRAS on Kac-Moody Lie algebras and Chapter QUANTUM GROUPS on Quantum Groups. Furthermore, Chapter REPRESENTATIONS OF LIE GROUPS AND ALGEBRAS deals with representations of Lie algebras and groups of Lie type. Of particular importance is Section Lie Algebras, dealing with the construction of representations of Lie algebras.