Elements Of Lie Algebra Research Articles

On inner ideals and ad-nilpotent elements of Lie algebras

An inner ideal of a Lie algebra L over a commutative ring k is a k-submodule B of L such that [ B [ B L ] ] ⊆ B [B[BL]] \subseteq B . This paper investigates properties of inner ideals and obtains results relating ad-nilpotent elements and inner ideals. For example, let L be a simple Lie algebra in which D y 2 = 0 D_y^2 = 0 implies y = 0 y = 0 , where D y {D_y} denotes the adjoint mapping determined by y. If L satisfies the descending chain condition on inner ideals and has proper inner ideals, then L contains a subalgebra S = ⟨ e , f , h ⟩ S = \langle e,f,h\rangle , isomorphic to the split 3-dimensional simple Lie algebra, such that D e 3 = D f 3 = 0 D_e^3 = D_f^3 = 0 . Lie algebras having such 3-dimensional subalgebras decompose into the direct sum of two copies of a Jordan algebra, two copies of a special Jordan module, and a Lie subalgebra of transformations of the Jordan algebra and module. The main feature of this decomposition is the correspondence between the Lie and the Jordan structures. In the special case when L is a finite dimensional, simple Lie algebra over an algebraically closed field of characteristic p > 5 p > 5 this decomposition yields: Theorem. L is classical if and only if there is an x ≠ 0 x \ne 0 in L such that D x p − 1 = 0 D_x^{p - 1} = 0 and if D y 2 = 0 D_y^2 = 0 implies y = 0 y = 0 . The proof involves actually constructing a Cartan subalgebra which has 1-dimensional root spaces for nonzero roots and then using the Block axioms.

Spectral resolution of the identity for matrices of elements of a lie algebra

AbstractThis is an application of the characteristic identity satisfied by matrices whose elements are also elements of a semi-simple Lie algebra. Generalized eigenvectors are determined for matrices consisting of generators of GL(n), O(n) and Sp(n), and it is shown how to resolve the identity into idempotents constructed from such eigenvectors. By this means rather general functions of the matrices may be defined. It is also shown how to determine traces of such functions, in terms of the invariants of the Lie algebra.

Restricted Lie algebras and simple associative algebras of characteristic 𝑝

Introduction. Let F be a field of characteristic p, and let C be a finite algebraic extension field of F such that CPCF. It has been shown in [4] that there is a one to one correspondence (up to isomorphisms) between the simple finite dimensional algebras with center F and containing C as a maximal commutative subring and the regular restricted Lie algebra extensions of C by the derivation algebra of C/F. This led to a very simple description of the group of similarity classes of these algebras. The main task of the present paper is to investigate the connection between restricted Lie algebras and simple associative algebras of characteristic p quite generally. For this purpose, we generalize the notion of a regular extension of C by the derivation algebra of C/F to that of a regular extension of a simple algebra A with center C by the derivation algebra of C/F. The correspondence mentioned above is then generalized to a one to one correspondence between these extensions and the simple algebras with center F which contain A as the commutator algebra of C (Theorem 3). We shall then show how every simple algebra containing a purely inseparable extension field of its center as a maximal commutative subring can be built up in a series of steps from regular Lie algebra extensions. With a suitable composition of regular extensions, which we define in ?2, this reduces the structure of the group of algebra classes with a fixed purely inseparable splitting field to the structural elements of restricted Lie algebras, at least in principle. However, it does not yield a direct description of this group, as was the case for splitting fields of exponent one. In ?3 we show that the tensor multiplication with a purely inseparable extension field C of F maps the Brauer group of algebra classes over F onto the Brauer group over C (Theorem 5). Although this result seems not to have been stated before, it may well be regarded as one of the culminating points of Albert's theory of p-algebras. In fact, Chapter VII of [1] contains all the essential points of a proof along classical lines, which is the first proof we give in ?3. Here, the main tools are Galois theory and the theory of cyclic p-extensions. Our second proof is entirely different. It proceeds within the framework of the first part of this paper, replacing the field theory with the theory of Lie algebras. In particular, we use the theory of restricted Lie algebra kernels, [3], in order to show that if A is a simple algebra with center C and CPC F then there always exists a regular extension of A by the derivation algebra of C/F, whence (by Theorem 3) A can be imbedded as the