Nonassociative rings

Abstract

Results in the theory of nonassociative rings and related directions reviewed in RZhMatematika during the period 1972-1978 are discussed. Special attention is given to infinite-dimensional Lie algebras, Jordan algebras, alternative rings, Mal'tsev algebras, and varieties, representations, and radicals of nonassociative rings.Results in the theory of nonassociative rings and related directions reviewed in RZhMatematika during the period 1972-1978 are discussed. Special attention is given to infinite-dimensional Lie algebras, Jordan algebras, alternative rings, Mal'tsev algebras, and varieties, representations, and radicals of nonassociative rings. I. Infinite-DimensionalLie Algebras I. Books. In the Russian language there appeared during this period the successive parts of the book Lie Algebras of the series of Bourbaki [26, 27], the book of Dixmier [55], and the book of Kaplansky [91]. Two books are devoted to the theory of infinite-dimensional Lie algebras: Amayo and Stewart [240] and Bakhturin [257]. The books [365, 400, 716, 718] are written in a more traditional style. Universal enveloping algebras are also discussed in [284].2. Varieties of Lie Algebras. Although the first works properly devoted to varieties of Lie algebras appeared only in 1967 (V. A. Parfenov) and in 1968 (Yu. A. Bakhturin), the machinery and first major results of this theory actually appeared earlier in the works of A. I. Mal'tsev, A. L Kostrikin, and A. I. Shirshov. Moreover, in the hands of these specialists in this theory the theories of varieties of groups and associative algebras were advanced. The latter circumstance led to the situation that at first analogues of results in these theories were sought in varieties of Lie algebras. Thus, theorems were proved on the freeness of a semigroup of varieties of Lie algebras over a field of characteristic 0 [148], on the description of Sehreier varieties over such a field [ii], and a monotonicity theorem for verbal ideals [ii, 146, 49]. It was noted in [14] that the theorems on a semigroup of varieties and Sehreier varieties are also true in the case of an arbitrary infinitedimensional field; it is also shown there that in the case of a field of two elements the multiplication of varieties of Lie algebras is nonassociative. A description of Schreier varieties of Lie algebras, i.e., varieties in which any subalgebra of a free algebra is free, was given by M. V. Zaitsev for the case of any commutative ring: it is shown there that the Sehreier varieties are trivial; e.g., if the base ring is a field these are the varieties D, ~i, @ in standard notation.The problem of a finite basis was solved negatively in [705]. This example, which pertains to fields of characteristic 2, was later extended to the case of any prime characteristic p [67,68]. In [67] an example is also given of a (2p + 3)-dimensional Lie algebra over an infinite field of characteristic p > 0 without a finite basis of identities (in the case p = 2 there is also such an example in [705]). In [707] an example was constructed of an ...

Generalized Schrödinger equations and Jordan pairs

“…It's easy to prove the next statement. It is natural to expect that the algebraic properties of (0.1) such as the existence of symmetries, conservation laws and so on are completely defined through the structure of the corresponding pair of vector space (V, V [6]). Evidently the name "Jordan pair" has been proposed in [7] and was motivated by a close connection with Jordan algebras.…”

“…where a superscript l means the transposition. One can get the examples of Jordan pairs by means of the following construction (see for instance [6]) associated with the graded expansions of Lie algebras. Let G = G : + G 0 + G_ ί be the Lie algebra with [G t , G/] £ G ί+j .…”

A note on the structure of complete alternative local algebras

“…We now outline an introduction to our objects of interest; for further details, see [2], [4], [5], [9]. As we are going to be dealing with potentially non-unital multiplicative structures, we shall use the term algebra to mean a Z-algebra; by contrast, rings are always assumed to be unital.…”

Shirshov composition techniques in Lie superalgebras (noncommutative Gröbner bases)

“…The theory of canonical bases in algebraic systems goes back to the theory of groups and graphs ([1-3]). This method became an important tool in solving algorithmic problems in the theory of Lie algebras (see, for example, [8][9][10]). It gives the possibility of obtaining a series of effective algorithms for symbolic calculations (see, for example, [4]).…”