On classical artinian localizations

ABSTRACT

We present some results about Lie algebras, which can be written as the sum of two subalgebras in two cases: where both subalgebras are simple or both are nilpotent. In the first case we suggest new examples of simple Lie algebras admitting decomposition into the sum of simple subalgebras and give explicit realizations where the existence of such decompositions was established earlier. We single out cases where such decomposition is not possible. We also construct examples of solvable Lie algebras, which are the sums of two nilpotent subalgebras, and the derived length of the sum is greater than the sum of the nilpotent indexes of the summands.     

The structure of algebraic jordan algebras without nonzero nilpotent elements

An algebraic, linear Jordan algebra without nonzero nil-potent elements is proved to be a subdirect sum of prime Jordan algebras each of which has finite capacity or contains simple subalgebras of arbitrary capacity. If in addition the base field has nonzero character-istic or the algebra satisfies a polynomial identity, then each of the summands is determined to be simple of finite capacity. Further, it is proved that algebraic, PI Jordan algebras without nonzero nilpotent elements are locally finite in the sense that any finitely generated subalgebra has finite capacity.     

关于Hom-李代数的结构

Let A be a multiplicative Hom-associative algebra and L a multiplicative Hom-Lie algebra together with surjective twisting maps. We show that if A is a sum of two commutative Hom-associative subalgebras, then the commutator Hom-ideal is nilpotent. Furthermore, we obtain an analogous result for Hom-Lie algebra L extending Kegel＇s Theorem. Finally, we discuss the Hom-Lie ideal structure of a simple Hom-associative algebra A by showing that any non-commutative Hom-Lie ideal of A must contain [A, A].     

Twisted rational functions and series

We study rings of twisted rational functions and twisted Laurent series over simple artinian rings. We determine the centers of such rings and investigate the structure of subalgebras of these rings. We extend to infinite dimensional division rings and to simple artinian rings results proven in a previous paper for finite dimensional division algebras. We investigate “Galois subrings” of rings of fractions.     

Comtrans algebras, Thomas sums, and bilinear forms

A comtrans algebra is said to decompose as the Thomas sum of two subalgebras if it is a direct sum at the module level, and if its algebra structure is obtained from the subalgebras and their mutual interactions as a sum of the corresponding split extensions. In this paper, we investigate Thomas sums of comtrans algebras of bilinear forms. General necessary and sufficient conditions are given for the decomposition of the comtrans algebra of a bilinear form as a Thomas sum. Over rings in which 2 is not a zero divisor, comtrans algebras of symmetric bilinear forms are identified as Thomas summands of algebras of infinitesimal isometries of extended spaces, the complementary Thomas summand being the algebra of infinitesimal isometries of the original space. The corresponding Thomas duals are also identified. These results represent generalizations of earlier results concerning the comtrans algebras of finite-dimensional Euclidean spaces, which were obtained using known properties of symmetric spaces. By contrast, the methods of the current paper involve only the theory of comtrans algebras.Received: 30 March 2004     

THE TRIANGULAR MODEL OF SEMI-SIMPLE L~*-ALGEBRA OF H-S OPERATORS

In this paper,some properties of semi-simple L-algebras are considered.At first,applying Cartan decomposition,the author constructs a family of nilpotentsubalgebras in a semi-simple L-algebra and proves that whole algebra can be spanned bythese subalgebras,their conjugations and Cartan subalgebras.Then,the author proves that every nonzero root vector of semi-simple L-algebra ofH-S operators is a finite rank operator and presents the triangular model of the algebra.Finally,non-Voltera property of the algebra is shown.     

B型和C型Weyl模的一个新重数公式

A monomial basis and a filtration of subalgebras for the universal enveloping algebra U(gl) of a complex simple Lie algebra gl of type Bl and Cl are given, and the decomposition of the Weyl module V (λ) as a U(gl)-module into a direct sum of Weyl modules V (μ)’s as U(gl-1)modules is described. In particular, a new multiplicity formula for the Weyl module V (λ) is obtained in this note.     

Engel Subalgebras of n-Lie Algebras

Engel subalgebras of finite-dimensional n-Lie algebras are shown to have similar properties to those of Lie algebras. Using these, it is shown that an n-Lie algebra, all of whose maximal subalgebras are ideals, is nilpotent. A primitive 2-soluble n-Lie algebra is shown to split over its minimal ideal, and all the complements to its minimal ideal are conjugate. A subalgebra is shown to be a Cartan subalgebra if and only if it is minimal Engel, provided that the field has sufficiently many elements. Cartan subalgebras are shown to have a property analogous to intravariance.     

Lower bounds for algebraic complexity of nilpotent associative algebras

Exact algebraic algorithms for calculating the product of two elements of nilpotent associative algebras over fields of characteristic zero are considered (this is a particular case of simultaneous calculation of several multinomials). The complexity of an algebra in this computational model is defined as the number of nonscalar multiplications of an optimal algorithm. Lower bounds for the tensor rank of nilpotent associative algebras (in terms of dimensions of certain subalgebras) are obtained, which give lower bounds for the algebraic complexity of this class of algebras. Examples of reaching these estimates for different dimensions of nilpotent algebras are presented.     

Computational Experiments with Nilpotent Lie Algebras

Results of computer experiments on the study of properties of generic Lie subalgebras with two generators in the Lie algebra of nilpotent matrices whose order does not exceed 10 are presented. The calculations carried out have made it possible to formulate several statements (so-called virtual theorems) on properties of the Lie subalgebras in question. The dimensions of the lower and upper central series and of the series of commutator subalgebras and the characteristic nilpotency property of the Lie subalgebras constructed here and of generic Lie subalgebras of codimension 1 in these Lie subalgebras are studied.

     

On the endomorphism ring of a module with relative chain conditions

A well-known result of Small states that if M is a noetherian left R-module having endomorphism ring S then any nil subring of S is nilpotent. Fisher [4] dualized this result and showed that if M is left artinian then any nil ideal of S is nilpotent. He gave a bound on the indices of nilpotency of nil subrings of the endomorphism rings of noetherian modules and raised the dual question of whether there are such bounds in the case of artinian modules. He gave an affirmative answer if the module is also assumed to be finitely-generated. Similar affirmative answers for modules with finite homogeneous length were given in [10] and [15]. On the other hand, the nilpotence of certain ideals of the endomorphism rings of modules noetherian relative to a torsion theory has been extensively studied. See [2,6,8,12,15,17]. Jirasko [11] dualized, in some sense, some of the results of [6] to torsion modules satisfying the descending chain conditions with respect to some radical.

In this paper we give a bound of indices of nilpotency on nil subrings of the endomorphism ring of a left R-module which is T-torsionfree with respect to some torsion theory T on R-mod. As a special case, we obtain an affirmative answer to Fisher's question. We also note that our results can be stated in an arbitrary Grothendieck category.     

The work of Vladimir Morozov on Lie algebras

We survey the works of V. V. Morozov on Lie algebras concentrating on the following three results on subalgebras of a semisimple complex Lie algebra: the theorem on a nilpotent element, the triangulizability theorem for solvable subalgebras, and the regularity theorem for nonsemisimple maximal subalgebras.     

Applications of perturbation theory to iterated fibrations

For a class of spaces including simply connected spaces and classifying spaces of nilpotent groups, relatively small differential graded algebras are constructed over commutative rings with 1 which are chain homotopy equivalent to the singular cochain algebra. An application to finitely generated torsion-free nilpotent groups over the integers is given.     

GROUPS OF AUTOMORPHISMS OF CHEVALLEY ALGEBRAS

In the paper we present an extension of Segal theorem on Lie algebra automorphisms of upper triangular nilpotent matrices to the whole class of nilpotent subalgebras generated by positive parts of indecomposable Chevalley algebras.     

A generalization of anti-commutative rings arising from 2-varieties

Let Rbe a nonassociative ring of characteristic not 2 or 3 which satisfies the identities (ab=ba) = (ac+ca)b, a(ac+ca) = b(ac+ca) and a²a = aa². We show that these rings are characterized as associative commutative rings with a type of biadditive mapping. From this characterization we show easily that simple rings are associative-commutative, or anti-commutative. Among the examples given, is a finite dimensional algebra which is solvable but not nilpotent.     

Representations and cohomology for Frobenius-Lusztig kernels

Let U_ζ be the quantum group (Lusztig form) associated to the simple Lie algebra g, with parameter ζ specialized to an ?-th root of unity in a field of characteristic p>0. In this paper we study certain finite-dimensional normal Hopf subalgebras U_ζ(G_r) of U_ζ, called Frobenius-Lusztig kernels, which generalize the Frobenius kernels G_r of an algebraic group G. When r=0, the algebras studied here reduce to the small quantum group introduced by Lusztig. We classify the irreducible U_ζ(G_r)-modules and discuss their characters. We then study the cohomology rings for the Frobenius-Lusztig kernels and for certain nilpotent and Borel subalgebras corresponding to unipotent and Borel subgroups of G. We prove that the cohomology ring for the first Frobenius-Lusztig kernel is finitely-generated when g has type A or D, and that the cohomology rings for the nilpotent and Borel subalgebras are finitely-generated in general.     

Infinite-dimensional diagonalization and semisimplicity

We characterize the diagonalizable subalgebras of End(V), the full ring of linear operators on a vector space V over a field, in a manner that directly generalizes the classical theory of diagonalizable algebras of operators on a finite-dimensional vector space. Our characterizations are formulated in terms of a natural topology (the “finite topology”) on End(V), which reduces to the discrete topology in the case where V is finite-dimensional. We further investigate when two subalgebras of operators can and cannot be simultaneously diagonalized, as well as the closure of the set of diagonalizable operators within End(V). Motivated by the classical link between diagonalizability and semisimplicity, we also give an infinite-dimensional generalization of the Wedderburn–Artin theorem, providing a number of equivalent characterizations of left pseudocompact, Jacoboson semisimple rings that parallel various characterizations of artinian semisimple rings. This theorem unifies a number of related results in the literature, including the structure of linearly compact, Jacobson semsimple rings and cosemisimple coalgebras over a field.