限制李三系的半单元

研究了限制李三系的半单元的一些重要性质,给出了若干个限制李三系是可换的条件,得到了限制李三系的有环面元基的几个条件,刻划了限制李三系的Frattini p-子系的一些性质.同时,研究了中心为零的所有元素是半单元的限制李三系的一些重要性质.     

Finite quantum groups over abelian groups of prime exponent

We classify pointed finite-dimensional complex Hopf algebras whose group of group-like elements is abelian of prime exponent p, p>17. The Hopf algebras we find are members of a general family of pointed Hopf algebras we construct from Dynkin diagrams. As special cases of our construction we obtain all the Frobenius-Lusztig kernels of semisimple Lie algebras and their parabolic subalgebras. An important step in the classification result is to show that all these Hopf algebras are generated by group-like and skew-primitive elements.     

Invariant Lie Algebras and Lie Algebras with a Small Centroid

A subalgebra of a Lie algebra is said to be invariant if it is invariant under the action of some Cartan subalgebra of that algebra. A known theorem of Melville says that a nilpotent invariant subalgebra of a finite-dimensional semisimple complex Lie algebra has a small centroid. The notion of a Lie algebra with small centroid extends to a class of all finite-dimensional algebras. For finite-dimensional algebras of zero characteristic with semisimple derivations in a sufficiently broad class, their centroid is proved small. As a consequence, it turns out that every invariant subalgebra of a finite-dimensional reductive Lie algebra over an arbitrary definition field of zero characteristic has a small centroid.     

Classical and Exceptional Root Systems and Quantizations

We present a method for quantizing semisimple Lie algebras. In Huru (Russ. Math. [2007]) we defined quantizations of the braided Lie algebra structure on a finite dimensional graded vector space V by quantizations of braided derivations on the exterior algebra of V ^* . We find quantizations of semisimple Lie algebras in this setting using the grading by their roots and shall go through all root systems, classical and exceptional.      

Strongly Flat and PO-Flat S-Posets

Abstract

The main purpose of this paper is to study Lie algebras L such that if a subalgebra U of L has a maximal subalgebra of dimension one then every maximal subalgebra of U has dimension one. Such an L is called lm(0)-algebra. This class of Lie algebras emerges when it is imposed on the lattice of subalgebras of a Lie algebra the condition that every atom is lower modular. We see that the effect of that condition is highly sensitive to the ground field F. If F is algebraically closed, then every Lie algebra is lm(0). By contrast, for every algebraically non-closed field there exist simple Lie algebras which are not lm(0). For the real field, the semisimple lm(0)-algebras are just the Lie algebras whose Killing form is negative-definite. Also, we study when the simple Lie algebras having a maximal subalgebra of codimension one are lm(0), provided that char(F) ≠ 2. Moreover, lm(0)-algebras lead us to consider certain other classes of Lie algebras and the largest ideal of an arbitrary Lie algebra L on which the action of every element of L is split, which might have some interest by themselves.     

ON QUASI-TORAL RESTRICTED LIE ALGEBRAS

§1. IntroductionIn the theory of associative ring there are a number of known su?cient conditions forthe commutativity of an associative ring R, some in the form of a polynomial identity.For instance, it is a well-known result that if every element x of …     

The Frattini p-subsystem of a solvable restricted Lie triple system

As a natural generalization of a restricted Lie algebra, a restricted Lie triple system was defined by Hodge. In this paper, we develop initially the Frattini theory for restricted Lie triple systems, generalize some results of Frattini p-subalgebra for restricted Lie algebras, obtain some properties of the Frattini p-subsystem and give the relationship between Фp（T） and Ф（T） for solvable Lie triple systems.     

On the zeros of polynomials over arbitrary finite dimensional algebras

In [3] it was shown that a polynomial of degree n with coefficients in an associative division algebra, which is d-dimensional over its center, has either infinitely many or at most n^d zeros. In this paper we raise the same question for arbitrary m-ary F-algebras A which are d-dimensional over the algebraically closed field F. Our main result states that in the affine space of m-ary algebras of dimension d there is a non-empty Zariski-open set whose elements A have the following property: in the space of polynomial of precise degree n with coefficients in A there is a non-empty Zariski-open set whose elements have precisely n^d zeros. It is shown that all simple algebras, all semi-simple associative algebras, all semisimple Jordan algebras (char F2), all semi-simple Lie algebras (char F=0), and the generic algebra possess this property.     

Splints of classical root systems

We define and classify splints of root systems of complex semisimple Lie algebras. In a few instances, splints play a role in determining branching rules of a module over a complex semisimple Lie algebra when restricted to a subalgebra. In these particular cases, the set of submodules with respect to the subalgebra themselves may be regarded as the character of an auxiliary Lie algebra which may or may not be another Lie subalgebra.