共查询到20条相似文献,搜索用时 31 毫秒
1.
Complemented Lie algebras are introduced in this paper (a notion similar to that studied by O. Loos and E. Neher in Jordan pairs). We prove that a Lie algebra is complemented if and only if it is a direct sum of simple nondegenerate Artinian Lie algebras. Moreover, we classify simple nondegenerate Artinian Lie algebras over a field of characteristic 0 or greater than 7, and describe the Lie inner ideal structure of simple Lie algebras arising from simple associative algebras with nonzero socle. 相似文献
2.
Cristina Draper Antonio Fernández López Esther García Miguel Gómez Lozano 《Journal of Algebra》2008,319(6):2372-2394
We define the socle of a nondegenerate Lie algebra as the sum of all its minimal inner ideals. The socle turns out to be an ideal which is a direct sum of simple ideals, and satisfies the descending chain condition on principal inner ideals. Every classical finite dimensional Lie algebra coincides with its socle, while relevant examples of infinite dimensional Lie algebras with nonzero socle are the simple finitary Lie algebras and the classical Banach Lie algebras of compact operators on an infinite dimensional Hilbert space. This notion of socle for Lie algebras is compatible with the previous ones for associative algebras and Jordan systems. We conclude with a structure theorem for simple nondegenerate Lie algebras containing abelian minimal inner ideals, and as a consequence we obtain that a simple Lie algebra over an algebraically closed field of characteristic 0 is finitary if and only if it is nondegenerate and contains a rank-one element. 相似文献
3.
This paper aims at developing a “local-global” approach for various types of finite dimensional algebras, especially those related to Hecke algebras. The eventual intention is to apply the methods and applications developed here to the cross-characteristic representation theory of finite groups of Lie type. We first review the notions of quasi-hereditary and stratified algebras over a Noetherian commutative ring. We prove that many global properties of these algebras hold if and only if they hold locally at every prime ideal. When the commutative ring is sufficiently good, it is often sufficient to check just the prime ideals of height at most one. These methods are applied to construct certain generalized q-Schur algebras, proving they are often quasi-hereditary (the “good” prime case) but always stratified. Finally, these results are used to prove a triangular decomposition matrix theorem for the modular representations of Hecke algebras at good primes. In the bad prime case, the generalized q-Schur algebras are at least stratified, and a block triangular analogue of the good prime case is proved, where the blocks correspond to Kazhdan-Lusztig cells. 相似文献
4.
M.P. Benito 《代数通讯》2013,41(7):2529-2545
Relationships between the structure of a Lie algebra and that of its lattice of ideals is studied for those Lie algebras whose ideal lattice is very close to that of an almost-abelian Lie algebra. It is shown here that if the base field is algebraically closed, finite or the real one, for any n ≥3 the only solvable Lie algebra whose lattice of ideals is isomorphic to that of the (n+l)-dimensional almost-abelian Lie algebra is itself. 相似文献
5.
José A. Anquela Teresa Cortés Esther García Miguel Gómez Lozano 《Mediterranean Journal of Mathematics》2016,13(1):29-52
We show that, unlike alternative algebras, prime quotients of a nondegenerate Jordan system or a Lie algebra need not be nondegenerate, even if the original Jordan system is primitive, or the Lie algebra is strongly prime, both with nonzero simple hearts. Nevertheless, for Jordan systems and Lie algebras directly linked to associative systems, we prove that even semiprime quotients are necessarily nondegenerate. 相似文献
6.
Dietrich Burde 《manuscripta mathematica》1998,95(1):397-411
Left-symmetric algebras (LSAs) are Lie admissible algebras arising from geometry. The leftinvariant affine structures on a
Lie groupG correspond bijectively to LSA-structures on its Lie algebra. Moreover if a Lie group acts simply transitively as affine transformations
on a vector space, then its Lie algebra admits a complete LSA-structure. In this paper we studysimple LSAs having only trivial two-sided ideals. Some natural examples and deformations are presented. We classify simple LSAs
in low dimensions and prove results about the Lie algebra of simple LSAs using a canonical root space decomposition. A special
class of complete LSAs is studied. 相似文献
7.
Xiaoping Xu 《Journal of Pure and Applied Algebra》2008,212(6):1253-1309
We construct irreducible modules of centrally-extended classical Lie algebras over left ideals of the algebra of differential operators on the circle, through certain irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries. The structures of vertex algebras associated with the vacuum representations of these algebras are determined. Moreover, we prove that under certain conditions, the highest-weight irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries naturally give rise to the irreducible modules of the simple quotients of these vertex algebras. From vertex algebra and its representation point of view, our results with positive integral central charge are high-order differential operator analogues of the well-known WZW models in conformal field theory associated with affine Kac-Moody algebras. Indeed, when the left ideals are the algebra of differential operators, our Lie algebras do contain affine Kac-Moody algebras as subalgebras and our results restricted on them are exactly the representation contents in WZW models. Similar results with negative central charge are also obtained. 相似文献
8.
Dietrich Burde 《manuscripta mathematica》1998,95(3):397-411
Left-symmetric algebras (LSAs) are Lie admissible algebras arising from geometry. The leftinvariant affine structures on a
Lie group {G} correspond bijectively to LSA-structures on its Lie algebra. Moreover if a Lie group acts simply transitively as affine
transformations on a vector space, then its Lie algebra admits a complete LSA-structure. In this paper we study simple LSAs having only trivial two-sided ideals. Some natural examples and deformations are presented. We classify simple LSAs
in low dimensions and prove results about the Lie algebra of simple LSAs using a canonical root space decomposition. A special
class of complete LSAs is studied.
Received: 10 June 1997 / Revised version: 29 September 1997 相似文献
9.
M. Boucetta 《Differential Geometry and its Applications》2004,20(3):279-291
In a previous paper (C. R. Acad. Sci. Paris Sér. I 333 (2001) 763–768), the author introduced a notion of compatibility between a Poisson structure and a pseudo-Riemannian metric. In this paper, we introduce a new class of Lie algebras called pseudo-Riemannian Lie algebras. The two notions are closely related: we prove that the dual of a Lie algebra endowed with its canonical linear Poisson structure carries a compatible pseudo-Riemannian metric if and only if the Lie algebra is a pseudo-Riemannian Lie algebra. Moreover, the Lie algebra obtained by linearizing at a point a Poisson manifold with a compatible pseudo-Riemannian metric is a pseudo-Riemannian Lie algebra. We also give some properties of the symplectic leaves of such manifolds, and we prove that every Poisson manifold with a compatible Riemannian metric is unimodular. Finally, we study Poisson Lie groups endowed with a compatible pseudo-Riemannian metric, and we give the classification of all pseudo-Riemannian Lie algebras of dimension 2 and 3. 相似文献
10.
We study codimension growth of infinite dimensional Lie algebras over a field of characteristic zero. We prove that if a Lie algebra L is an extension of a nilpotent algebra by a finite dimensional semisimple algebra then the PI-exponent of L exists and is a positive integer. 相似文献
11.
Alexandros Patsourakos 《代数通讯》2013,41(3):927-932
The free Lie, right, and left Leibniz algebras are obtained as quotients of the free nonassociative algebra by suitable ideals. In this article, we prove some remarkable properties of these ideals. 相似文献
12.
Askar Dzhumadil’daev 《代数通讯》2018,46(1):129-142
We prove that assosymmetric algebras under the Jordan product are Lie triple algebras. A Lie triple algebra is called special if it is isomorphic to a subalgebra of the plus-algebra of some assosymmetric algebra. We establish that the Glennie identity of degree 8 is valid for special Lie triple algebras, but not for all Lie triple algebras. 相似文献
13.
Tubular algebras and affine Kac-Moody algebras 总被引:1,自引:0,他引:1
Zheng-xin CHEN & Ya-nan LIN School of Mathematics Computer Science Pujian Normal University Fuzhou China School of Mathematical Sciences Xiamen University Xiamen China 《中国科学A辑(英文版)》2007,50(4):521-532
The purpose of this paper is to construct quotient algebras L(A)1C/I(A) of complex degenerate composition Lie algebras L(A)1C by some ideals, where L(A)1C is defined via Hall algebras of tubular algebras A, and to prove that the quotient algebras L(A)1C/I(A) are isomorphic to the corresponding affine Kac-Moody algebras. Moreover, it is shown that the Lie algebra Lre(A)1C generated by A-modules with a real root coincides with the degenerate composition Lie algebra L(A)1C generated by simple A-modules. 相似文献
14.
Má tyá s Domokos Vesselin Drensky 《Transactions of the American Mathematical Society》1998,350(7):2797-2811
Nagata gave a fundamental sufficient condition on group actions on finitely generated commutative algebras for finite generation of the subalgebra of invariants. In this paper we consider groups acting on noncommutative algebras over a field of characteristic zero. We characterize all the T-ideals of the free associative algebra such that the algebra of invariants in the corresponding relatively free algebra is finitely generated for any group action from the class of Nagata. In particular, in the case of unitary algebras this condition is equivalent to the nilpotency of the algebra in Lie sense. As a consequence we extend the Hilbert-Nagata theorem on finite generation of the algebra of invariants to any finitely generated associative algebra which is Lie nilpotent. We also prove that the Hilbert series of the algebra of invariants of a group acting on a relatively free algebra with a non-matrix polynomial identity is rational, if the action satisfies the condition of Nagata.
15.
S. V. Pchelintsev 《Siberian Mathematical Journal》2018,59(2):341-356
We study the overalgebras and the ideals of the Jordan algebras possessing prime (?1, 1)-envelopings. If a Jordan algebra possesses a prime nonassociative (?1, 1)-enveloping then we prove that it is also prime; furthermore, its every ideal is a prime algebra. In particular, the overalgebras and metaideals of Jordan monsters are prime. 相似文献
16.
L. Foissy 《Journal of Pure and Applied Algebra》2012,216(2):480-494
We first prove that a graded, connected, free and cofree Hopf algebra is always self-dual. Then, we prove that two graded, connected, free and cofree Hopf algebras are isomorphic if and only if they have the same Poincaré–Hilbert formal series. If the characteristic of the base field is zero, we prove that the Lie algebra of the primitive elements of such an object is free, and we deduce a characterization of the formal series of free and cofree Hopf algebras by a condition of growth of the coefficients. We finally show that two graded, connected, free and cofree Hopf algebras are isomorphic as (nongraded) Hopf algebras if and only if the Lie algebras of their primitive elements have the same number of generators. 相似文献
17.
Novikov algebras and Novikov structures on Lie algebras 总被引:1,自引:0,他引:1
We study ideals of Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We present the first example of a three-step nilpotent Lie algebra which does not admit a Novikov structure. On the other hand we show that any free three-step nilpotent Lie algebra admits a Novikov structure. We study the existence question also for Lie algebras of triangular matrices. Finally we show that there are families of Lie algebras of arbitrary high solvability class which admit Novikov structures. 相似文献
18.
Let A be an algebra whose multiplication algebra M(A) is semiprime. We prove that, except in an exceptional case, the proper closed prime ideals of A are the maximal closed ideals of A, for the closure operations π and ?. In fact, these sets agree for both closures. The same can be said in M(A) for the closure operations π and ?′. Moreover, we establish the relationships between the proper closed prime ideals of A and the ones of the algebras M(A),U and A/U, for a given ideal U of A. 相似文献
19.
Quasi-hereditary algebras can be viewed as a Lie theory approach to the theory of finite dimensional algebras. Motivated by the existence of certain nice bases for representations of semisimple Lie algebras and algebraic groups, we will construct in this paper nice bases for (split) quasi-hereditary algebras and characterize them using these bases. We first introduce the notion of a standardly based algebra, which is a generalized version of a cellular algebra introduced by Graham and Lehrer, and discuss their representation theory. The main result is that an algebra over a commutative local noetherian ring with finite rank is split quasi-hereditary if and only if it is standardly full-based. As an application, we will give an elementary proof of the fact that split symmetric algebras are not quasi-hereditary unless they are semisimple. Finally, some relations between standardly based algebras and cellular algebras are also discussed. 相似文献
20.
Lie antialgebras is a class of supercommutative algebras recently appeared in symplectic geometry. We define the notion of
enveloping algebra of a Lie antialgebra and study its properties. We show that every Lie antialgebra is canonically related
to a Lie superalgebra and prove that its enveloping algebra is a quotient of the enveloping algebra of the corresponding Lie
superalgebra. 相似文献