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1.
The work in this paper is a continuation of an earlier paper of the second author (Acta Math. 34 (1991), 191-202). We discuss the properties of finite-dimensional complete Lie algebras with abelian nilpotent radical over the complex field . We solve the problems of isomorphism, classification and realization of complete Lie algebras with commutative nilpotent radical.
2.
Let n ≥ 3. The complex Lie algebra, which is attached to a unit form q(x 1, x 2,..., x n) = \({\sum\nolimits_{i = 1}^n {x_i^2 + \sum\nolimits_{1 \leqslant i \leqslant j \leqslant n} {\left( { - 1} \right)} } ^{j - i}}{x_i}{x_j}\) and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type A n , and realized by the Ringel-Hall Lie algebra of a Nakayama algebra of radical square zero. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra. 相似文献
3.
4.
A. Jaikin-Zapirain 《Proceedings of the American Mathematical Society》2005,133(10):2817-2820
In this work we solve a conjecture of Y. Barnea and M. Isaacs about centralizer sizes and the nilpotency class in nilpotent finite-dimensional Lie algebras and finite -groups.
5.
We study the exponential growth of the codimensions cn(L) of a finite-dimensional Lie algebra L over a field of characteristic zero. We show that if the solvable radical of L is nilpotent then
exists and is an integer. 相似文献
6.
A locally convex Lie algebra is said to be locally exponential if it belongs to some local Lie group in canonical coordinates. In this note we give criteria
for locally exponential Lie algebras of vector fields on an infinite-dimensional manifold to integrate to global Lie group
actions. Moreover, we show that all necessary conditions are satisfied if the manifold is finite-dimensional connected and
σ-compact, which leads to a generalization of Palais’ Integrability Theorem.
相似文献
7.
Infinite-dimensional Lie algebras of generalized Block type 总被引:2,自引:0,他引:2
J. Marshall Osborn Kaiming Zhao 《Proceedings of the American Mathematical Society》1999,127(6):1641-1650
This paper investigates a class of infinite-dimensional Lie algebras over a field of characteristic which are called here Lie algebras of generalized Block type, and which genereralize a class of Lie algebras originally defined by Richard Block. Under certain natural restrictions, this class of Lie algebras is shown to break into five subclasses. One of these subclasses contains all generalized Cartan type Lie algebras and some Lie algebras of generalized Cartan type , and a second one is the class of Lie algebras of type , which were previously defined and studied elsewhere by the authors. The other three types are hybrids of the first two types, and are new.
8.
A. Dosiev 《Journal of Mathematical Sciences》2004,124(2):4886-4908
Topological algebras of (convergent) power series of elements of a Lie algebra are introduced and the existence of continuous homomorphisms of these algebras into an operator algebra is studied. For the Slodkowski spectra, the spectral mapping theorem
is proved for generators a of a finite-dimensional nilpotent Lie algebra of bounded linear operators under the condition that a family f of elements of a power series algebra is finite-dimensional. Bibliography: 22 titles. 相似文献
9.
Shun-Jen Cheng Weiqiang Wang R. B. Zhang 《Transactions of the American Mathematical Society》2008,360(11):5883-5924
We establish a direct connection between the representation theories of Lie algebras and Lie superalgebras (of type ) via Fock space reformulations of their Kazhdan-Lusztig theories. As a consequence, the characters of finite-dimensional irreducible modules of the general linear Lie superalgebra are computed by the usual parabolic Kazhdan-Lusztig polynomials of type . In addition, we establish closed formulas for canonical and dual canonical bases for the tensor product of any two fundamental representations of type .
10.
We study relations between finite-dimensional representations of color Lie algebras and their cocycle twists. Main tools are the universal enveloping algebras and their FCR-properties (finite-dimensional representations are completely reducible.) Cocycle twist preserves the FCR-property. As an application, we compute all finite dimensional representations (up to isomorphism) of the color Lie algebra .Presented by A. Verschoren. 相似文献
11.
T. Skrypnyk 《Acta Appl Math》2007,99(3):261-282
We construct a family of special quasigraded Lie algebras
of functions of one complex variables with values in finite-dimensional Lie algebra
, labeled by the special 2-cocycles F on
. The main property of the constructed Lie algebras
is that they admit Kostant-Adler-Symes scheme. Using them we obtain new integrable finite-dimensional Hamiltonian systems
and new hierarchies of soliton equations. 相似文献
12.
For each even lattice \({\mathcal L}\), there is a canonical way to construct an infinite-dimensional Lie algebra via lattice vertex operator algebra theory, we call this Lie algebra and its subalgebras the Borcherds type Lie algebras associated to \({\mathcal L}\). In this paper, we apply this construction to even lattices arising from representation theory of finite-dimensional associative algebras. This is motivated by the different realizations of Kac-Moody algebras by Borcherds using lattice vertex operators and by Peng-Xiao using Ringel-Hall algebras respectively. For any finite-dimensional algebra \(A\) of finite global dimension, we associate a Borcherds type Lie algebra \(\mathfrak {BL}(A)\) to \(A\). In contrast to the Ringel-Hall Lie algebra approach, \(\mathfrak {BL}(A)\) only depends on the symmetric Euler form or Tits form but not the full representation theory of \(A\). However, our results show that for certain classes of finite-dimensional algebras whose representation theory is ’controlled’ by the Euler bilinear forms or Tits forms, their Borcherds type Lie algebras do have close relations with the representation theory of these algebras. Beyond the class of hereditary algebras, these algebras include canonical algebras, representation-directed algebras and incidence algebras of finite prinjective types. 相似文献
13.
David A. Towers 《Algebras and Representation Theory》2017,20(3):735-750
The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call minimal non- \({\mathcal N}\). To facilitate this we investigate solvable Lie algebras of nilpotent length k, and of nilpotent length ≤k, and extreme Lie algebras, which have the property that their nilpotent length is equal to the number of conjugacy classes of maximal subalgebras. We characterise the minimal non-\({\mathcal N}\) Lie algebras in which every nilpotent subalgebra is abelian, and those of solvability index ≤3. 相似文献
14.
For any grading by an abelian group G on the exceptional simple Lie algebra \(\mathcal {L}\) of type E 6 or E 7 over an algebraically closed field of characteristic zero, we compute the graded Brauer invariants of simple finite-dimensional modules, thus completing the computation of these invariants for simple finite-dimensional Lie algebras. This yields the classification of finite-dimensional G-graded simple \(\mathcal {L}\)-modules, as well as necessary and sufficient conditions for a finite-dimensional \(\mathcal {L}\)-module to admit a G-grading compatible with the given G-grading on \(\mathcal {L}\). 相似文献
15.
D. V. Alekseevskii 《Journal of Mathematical Sciences》1985,28(6):924-949
The survey deals with the investigations reviewed inReferativnyi Zhurnal Matematika between 1977–1981. In the survey there are reflected the investigations on the structure of Lie groups and Lie algebras, on their finite-dimensional linear representations and universal enveloping algebras, on the theory of invariants and Lie groups of transformations, and also on continuous and discrete subgroups of Lie groups.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 20, No. 153–192, 1982. 相似文献
16.
Helge Glöckner 《Journal of Functional Analysis》2007,245(1):19-61
Let G be a Lie group which is the union of an ascending sequence G1⊆G2⊆? of Lie groups (all of which may be infinite-dimensional). We study the question when in the category of Lie groups, topological groups, smooth manifolds, respectively, topological spaces. Full answers are obtained for G the group Diffc(M) of compactly supported C∞-diffeomorphisms of a σ-compact smooth manifold M; and for test function groups of compactly supported smooth maps with values in a finite-dimensional Lie group H. We also discuss the cases where G is a direct limit of unit groups of Banach algebras, a Lie group of germs of Lie group-valued analytic maps, or a weak direct product of Lie groups. 相似文献
17.
Dimension formula for graded Lie algebras and its applications 总被引:1,自引:0,他引:1
Seok-Jin Kang Myung-Hwan Kim 《Transactions of the American Mathematical Society》1999,351(11):4281-4336
In this paper, we investigate the structure of infinite dimensional Lie algebras graded by a countable abelian semigroup satisfying a certain finiteness condition. The Euler-Poincaré principle yields the denominator identities for the -graded Lie algebras, from which we derive a dimension formula for the homogeneous subspaces . Our dimension formula enables us to study the structure of the -graded Lie algebras in a unified way. We will discuss some interesting applications of our dimension formula to the various classes of graded Lie algebras such as free Lie algebras, Kac-Moody algebras, and generalized Kac-Moody algebras. We will also discuss the relation of graded Lie algebras and the product identities for formal power series.
18.
Ivan Dimitrov Olivier Mathieu Ivan Penkov 《Transactions of the American Mathematical Society》2000,352(6):2857-2869
Given any simple Lie superalgebra , we investigate the structure of an arbitrary simple weight -module. We introduce two invariants of simple weight modules: the shadow and the small Weyl group. Generalizing results of Fernando and Futorny we show that any simple module is obtained by parabolic induction from a cuspidal module of a Levi subsuperalgebra. Then we classify the cuspidal Levi subsuperalgebras of all simple classical Lie superalgebras and of the Lie superalgebra W. Most of them are simply Levi subalgebras of , in which case the classification of all finite cuspidal representations has recently been carried out by one of us (Mathieu). Our results reduce the classification of the finite simple weight modules over all classical simple Lie superalgebras to classifying the finite cuspidal modules over certain Lie superalgebras which we list explicitly.
19.
This paper examines the problem of classifying finite-dimensional Lie algebras over the field C with a given radical \(\mathfrak{r}\) and also the problem of classifying algebraic Lie algebras with a given nilpotent radical \(\mathfrak{r}\) . A detailed study is made of the case when \(\mathfrak{r}\) is the nilpotent radical of a parabolic subalgebra of a semisimple Lie algebra. 相似文献