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We introduce a general approach to the study of left-invariant K-contact structures on Lie groups and we obtain a full classification in dimension five. We show that Sasakian structures on five-dimensional Lie algebras with non-trivial center are a relatively rare phenomenon with respect to K-contact structures. We also prove that a five-dimensional solvmanifold with a left-invariant K-contact (not Sasakian) structure is a \mathbb S1{\mathbb S^1} -bundle over a symplectic solvmanifold. Rigidity results are then obtained for five-dimensional K-contact Lie algebras with trivial center and for K-contact η-Einstein structures. Moreover, five-dimensional Sasakian φ-symmetric Lie algebras are completely classified, and some explicit examples of five-dimensional Sasakian pseudo-metric Lie algebras are provided. 相似文献
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Frobenius Lie algebras 总被引:2,自引:0,他引:2
A. G. Elashvili 《Functional Analysis and Its Applications》1982,16(4):326-328
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B. Enriquez 《Selecta Mathematica, New Series》2001,7(3):321-407
To any field
\Bbb K \Bbb K of characteristic zero, we associate a set
(\mathbbK) (\mathbb{K}) and a group
G0(\Bbb K) {\cal G}_0(\Bbb K) . Elements of
(\mathbbK) (\mathbb{K}) are equivalence classes of families of Lie polynomials subject to associativity relations. Elements of
G0(\Bbb K) {\cal G}_0(\Bbb K) are universal automorphisms of the adjoint representations of Lie bialgebras over
\Bbb K \Bbb K . We construct a bijection between
(\mathbbK)×G0(\Bbb K) (\mathbb{K})\times{\cal G}_0(\Bbb K) and the set of quantization functors of Lie bialgebras over
\Bbb K \Bbb K . This construction involves the following steps.? 1) To each element v \varpi of
(\mathbbK) (\mathbb{K}) , we associate a functor
\frak a?\operatornameShv(\frak a) \frak a\mapsto\operatorname{Sh}^\varpi(\frak a) from the category of Lie algebras to that of Hopf algebras;
\operatornameShv(\frak a) \operatorname{Sh}^\varpi(\frak a) contains
U\frak a U\frak a .? 2) When
\frak a \frak a and
\frak b \frak b are Lie algebras, and
r\frak a\frak b ? \frak a?\frak b r_{\frak a\frak b} \in\frak a\otimes\frak b , we construct an element
?v (r\frak a\frak b) {\cal R}^{\varpi} (r_{\frak a\frak b}) of
\operatornameShv(\frak a)?\operatornameShv(\frak b) \operatorname{Sh}^\varpi(\frak a)\otimes\operatorname{Sh}^\varpi(\frak b) satisfying quasitriangularity identities; in particular,
?v(r\frak a\frak b) {\cal R}^\varpi(r_{\frak a\frak b}) defines a Hopf algebra morphism from
\operatornameShv(\frak a)* \operatorname{Sh}^\varpi(\frak a)^* to
\operatornameShv(\frak b) \operatorname{Sh}^\varpi(\frak b) .? 3) When
\frak a = \frak b \frak a = \frak b and
r\frak a ? \frak a?\frak a r_\frak a\in\frak a\otimes\frak a is a solution of CYBE, we construct a series
rv(r\frak a) \rho^\varpi(r_\frak a) such that
?v(rv(r\frak a)) {\cal R}^\varpi(\rho^\varpi(r_\frak a)) is a solution of QYBE. The expression of
rv(r\frak a) \rho^\varpi(r_\frak a) in terms of
r\frak a r_\frak a involves Lie polynomials, and we show that this expression is unique at a universal level. This step relies on vanishing
statements for cohomologies arising from universal algebras for the solutions of CYBE.? 4) We define the quantization of a
Lie bialgebra
\frak g \frak g as the image of the morphism defined by ?v(rv(r)) {\cal R}^\varpi(\rho^\varpi(r)) , where
r ? \mathfrakg ?\mathfrakg* r \in \mathfrak{g} \otimes \mathfrak{g}^* .<\P> 相似文献
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Anselm Eggert 《Semigroup Forum》1990,41(1):115-121
This paper gives some basic facts on Lie semialgebras and shows the crucial steps that lead to a classification of semialgebras
in a class of Lie algebras that contains the reductive ones. The classification of invariant wedges by Hilgert and Hofmann
is a prerequisite.
This paper was presented at the Conference on “The analytical and topological theory of semigroups” in Oberwolfach, January
29 through February 4, 1989 相似文献
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Yanan Lin 《Advances in Mathematics》2005,196(2):487-530
This article is to study relations between tubular algebras of Ringel and elliptic Lie algebras in the sense of Saito-Yoshii. Using the explicit structure of the derived categories of tubular algebras given by Happel-Ringel, we prove that the elliptic Lie algebra of type , , or is isomorphic to the Ringel-Hall Lie algebra of the root category of the tubular algebra with the same type. As a by-product of our proof, we obtain a Chevalley basis of the elliptic Lie algebra following indecomposable objects of the root category of the corresponding tubular algebra. This can be viewed as an analogue of the Frenkel-Malkin-Vybornov theorem in which they described a Chevalley basis for each untwisted affine Kac-Moody Lie algebra by using indecomposable representations of the corresponding affine quiver. 相似文献
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本文旨在讨论有限维李代数及某些广义李代数的结构常数及相应的立方阵 ,从而获得一种新的从元素运算的角度的方法来刻划它们 ,即将对有限维李代数及某些广义李代数的讨论可转化为对 n× n× n立方阵的讨论。 相似文献
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N. A. Koreshkov 《Mathematical Notes》2010,88(1-2):39-47
In the paper, some properties of algebras of associative type are studied, and these properties are then used to describe the structure of finite-dimensional semisimple modular Lie algebras. It is proved that the homogeneous radical of any finite-dimensional algebra of associative type coincides with the kernel of some form induced by the trace function with values in a polynomial ring. This fact is used to show that every finite-dimensional semisimple algebra of associative type A = ⊕ αεG A α graded by some group G, over a field of characteristic zero, has a nonzero component A 1 (where 1 stands for the identity element of G), and A 1 is a semisimple associative algebra. Let B = ⊕ αεG B α be a finite-dimensional semisimple Lie algebra over a prime field F p , and let B be graded by a commutative group G. If B = F p ? ? A L , where A L is the commutator algebra of a ?-algebra A = ⊕ αεG A α ; if ? ? ? A is an algebra of associative type, then the 1-component of the algebra K ? ? B, where K stands for the algebraic closure of the field F p , is the sum of some algebras of the form gl(n i ,K). 相似文献
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O. K. Sheinman 《Functional Analysis and Its Applications》1990,24(3):210-219
G. M. Krzhizhanovskii Energetics Institute. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 24, No. 3, pp. 51–61, July–September, 1990. 相似文献
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Christine Riedtmann 《Commentarii Mathematici Helvetici》1994,69(1):291-310