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The Lie algebra of Cartan type H which occurs as a subalgebra of the Lie algebra of derivations of the polynomial algebra was generalized by the first author to a class which included a subalgebra of the derivations of the Laurent polynomials
. We show in this paper that these generalizations of Cartan type H algebras are isomorphic to certain generalizations of the classical algebra of Poisson brackets, and that it can be generalized
further. In turn, these algebras can be recast in a form that is an adaption of a class of Lie algebras of characteristic
p that was defined in 1958 be R. Block. A further generalization of these algebras is the main topic of this paper. We show
when these algebras are simple, find their derivations, and determine all possible isomorphisms between two of these algebras.
Received December 20, 1996; in final form September 15, 1997 相似文献
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Lie algebras which are isomorphic to central quotients of quaternion division algebras are investigated. 相似文献
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Andriy Panasyuk 《Differential Geometry and its Applications》2006,24(5):482-491
We give a criterion of (micro-)kroneckerity of the linear Poisson pencil on g∗ related to an algebraic Nijenhuis operator on a finite-dimensional Lie algebra g. As an application we get a series of examples of completely integrable systems on semisimple Lie algebras related to Borel subalgebras and a new proof of the complete integrability of the free rigid body system on gln. 相似文献
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Anton Izosimov 《Differential Geometry and its Applications》2013,31(5):557-567
A Poisson pencil is called flat if all brackets of the pencil can be simultaneously locally brought to a constant form. Given a Poisson pencil on a 3-manifold, we study under which conditions it is flat. Since the works of Gelfand and Zakharevich, it is known that a pencil is flat if and only if the associated Veronese web is trivial. We suggest a simpler obstruction to flatness, which we call the curvature form of a Poisson pencil. This form can be defined in two ways: either via the Blaschke curvature form of the associated web, or via the Ricci tensor of a connection compatible with the pencil.We show that the curvature form of a Poisson pencil can be given by a simple explicit formula. This allows us to study flatness of linear pencils on three-dimensional Lie algebras, in particular those related to the argument translation method. Many of them appear to be non-flat. 相似文献
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Zhiqi Chen 《Czechoslovak Mathematical Journal》2011,61(2):323-328
Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and the Hamiltonian operators
in formal variational calculus. In this note we prove that the underlying Lie algebras of quadratic Novikov algebras are 2-step
nilpotent. Moreover, we give the classification up to dimension 10. 相似文献
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A. A. Kucherov O. A. Pikhtilkova S. A. Pikhtilkov 《Journal of Mathematical Sciences》2012,186(4):651-654
Sufficient conditions for Lie algebra primitiveness and examples of primitive Lie algebras and nonprimitive Lie algebras are given. 相似文献
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Nora C. Hopkins 《代数通讯》2013,41(3):767-775
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Vincenzo de Filippis Giovanni Scudo Mohammad S. Tammam El-Sayiad 《Czechoslovak Mathematical Journal》2012,62(2):453-468
Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R, L a non-central Lie ideal of R, F a non-zero generalized derivation of R. Suppose that [F(u), u]F(u) = 0 for all u ε L, then one of the following holds:
- there exists α ε C such that F(x) = α x for all x ε R
- R satisfies the standard identity s 4 and there exist a ε U and α ε C such that F(x) = ax + xa + αx for all x ε R.
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Schur showed that if the center of a group has finite index, then the commutator subgroup is finite. By replacing inner automorphisms by automorphisms, Hegarty obtained a variation of Schur's result. The Lie algebra analogue of Schur's result is well-known. We obtain a strong Lie algebra version of Hegarty's result. 相似文献
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M.-A. Knus 《Transformation Groups》2009,14(2):361-386
Over an algebraically closed field of characteristic zero simple Lie algebras admit outer automorphisms of order 3 if and
only if they are of type D4. Moreover, thereare two conjugacy classes of such automorphisms. Among orthogonal Lie algebras over arbitrary fields of characteristic
zero, only orthogonal Lie algebras relative to quadratic norm forms of Cayley algebras admit outer automorphisms of order
3. We give a complete list of conjugacy classes of outer automorphisms of order 3 for orthogonal Lie algebras over arbitrary
fields of characteristic zero. For the norm form of a given Cayley algebra, one class is associated with the Cayley algebra
and the others with central simple algebras of degree 3 with involution of the second kind such that the cohomological invariant
of the involution is the norm form. 相似文献