Some equations in algebras obtained by the Cayley-Dickson process.
Some examples of nil Lie algebras
Generalizing Petrogradsky’s construction, we give examples of infinite-dimensional nil Lie algebras of finite Gelfand–Kirillov dimension over any field of positive characteristic.
Some examples of real division algebras.
Some exceptional compact matrix pseudogroups
Some lagrangian invariants of symplectic manifolds
The KV-homology theory is a new framework which yields interesting properties of lagrangian foliations. This short note is devoted to relationships between the KV-homology and the KV-cohomology of a lagrangian foliation. Let us denote by (resp. ) the KV-algebra (resp. the space of basic functions) of a lagrangian foliation F. We show that there exists a pairing of cohomology and homology to . That is to say, there is a bilinear map , which is invariant under F-preserving symplectic diffeomorphisms....
Some Lie rings associated with Burnside groups.
Some necessary and sufficient conditions for nilpotent -Lie superalgebras
The paper studies nilpotent -Lie superalgebras over a field of characteristic zero. More specifically speaking, we prove Engel’s theorem for -Lie superalgebras which is a generalization of those for -Lie algebras and Lie superalgebras. In addition, as an application of Engel’s theorem, we give some properties of nilpotent -Lie superalgebras and obtain several sufficient conditions for an -Lie superalgebra to be nilpotent by using the notions of the maximal subalgebra, the weak ideal and the...
Some notes about -Lie algebras.
Some Primitive Differential Operator Rings.
Some properties of complex filiform Lie algebras.
The purpose of this paper is to study some properties of Filiform Lie Algebras (FLA) and to prove the following theorem: a FLA, of dimension n, is either derived from a Solvable Lie Algebra (SLA) of dimension n+1 or not derived from any LA.
Some properties of generalized reduced Verma modules over -graded modular Lie superalgebras
We study some properties of generalized reduced Verma modules over -graded modular Lie superalgebras. Some properties of the generalized reduced Verma modules and coinduced modules are obtained. Moreover, invariant forms on the generalized reduced Verma modules are considered. In particular, for -graded modular Lie superalgebras of Cartan type we prove that generalized reduced Verma modules are isomorphic to mixed products of modules.
Some properties of octonion and quaternion algebras.
Some properties of the composition algebras.
Some properties of the family of modular Lie superalgebras
In this paper, we continue to investigate some properties of the family of finite-dimensional simple modular Lie superalgebras which were constructed by X. N. Xu, Y. Z. Zhang, L. Y. Chen (2010). For each algebra in the family, a filtration is defined and proved to be invariant under the automorphism group. Then an intrinsic property is proved by the invariance of the filtration; that is, the integer parameters in the definition of Lie superalgebras are intrinsic. Thereby, we classify these Lie...
Some Remarks on Dirac Structures and Poisson Reductions
Dirac structures are characterized in terms of their characteristic pairs defined in this note and then Poisson reductions are discussed from the point of view of Dirac structures.
Some remarks on quantum and braided group gauge theory
We clarify some aspects of quantum group gauge theory and its recent generalisations (by T. Brzeziński and the author) to braided group gauge theory and coalgebra gauge theory. We outline the diagrammatic version of the braided case. The bosonisation of any braided group provides us a trivial principal bundle in three ways.
Some remarks on the Akivis algebras and the Pre-Lie algebras
In this paper, by using the Composition-Diamond lemma for non-associative algebras invented by A. I. Shirshov in 1962, we give Gröbner-Shirshov bases for free Pre-Lie algebras and the universal enveloping non-associative algebra of an Akivis algebra, respectively. As applications, we show I. P. Shestakov’s result that any Akivis algebra is linear and D. Segal’s result that the set of all good words in forms a linear basis of the free Pre-Lie algebra generated by the set . For completeness,...
Some remarks on the cohomology of the Poisson algebra
Some Theorems on Saturated Homomorphs of Soluble Lie Algebras.
Something about Frattini subalgebras of a class of solvable Lie algebras.