On the geometry of para-Kählerian Cartan spaces

We consider Cartan Spaces with almost product structure. We prove that, in such spaces, there exist para-Kählerian structures and establish criteria for semisymmetric and Ricci-semisymmetric spaces.     

Jordan Algebras and Dual Affine Connections on Symmetric Cones

We study dually flat structures on symmetric cones associated with Jordan algebras. We give an interpretation of connections, a geometrical concept, in terms of Jordan algebras and show a relation between doubly autoparallel submanifolds and Jordan subalgebras.     

Dual affine-metrically connected spaces

In this paper we study the dual geometry of a normalized affinely connected space A _n,n . In particular, we consider the dual affine-metrically connected spaces \(\mathop M\limits^p _{n,n} \) induced by a nondegenerate normalization of an affine-metrically connected space M _n,n .     

Linear connections of a framed distribution of hyperplane elements in conformal space

We study the geometry of affine and normal connections induced by a complete normalization of mutually orthogonal distributions $ \mathcal{M} We study the geometry of affine and normal connections induced by a complete normalization of mutually orthogonal distributions and in conformal space C _n , where is a distribution of hyperplane elements, and is a distribution of line elements. We consider invariant fields of pencils that are parallel with respect to the normal connection along any curve belonging to the distribution . Original Russian Text ? A.M. Matveeva, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 7, pp. 79–84.     

伪仿射对偶框架

对于给定冗余的仿射框架X（Ф）={Ф1,…,ФL）,存在对偶Y（ψ）={ψ1,…,ψL）,这个对偶可能不是仿射框架,因为它不满足Bessel条件．这样的对偶我们称为伪仿射对偶,并给出伪仿射对偶框架的一种构造公式．     

Cartan Geometries and Dynamics

We show how some basic dynamical ideas can be brought to bear on the study of Cartan geometries. In our main results, we give conditions for certain types of Cartan geometries to have constant curvature. We also consider ergodic actions of `higher-rank' semisimple groups on bundles supporting (not necessarily invariant) Cartan connections and show that these are `standard locally homogeneous' actions provided that some noncompact 1-parameter subgroup `preserves' a Cartan geometry.     

Cartan Subalgebras of Leibniz n-Algebras

The present article is devoted to the investigation of properties of Cartan subalgebras and regular elements in Leibniz n-algebras. The relationship between Cartan subalgebras and regular elements of given Leibniz n-algebra and Cartan subalgebras and regular elements of the corresponding factor n-Lie algebra is established.     

A realization of Cartan extensions

We revise the notion of a Cartan extension and consider in detail simple examples of known Cartan extensions.     

Dual affine connections on a quadratic hyperband distribution in a projective-metric space and their applications

In this paper, we develop fundamentals of the dual theory of quadratic hyperband distributions H of m-dimensional line elements in a projective-metric space K _n (m < n ? 1). In particular, we show that, on a dual normalized distribution H, there are induced two dual affine connections and indicate some applications of these connections to the geometry of m-webs on H.     

The Geometry of Lightlike Hypersurfaces of the de Sitter Space

It is proved that the geometry of lightlike hypersurfaces of the de Sitter space Sn+11 is directly connected with the geometry of hypersurfaces of the conformal space Cn. This connection is applied for a construction of an invariant normalization and an invariant affine connection of lightlike hypersurfaces as well as for studying singularities of lightlike hypersurfaces.     

Automorphisms of infinite dimensional Lie algebras of Cartan type

In this paper it is proved that any automorphism of an infinite dimensional Lie algebra of Cartan type over an algebraically closed field of characteristicp>2 can be induced from an automorphism of the associated divided power algebra.     

Cartan geometries and multiplicative forms

In this paper we show that Cartan geometries can be studied via transitive Lie groupoids endowed with a special kind of vector-valued multiplicative 1-forms. This viewpoint leads us to a more general notion, that of Cartan bundle, which encompasses both Cartan geometries and G-structures.     

Cartan矩阵,分解数与单模的顶

设Ｂ是有限群Ｇ的ｐ－块代数，考虑以Ｂ的Ｂｒａｕｅｒ不可约特征标为基底生成的自由Ａｂｅｌ群，以及由Ｂ的Ｃａｒｔａｎ矩阵决定的这个群的商群．通过研究这个商群中元素的阶数，我们得到了Ｃａｒｔａｎ矩阵和Ｃａｒｔａｎ数与单模的顶点及亏群的一些新关系式．对于常不可约特征标的高度，　我们得到了它与分解数以及Ｃａｒｔａｎ不变量的一些新关系式．     

Cartan矩阵的分解和Brauer特征标次数猜想

设G为有限群,N是G的正规子群.记J=J(F[N])为F[N]的Jacobson根,I=Ann(J)={α∈F[G]|Jα=0}为J在F[G]中的零化子.本文主要研究了,根据F[G/N]和F[G]/I的Cartan矩阵,分解F[G]的Cartan矩阵.这种分解在Cartan不变量和G的合成因子之间建立了一些联系.本文指出N中p-亏零块的存在性依赖于Cartan不变量或者I在F[G]中的性质,证明了Cartan矩阵的分解部分地依赖于B所覆盖的N中的块的性质.本文研究了b为N上的块且l(b)=1时,覆盖b的G中的块B的性质.在两类情形下,本文证明了块代数上关于Brauer特征标次数的猜想成立,涵盖了Holm和Willems研究的某些情形.进而对Holm和Willems提出的问题给出了肯定的回答.另外,本文还给出了Cartan不变量的一些其它结果.     

Cartan Subalgebras with Engel Decomposition

Mathematical Notes -     

On realization of fusion rings from generalized Cartan matrices

The Casimir element of a fusion ring (R, B) gives rise to the so called Casimir matrix C of (R, B). This enables us to construct a generalized Cartan matrix D-C in the sense of Kac for a suitable diagonal matrix D. In this paper, we study some elementary properties of the Casimir matrix C and use them to realize certain fusion rings from the generalized Cartan matrix D-C of finite (resp. affine) type. It turns out that there exists a fusion ring with D-C being of finite (resp. affine) type if and only if D-C has only the form A₂ (resp. A₁⁽¹⁾. We also realize all fusion rings with D-C being a particular generalized Cartan matrix of indefinite type.     

Invariant measures for the continual Cartan subgroup

We construct and study the one-parameter semigroup of σ-finite measures Lθ, θ>0, on the space of Schwartz distributions that have an infinite-dimensional abelian group of linear symmetries; this group is a continual analog of the classical Cartan subgroup of diagonal positive matrices of the group SL(n,R). The parameter θ is the degree of homogeneity with respect to homotheties of the space, we prove uniqueness theorem for measures with given degree of homogeneity, and call the measure with degree of homogeneity equal to one the infinite-dimensional Lebesgue measure L. The structure of these measures is very closely related to the so-called Poisson-Dirichlet measures PD(θ), and to the well-known gamma process. The nontrivial properties of the Lebesgue measure are related to the superstructure of the measure PD(1), which is called the conic Poisson-Dirichlet measure—CPD. This is the most interesting σ-finite measure on the set of positive convergent monotonic real series.     

Extensions of Modules over Hopf Algebras Arising from Lie Algebras of Cartan Type

In this paper we prove that there are no self-extensions of simple modules over restricted Lie algebras of Cartan type. The proof given by Andersen for classical Lie algebras not only uses the representation theory of the Lie algebra, but also representations of the corresponding reductive algebraic group. The proof presented in the paper follows in the same spirit by using the construction of a infinite-dimensional Hopf algebra D(G) u( ) containing u( ) as a normal Hopf subalgebra, and the representation theory of this algebra developed in our previous work. Finite-dimensional hyperalgebra analogs D(G _r ) u( ) have also been constructed, and the results are stated in this setting.     

Simple Modules for the Restricted Cartan Type Lie Algebras

The simple modules with homogeneous characters are considered, their dimension formulas are determined.Presented by D. Passman.