and similarly for the right topological centre Z(r)(A′′). The algebra A is said to be Arens regular if Z(`)(A′′) = Z(r)(A′′) = A′′ and strongly Arens irregular if Z(`)(A′′) = Z(r)(A′′) = A. For example, every C∗-algebra is Arens regular [2]. There has been a great deal of study of these two algebras, especially in the case where A is the group algebra L(G) for a locally compact group G. Results...

We give a example of non nilpotent faithful representation of a filiform Lie algebra. This gives one counter-example of the conjecture saying that every affine connection on a filiform Lie group is complete. 1. Affine connection on a nilpotent Lie algebra 1.1. Affine connection on nilpotent Lie algebras. Definition 1. Let g be a n-dimensional Lie algebra over R. It is called affine if there is ...

To any finite group Γ ⊂ Sp(V ) of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, Hκ, of the algebra C[V ]#Γ, smash product of Γ with the polynomial algebra on V . The parameter κ runs over points of CP , where r =number of conjugacy classes of symplectic reflections in Γ. The algebra Hκ, called a symplectic reflection algebra, is expected to be rela...

The Cartan decomposition in a semisimple Lie group is a generalization of the polar decomposition of matrices. In this paper we consider an even more general setting in which one obtains an analogous decomposition. In the semisimple case, this decomposition was worked out in a seminal paper of G. I. Ol'shanski 13]. In this paper we give general necessary and suucient conditions for this decompo...

This paper has four interrelated themes: (1) express Laplace and Mellin transforms of sums of positive random variables in terms of the Mellin transform of the summands; (2) show the equivalence of the two Barnes’ lemmas with known properties of gamma distributions; (3) establish properties of the sum of two reciprocal gamma variables, and related results; (4) study the G distributions (whose M...

It is shown that $A:=H_{1,\eta}(G)$, the Sympectic Reflection Algebra, has $T_G$ independent traces, where number of conjugacy classes elements without eigenvalue 1 belonging to finite group $G$ generated by system symplectic reflections. Simultaneously, we show algebra $A$, considered as a superalgebra with natural parity, $S_G$ supertraces, -1 $G$. We consider also $A$ Lie $A^L$ and $A^S$. if...

Given a free action of compact Lie group G on unital C?-algebra A and spectral triple the corresponding fixed point algebra AG, we present systematic in-depth construction that is build upon geometry AG G. We compare our with selection established examples.

Let G be a finite group and p a prime number. The Plesken Lie algebra is a subalgebra of the complex group algebra C[G] and admits a directsum decomposition into simple Lie algebras. We describe finite-field versions of the Plesken Lie algebra via traditional and computational methods. The computations motivate our conjectures on the general structure of the modular Plesken Lie algebra.

To any finite group Γ ⊂ Sp(V ) of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, Hκ, of the algebra C[V ]#Γ, smash product of Γ with the polynomial algebra on V . The parameter κ runs over points of CP , where r =number of conjugacy classes of symplectic reflections in Γ. The algebra Hκ, called a symplectic reflection algebra, is expected to be rela...

To any finite group Γ ⊂ Sp(V ) of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, Hκ, of the algebra C[V ]#Γ, smash product of Γ with the polynomial algebra on V . The parameter κ runs over points of CP, where r =number of conjugacy classes of symplectic reflections in Γ. The algebra Hκ, called a symplectic reflection algebra, is related to the coord...