In the fundamental paper [11], J. Dixmier posed six problems for the Weyl algebra A1 over a field K of characteristic zero. Problem 3 was solved by Joseph and Stein [15] (using results of McConnel and Robson [20]), problem 5 was solved by the author in [7]. Using a (difficult) polarization theorem for the Weyl algebra A1 Joseph [15] solved problem 6. Problems 1, 2, and 4 are still open. Note th...

In a perfect category every object has a minimal projective resolution. We give a sufficient condition for the category of modules over a category-graded algebra to be perfect. AMS Subject Classification (2000): 18E15, 16W50. In [6] the second author explored homological properties of algebras graded over a small category. Our interest in these algebras arose from our research on the homologica...

H. Weyl has defined a semi-simple algebra (of finite rank) to be an algebra which admits a faithful semi-simple linear representation. Now, algebras are rings with a field of operators; Artin and others have shown that the theory of semi-simple algebras can be generalized to a theory of semi-simple rings (without the field of operators) provided we replace the condition of finite rank by suitab...

in this paper, we introduce the notion of multiplier in -algebra and study relationships between multipliers and some special mappings, likeness closure operators, homomorphisms and ( -derivations in -algebras. we introduce the concept of idempotent multipliers in bl-algebra and weak congruence and obtain an interconnection between idempotent multipliers and weak congruences. also, we introduce...

We show that the non-linear semi-quantum Hamiltonians which may be expressed as ( ) ( ) ˆ ˆ ∑ N j j j p H a q p O V q m 2 1 , 2 = = + + (where ˆ j O is the set of generators of some Lie algebra and ( ) q p , are the classical conjugated canonical variables) always close a partial semi Lie algebra under commutation and, because of this, it is always possible to integrate the mean values of the q...

We characterize the structure of a seven-dimensional Lie algebra with non-trivial center endowed closed G$_2$-structure. Using this result, we classify all unimodular algebras admitting G$_2$-structures, up to isomorphism, and show that six them arise as contactization symplectic algebra. Finally, prove every semi-algebraic soliton on must be expanding, determine dimension at least two admit so...

After introducing double derivations of $n$-Lie algebra $L$ we‎ ‎describe the relationship between the algebra $mathcal D(L)$ of double derivations and the usual‎ ‎derivation Lie algebra $mathcal Der(L)$‎. ‎In particular‎, ‎we prove that the inner derivation algebra $ad(L)$‎ ‎is an ideal of the double derivation algebra $mathcal D(L)$; we also show that if $L$ is a perfect $n$-Lie algebra‎ ‎wit...

let a be a banach algebra and e be a banach a-bimodule then s = a  e, the l1-direct sum of a and e becomes a module extension banach algebra when equipped with the algebras product (a; x):(a′; x′) = (aa′; a:x′ + x:a′). in this paper, we investigate △-amenability for these banach algebras and we show that for discrete inverse semigroup s with the set of idempotents es, the module extension bana...