We determine the Hochschild homology and cohomology of the generalized Weyl algebras of rank one which are of ‘quantum’ type in all but a few exceptional cases.

Let G be the Banach-Lie group of all holomorphic automorphisms of the open unit ball $$B_{𝔄}$$ in a J*-algebra $$𝔄$$ of operators. Let $$𝔉$$ be the family of all collectively compact subsets W contained in $$B_{𝔄}$$ . We show that the subgroup F ⊂ G of all those g ∈ G that preserve the family $$𝔉$$ is a closed Lie subgroup of G and characterize its Banach-Lie algebra. We make a detailed study of F when $$𝔄$$ is a Cartan factor.

In an earlier paper, the first two authors have shown that the convolution of a function $f$ continuous on the closure of a Cartan domain and a $K$-invariant finite measure $\mu$ on that domain is again continuous on the closure, and, moreover, its restriction to any boundary face $F$ depends only on the restriction of $f$ to $F$ and is equal to the convolution, in $F$, of the latter restriction with some measure ${\mu }_F$ on $F$ uniquely determined by $\mu$. In this article, we give an explicit formula for ${\mu }_F$ in terms of $F$,...

Hom-Akivis algebras are introduced. The commutator-Hom-associator algebra of a non-Hom-associative algebra (i.e. a Hom-nonassociative algebra) is a Hom-Akivis algebra. It is shown that Hom-Akivis algebras can be obtained from Akivis algebras by twisting along algebra endomorphisms and that the class of Hom-Akivis algebras is closed under self-morphisms. It is pointed out that a Hom-Akivis algebra associated to a Hom-alternative algebra is a Hom-Malcev algebra.

For each simple symplectic triple system over the real numbers, the standard enveloping Lie algebra and the algebra of inner derivations of the triple provide a reductive pair related to a semi-Riemannian homogeneous manifold. It is proved that this is an Einstein manifold.

We construct irreducible graded representations of simply laced Khovanov–Lauda algebras which are concentrated in one degree. The underlying combinatorics of skew shapes and standard tableaux corresponding to arbitrary simply laced types has been developed previously by Peterson, Proctor and Stembridge. In particular, the Peterson–Proctor hook formula gives the dimensions of the homogeneous irreducible modules corresponding to straight shapes.

Let $𝔐$ be a Jordan-Banach algebra with identity 1, whose norm satisfies:(i) $\parallel ab\parallel \le \parallel a\parallel \parallel b\parallel$,   $a,b\in 𝔐$(ii) $\parallel a^2\parallel =\parallel a{\parallel }^2$(iii) $\parallel a^2\parallel \le \parallel a^2+b^2\parallel$.$𝔐$ is called a JB algebra (E.M. Alfsen, F.W. Shultz and E. Stormer, Oslo preprint (1976)). The set ${𝔐}^{+}$ of squares in $𝔐$ is a closed convex cone. $(𝔐,{𝔐}^{+},\mathbf{1})$ is a complete ordered vector space with $\mathbf{1}$ as a order unit. In addition, we assume $𝔐$ to be monotone complete (i.e. $𝔐$ coincides with the bidual ${𝔐}^{**}$), and that there exists a finite normal faithful trace $\phi$ on $𝔐$.Then the completion $\{{𝔐}^{+}{\}}_{\phi }$ of ${𝔐}^{+}$ with respect to the Hilbert structure...

On étudie ici les notions d’algèbre de Gerstenhaber à homotopie près et d’homologie des algèbres de Gerstenhaber du point de vue de la théorie des opérades. Précisément, on donne une description explicite des $𝒢$-algèbres à homotopie près (c’est-à-dire d’algèbres sur le modèle minimal de l’opérade $𝒢$ des algèbres de Gerstenhaber). On décrit également le complexe calculant l’homologie des $𝒢$-algèbres. On donne une suite spectrale qui converge vers cette homologie et quelques exemples de calculs. Enfin...

Soit $g$ une $p$-algèbre de Lie parfaite au sens des algèbres de Lie (i.e. $g/[g,g]=0)$. Nous déterminons, en degré deux, le groupe d’homologie restreinte de $g$ en fonction de son groupe d’homologie d’algèbre de Lie. Nous appliquons ce résultat à l’algèbre de Lie $s{l}_n\left(A\right)$ des matrices de trace nulle sur une algèbre commutative, et nous montrons que pour sa structure de $p$-algèbre de Lie, le groupe d’homologie restreinte de dimension deux ne se stabilise pas, contrairement au groupe d’homologie d’algèbre de Lie étudié par...

For a Lie algebroid, divergences chosen in a classical way lead to a uniquely defined homology theory. They define also, in a natural way, modular classes of certain Lie algebroid morphisms. This approach, applied for the anchor map, recovers the concept of modular class due to S. Evens, J.-H. Lu, and A. Weinstein.