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Highest Weight Modules of W1+∞, Darboux Transformations and the Bispectral Problem

Bakalov, B., Horozov, E., Yakimov, M. (1997)

Serdica Mathematical Journal

This paper is a survey of our recent results on the bispectral problem. We describe a new method for constructing bispectral algebras of any rank and illustrate the method by a series of new examples as well as by all previously known ones. Next we exhibit a close connection of the bispectral problem to the representation theory of W1+∞–algerba. This connection allows us to explain and generalise to any rank the result of Magri and Zubelli on the symmetries of the manifold of the bispectral operators...

Hochschild homology and cohomology of generalized Weyl algebras

Marco A. Farinati, Andrea L. Solotar, Mariano Suárez-Álvarez (2003)

Annales de l’institut Fourier

We compute Hochschild homology and cohomology of a class of generalized Weyl algebras, introduced by V. V. Bavula in St. Petersbourg Math. Journal, 4 (1) (1999), 71-90. Examples of such algebras are the n-th Weyl algebras, 𝒰 ( 𝔰 𝔩 2 ) , primitive quotients of 𝒰 ( 𝔰 𝔩 2 ) , and subalgebras of invariants of these algebras under finite cyclic groups of automorphisms. We answer a question of Bavula–Jordan (Trans. A.M.S., 353 (2) (2001), 769-794) concerning the generators of the group of automorphisms of a generalized Weyl...

Holomorphic automorphisms and collective compactness in J*-algebras of operator

José Isidro (2007)

Open Mathematics

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.

Holomorphic retractions and boundary Berezin transforms

Jonathan Arazy, Miroslav Engliš, Wilhelm Kaup (2009)

Annales de l’institut Fourier

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 μ 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 μ F on F uniquely determined by  μ . In this article, we give an explicit formula for μ F in terms of  F ,...

Hom-Akivis algebras

A. Nourou Issa (2011)

Commentationes Mathematicae Universitatis Carolinae

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.

Homogeneous Einstein manifolds based on symplectic triple systems

Cristina Draper Fontanals (2020)

Communications in Mathematics

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.

Homogeneous representations of Khovanov–Lauda Algebras

Alexander Kleshchev, Arun Ram (2010)

Journal of the European Mathematical Society

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.

Homogeneous self dual cones versus Jordan algebras. The theory revisited

Jean Bellissard, B. Iochum (1978)

Annales de l'institut Fourier

Let 𝔐 be a Jordan-Banach algebra with identity 1, whose norm satisfies:(i) ∥ a b ∥ ≤ ∥ a ∥ ∥ b ∥ ,    a , b ∈ 𝔐 (ii) ∥ a 2 ∥ = ∥ a ∥ 2 (iii) ∥ a 2 ∥ ≤ ∥ a 2 + b 2 ∥ . 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 + , 1 ) is a complete ordered vector space with 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 φ on 𝔐 .Then the completion { 𝔐 + } φ of 𝔐 + with respect to the Hilbert structure...

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