Irreducible bimodules over alternative algebras and superalgebras

Shestakov, I. P.; Trushina, Maria

doi:10.1090/tran/6475

Cited by 16 publications

(6 citation statements)

References 32 publications

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“…The organization of this article is as follows. In Section 2, we give a brief overview of the algebra of octonions, and review some definitions and basic properties on the theory of octonionic modules; more details can be found in [12,21,22]. In Section 3, we introduce the notion of O-para-linear maps and give a detailed discussion on this concept.…”

Section: It Turns Out That the Two Functors (M ⊗ Llmentioning

confidence: 99%

“…Subsequently, Jacobson [14] determined the irreducible representations for finite dimensional semisimple alternative algebras. A more general study for alternative bimodules is given in [22]. Recently, only for the octonionic case, we [12] introduce the new notions of associative elements and conjugate associative elements, which give rise to a new description of the structure of O-modules.…”

Section: O-modulesmentioning

confidence: 99%

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Non-associative Categories of Octonionic Bimodules

Huo

Ren²

2023

Commun. Math. Stat.

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Category is put to work in the non-associative realm in the article. We focus on a typical example of non-associative category. Its objects are octonionic bimodules, morphisms are octonionic para-linear maps, and compositions are non-associative in general. The octonionic para-linear map is the main object of octonionic Hilbert theory because of the octonionic Riesz representation theorem. An octonionic para-linear map f is in general not octonionic linear since it subjects to the rule Re f (px) − pf (x) = 0. The composition should be modified asso that it preserves the octonionic para-linearity.In this non-associative category, we introduce the Hom and Tensor functors which constitute an adjoint pair. We establish the Yoneda lemma in terms of the new notion of weak functor. To define the exactness in a non-associative category, we introduce the notion of the enveloping category via a universal property. This allows us to establish the exactness of the Hom functor and Tensor functor.

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Section: It Turns Out That the Two Functors (M ⊗ Llmentioning

confidence: 99%

Section: O-modulesmentioning

confidence: 99%

Non-associative Categories of Octonionic Bimodules

Huo

Ren²

2023

Commun. Math. Stat.

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“…We recall some basic notations and results on O-modules in this subsection. We refer to [11,20,19,10] for octonionic modules and even for modules of alternative algebras. A left O-module M is a real vector space M endowed with an O-scalar multiplication such that the left associator is left alternative, i.e., for all p, q ∈ O and m ∈ M we have…”

Section: Preliminariesmentioning

confidence: 99%

Para-linearity as the nonassociative counterpart of linearity

Huo¹,

Ren²

2021

Preprint

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In an octonionic Hilbert space H, the octonionic linearity is taken to fail for the maps induced by the octonionic inner products, and it should be replaced with the octonionic para-linearity. However, to introduce the notion of the octonionic para-linearity we encounter an insurmountable obstacle. That is, the axiom pu, u = p u, u for any octonion p and element u ∈ H introduced by Goldstine and Horwitz in 1964 can not be interpreted as a property to be obeyed by the octonionic para-linear maps. In this article, we solve this critical problem by showing that this axiom is in fact non-independent from others. This enables us to initiate the study of octonionic para-linear maps. We can thus establish the octonionic Riesz representation theorem which, up to isomorphism, identifies two octonionic Hilbert spaces with one being the dual of the other. The dual space consists of continuous left para-linear functionals and it becomes a right O-module under the multiplication defined in terms of the second associators which measures the failure of O-linearity. This right multiplication has an alternative expression (f ⊙ p)(x) = pf (p −1 x)p, which is a generalized Moufang identity. Remarkably, the multiplication is compatible with the canonical norm, i.e., ||f ⊙ p|| = ||f || |p| . Our final conclusion is that para-linearity is the nonassociative counterpart of linearity.

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“…An octonionic module is a specific alternative module since the octonionic algebra is an alternative algebra. The general representation theory over alternative algebras has been fully studied; see [18,12,20,19,10]. We recall some basic notations and results on O-modules in this subsection.…”

Section: O-modulesmentioning

confidence: 99%

Structure of octonionic Hilbert spaces with applications in the Parseval equality and Cayley-Dickson algebras

Huo,

Ren

2021

Preprint

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Contrary to the simple structure of the tensor product of the quaternionic Hilbert space, the octonionic situation becomes more involved. It turns out that an octonionic Hilbert space can be decomposed as an orthogonal direct sum of two subspaces, each of them isomorphic to a tensor product of an irreducible octonionic Hilbert space with a real Hilbert space. As an application, we find that for a given orthogonal basis the octonionic Parseval equality holds if and only if the basis is weak associative. Fortunately, there always exists a weak associative orthogonal basis in an octonionic Hilbert space. This completely removes the obstacles caused by the failure of the octonionic Parseval equality. As another application, we provide a new approach to study the Cayley-Dickson algebras, which turn out to be specific examples of octonionic Hilbert spaces. An explicit weak associative orthonormal basis is constructed in each Cayley-Dickson algebra.

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Irreducible bimodules over alternative algebras and superalgebras

Abstract: The irreducible alternative superbimodules are studied. The complete classification is obtained for even bimodules of arbitrary dimension and for finite-dimensional irreducible superbimodules over an algebraically closed field.

Cited by 16 publications

References 32 publications

Non-associative Categories of Octonionic Bimodules

Non-associative Categories of Octonionic Bimodules

Para-linearity as the nonassociative counterpart of linearity

Structure of octonionic Hilbert spaces with applications in the Parseval equality and Cayley-Dickson algebras

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