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A 4 3 -grading on a 56 -dimensional simple structurable algebra and related fine gradings on the simple Lie algebras of type E

Diego Aranda-Orna, Alberto Elduque, Mikhail Kochetov (2014)

Commentationes Mathematicae Universitatis Carolinae

We describe two constructions of a certain 4 3 -grading on the so-called Brown algebra (a simple structurable algebra of dimension 56 and skew-dimension 1 ) over an algebraically closed field of characteristic different from 2 . The Weyl group of this grading is computed. We also show how this grading gives rise to several interesting fine gradings on exceptional simple Lie algebras of types E 6 , E 7 and E 8 .

A class of fermionic Novikov superalgebras which is a class of Novikov superalgebras

Huibin Chen, Shaoqiang Deng (2018)

Czechoslovak Mathematical Journal

We construct a special class of fermionic Novikov superalgebras from linear functions. We show that they are Novikov superalgebras. Then we give a complete classification of them, among which there are some non-associative examples. This method leads to several new examples which have not been described in the literature.

A new approach to Hom-left-symmetric bialgebras

Qinxiu Sun, Qiong Lou, Hongliang Li (2021)

Czechoslovak Mathematical Journal

The main purpose of this paper is to consider a new definition of Hom-left-symmetric bialgebra. The coboundary Hom-left-symmetric bialgebra is also studied. In particular, we give a necessary and sufficient condition that s -matrix is a solution of the Hom- S -equation by a cocycle condition.

A review on δ-structurable algebras

Noriaki Kamiya, Daniel Mondoc, Susumu Okubo (2011)

Banach Center Publications

In this paper we give a review on δ-structurable algebras. A connection between Malcev algebras and a generalization of δ-structurable algebras is also given.

Conservative algebras and superalgebras: a survey

Yury Popov (2020)

Communications in Mathematics

We give a survey of results obtained on the class of conservative algebras and superalgebras, as well as on their important subvarieties, such as terminal algebras.

Embedding of dendriform algebras into Rota-Baxter algebras

Vsevolod Gubarev, Pavel Kolesnikov (2013)

Open Mathematics

Following a recent work [Bai C., Bellier O., Guo L., Ni X., Splitting of operations, Manin products, and Rota-Baxter operators, Int. Math. Res. Not. IMRN (in press), DOI: 10.1093/imrn/rnr266] we define what is a dendriform dior trialgebra corresponding to an arbitrary variety Var of binary algebras (associative, commutative, Poisson, etc.). We call such algebras di- or tri-Var-dendriform algebras, respectively. We prove in general that the operad governing the variety of di- or tri-Var-dendriform...

Fermionic Novikov algebras admitting invariant non-degenerate symmetric bilinear forms

Zhiqi Chen, Xueqing Chen, Ming Ding (2020)

Czechoslovak Mathematical Journal

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. Fermionic Novikov algebras correspond to a certain Hamiltonian superoperator in a supervariable. In this paper, we show that fermionic Novikov algebras equipped with invariant non-degenerate symmetric bilinear forms are Novikov algebras.

F-quasigroups and generalized modules

Tomáš Kepka, Michael K. Kinyon, Jon D. Phillips (2008)

Commentationes Mathematicae Universitatis Carolinae

In Kepka T., Kinyon M.K., Phillips J.D., The structure of F-quasigroups, J. Algebra 317 (2007), 435–461, we showed that every F-quasigroup is linear over a special kind of Moufang loop called an NK-loop. Here we extend this relationship by showing an equivalence between the class of (pointed) F-quasigroups and the class corresponding to a certain notion of generalized module (with noncommutative, nonassociative addition) for an associative ring.

F-quasigroups isotopic to groups

Tomáš Kepka, Michael K. Kinyon, Jon D. Phillips (2010)

Commentationes Mathematicae Universitatis Carolinae

In Kepka T., Kinyon M.K., Phillips J.D., The structure of F-quasigroups, math.GR/0510298, we showed that every loop isotopic to an F-quasigroup is a Moufang loop. Here we characterize, via two simple identities, the class of F-quasigroups which are isotopic to groups. We call these quasigroups FG-quasigroups. We show that FG-quasigroups are linear over groups. We then use this fact to describe their structure. This gives us, for instance, a complete description of the simple FG-quasigroups. Finally,...

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