arXiv subjects

Tarik Anowar

Publications and source records attributed to Tarik Anowar.

4 recordsLinked to original sources

arXiv2026

Affine Nilpotency and Engel's Theorem for Lie Affgebras

We study some structural properties of Lie affgebras as affine analogues of Lie algebras. We introduce the notion of an ideaf, the affine counterpart of an ideal, and establish characterizations of left and right ideafs. We further study centers, quotient structures, and product ideafs, proving, under suitable conditions, that the center of a Lie affgebra is an ideaf. We then develop the notion of affine nilpotency and establish connections between the nilpotency of Lie affgebras and that of their retracted Lie algebras. As an application, we prove an Engel-type theorem for Lie affgebras of the form $\mathfrak{a}(\mathfrak{g};\kappa=2\lambda,\lambda,s)$.

math.RA↗

arXiv2025

On Affine Version of Hom-Lie Algebras

This paper introduces Hom-type analogues of affine algebraic structures, termed Hom-affgebras. Extending Brzezi\'nski's theory of affgebras and the Hom-algebra framework developed by Hartwig-Larsson-Silvestrov, we define and study Hom-associative, Hom-pre-Lie, and Hom-Lie affgebras, where the classical identities are twisted by an affine self-map. We show how Hom-associative, Hom-pre-Lie, and Hom-Lie affgebras are related to one another. The main focus of this paper is on Hom-Lie affgebras and their fibers. We study the concept of generalized derivations for Hom-Lie algebras, extending the notion of generalized derivations for Lie algebras. We explore the close relationship between Hom-Lie affgebras and such derivations. We show that every Hom-Lie affgebra both determines and is determined by a Hom-Lie algebra together with such a generalized derivation and a constant. Furthermore, we establish that a homomorphism between Lie affgebras corresponds to a homomorphism between their associated Lie fibers along with a constant, and vice versa.

math.RA↗

arXiv2025

On Hom-Analogues of Heaps and Trusses

This paper introduces Hom-heaps, Hom-trusses, and Hom-braces as Hom-type analogues of their classical counterparts. We establish the correspondence between Hom-heaps and Hom-groups by showing that the retract of a Hom-heap at a point forms a Hom-group precisely when the point is fixed by the twisting map, and prove that translation maps induce isomorphisms between Hom-group retracts at different fixed base points. We introduce three equivalent notions of Hom-trusses and investigate their structural properties. We also propose three variants of Hom-braces and establish their correspondence with Hom-trusses, showing that certain Hom-trusses naturally give rise to Hom-braces and conversely. These results provide a unified framework extending heap and truss theory to the Hom-algebraic setting, with potential applications to the Yang--Baxter equation and non-associative geometry.

math.RA↗

arXiv2025

Cohomology and Extensions of $C_p$-Green Functors of Lie Type

We develop a theory of $C_p$-Green functors of Lie type, unifying the axiomatic framework of Green functors with the structure of Lie algebras under the action of a cyclic group $C_p$ of prime order. Extending classical notions from representation theory and topology, we define tensor and exterior products, introduce an equivariant Chevalley-Eilenberg cohomology, and construct cup products that endow the cohomology with a graded Green functor of Lie type structure. A key result establishes a correspondence between equivalence classes of singular extensions and second cohomology groups, generalizing classical Lie algebra extension theory to the equivariant setting. This framework enriches the toolkit for studying equivariant algebraic structures and paves the way for further applications in deformation theory, homotopical algebra, and representation theory.

math.RA↗