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Saeid Azam

Publications and source records attributed to Saeid Azam.

At least 19 recordsLinked to original sources
arXiv2025

Localization and filtration in extended affine Lie algebras

We investigate the notions of \emph{localization} and \emph{filtration} in the context of extended affine Lie algebras. Our primary objective is to develop a localization theory that facilitates the construction of meaningful local substructures, particularly local affine Lie subalgebras. These subalgebras play a crucial role in understanding the global structure of extended affine Lie algebras. It is noteworthy that the existence of appropriate local subalgebras, particularly affine Lie subalgebras, is also fundamental to the representation theory of extended affine Lie algebras. As a natural outcome of our localization approach, we also introduce a formal notion of filtration for a given extended affine Lie algebra. This study is motivated by our interest in modular theory, specifically the integral structures of extended affine Lie algebras.

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arXiv2025

Chevalley bases for extended affine Lie algebras

Claude Chevalley provided a basis for a {finite dimensional} simple complex Lie algebra called the Chevalley basis. This basis has the distinguishing property that all the structure constants are integers. Chevalley groups, which are similar to Lie groups but over finite fields, can be constructed using these bases. Parallel results also hold in affine Lie algebras. We develop a uniform theory of Chevalley bases for extended affine Lie algebras of an arbitrary type consistent with the ordinary theory for finite and affine cases. It explains how a Chevalley basis for a finite-dimensional simple Lie algebra or an affine Lie algebra can be extended to one for the covering extended affine Lie algebras.

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arXiv2025

Involutive root-graded Lie algebras and Lie tori of type $A$

We investigate the concept of a ``Chevalley involution'' within the framework of root-graded Lie algebras with compatible grading. We provide a characterization of all centerless Lie tori of type $A_\ell(\ell\geq2)$ admitting a Chevalley involution. This work extends and completes the earlier results regarding the existence of such involutions for Lie tori of reduced types.

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arXiv2025

Chevalley bases for elliptic extended affine Lie algebras of type $A_1$

We investigate Chevalley bases for extended affine Lie algebras of type $A_1$.The concept of integral structures for extended affine Lie algebras of rank greater than one has been successfully explored in recent years. However, for the rank one it has turned out that the situation becomes more delicate. In this work, we consider $A_1$-type extended affine Lie algebras of {nullity} $2$, known as elliptic extended affine Lie algebras. These Lie algebras are build using the Tits-Kantor-Koecher (TKK) construction by applying some specific Jordan algebras: the plus algebra of a quantum torus, the Hermitian Jordan algebra of the ring of Laurent polynomials equipped with an involution, and the Jordan algebra associated with a semilattice. By examining these ingredient we determine appropriate bases for null spaces of the corresponding elliptic extended affine Lie algebra leading to the establishment of Chevalley bases for these Lie algebras.

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arXiv2024

Characters for extended affine Lie algebras; a combinatorial approach

The behavior of objects associated with general extended affine Lie algebras is typically distinct from their counterparts in affine Lie algebras. Our research focuses on studying characters and Cartan automorphisms, which appear in the study of Chevalley involutions and Chevalley bases for extended affine Lie algebras. We show that for almost all extended affine Lie algebras, any finite order Cartan automorphism is diagonal, and its corresponding combinatorial map is a character.

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arXiv2023

A characterization of minimal extended affine root systems (Relations to Elliptic Lie Algebras)

Extended affine root systems appear as the root systems of extended affine Lie algebras. A subclass of extended affine root systems, whose elements are called ``minimal" turns out to be of special interest mostly because of the geometric properties of their Weyl groups; they possess the so-called ``presentation by conjugation". In this work, we characterize minimal extended affine root systems in terms of ``minimal reflectable bases" which resembles the concept of the ``base" for finite and affine root systems. As an application, we construct elliptic Lie algebras by means of Serre's type generators and relations.

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arXiv2023

Chevalley involutions for Lie tori and extended affine Lie algebras

In finite-dimensional simple Lie algebras and affine Kac-Moody Lie algebras, Chevalley involutions are crucial ingredients of the modular theory. Towards establishing the modular theory for extended affine Lie algebras, we investigate the existence of ``Chevalley involutions" for Lie tori and extended affine Lie algebras. We first discuss how to lift a Chevalley involution from the centerless core which is characterized to be a centerless Lie torus to the core and then to the entire extended affine Lie algebra. We then prove that each centerless Lie torus of reduced type admits a Chevalley involution.

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arXiv2021

Integral structures in extended affine Lie algebras

We construct certain integral structures for the cores of reduced tame extended affine Lie algebras of rank at least 2. One of the main tools to achieve this is a generalization of Chevalley automorphisms in the context of extended affine Lie algebras. As an application, groups of extended affine Lie type associated to the adjoint representation are defined over arbitrary fields.

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arXiv2019

Groups of extended affine Lie type

We construct certain Steinberg groups associated to extended affine Lie algebras and their root systems. Then by the integration methods of Kac and Peterson for integrable Lie algebras, we associate a group to every tame extended affine Lie algebra. Afterwards, we show that the extended affine Weyl group of the ground Lie algebra can be recovered as a quotient group of two subgroups of the group associated to the underlying algebra similar to Kac-Moody groups.

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arXiv2015

A new characterization of Kac-Moody-Malcev superalgebras

In the past two decades there has been a great attention to Lie (super)algebras which are extensions of affine Kac-Moody Lie (super)algebras, in certain typical or axiomatic approaches. These Lie (super)algebras have been mostly studied under variations of the name "extended affine Lie (super)algebras". We show that certain classes of Malcev (super)algebras also can be put in this framework. This in particular allows to provide new examples of Malcev (super)algebras which extend the known Kac-Moody Malcev (super)algebras.

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arXiv2014

Finite dimensional compact and unitary Lie superalgebras

Motivated by the theory of unitary representations of finite dimensional Lie supergroups, we describe those Lie superalgebras which have a faithful finite dimensional unitary representation. We call these Lie superalgebras unitary. This is achieved by describing the classification of real finite dimensional compact simple Lie superalgebras, and analyzing, in a rather elementary and direct way, the decomposition of reductive Lie superalgebras ($\g$ is a semisimple $\g_{\bar 0}$-module) over fields of characteristic zero into ideals.

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arXiv2013

Jordan tori for a torsion free abelian group

We classify Jordan $G$-tori, where $G$ is any torsion-free abelian group. Using the Zelmanov prime structure theorem, such a class divides into three types, namely, {the Hermitian type, the Clifford type and the Albert type.} We concretely describe Jordan $G$-tori of each type.

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arXiv2012

Weyl Groups Associated with Affine Reflection Systems of Type $A_1$ (Coxeter Type Defining Relations)

In this paper, we offer a presentation for the Weyl group of an affine reflection system $R$ of type $A_1$ as well as a presentation for the so called hyperbolic Weyl group associated with an affine reflection system of type $A_1$. Applying these presentations to extended affine Weyl groups, and using a description of the center of the hyperbolic Weyl group, we also give a new finite presentation for an extended affine Weyl group of type $A_1$. Our presentation for the (hyperbolic) Weyl group of an affine reflection system of type $A_1$ is the first non-trivial presentation given in such a generality, and can be considered as a model for other types.

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arXiv2012

A Length Function for Weyl Groups of extended affine root systems of Type $A_1$

In this work, we study the concept of the length function and some of its combinatorial properties for the class of extended affine root systems of type $A_1$. We introduce a notion of root basis for these root systems, and using a unique expression of the elements of the Weyl group with respect to a set of generators for the Weyl group, we calculate the length function with respect to a very specific root basis.

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arXiv2011

Extended affinization of Invariant Affine Reflection Algebras

Invariant affine reflection algebras are the last and the most general known extension of affine Kac-Moody Lie algebras, introduced in recent years. We develop a method known as "affinization" to the class of invariant affine reflection algebras, and show that starting with an algebra from this class together with a certain finite order automorphism, and applying the so called "loop construction", we obtain again an invariant affine reflection algebra. This can be considered as an important step towards the realization of invariant affine reflection algebras.

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arXiv2011

Reflectable bases for affine reflection systems

The notion of a "root base" together with its geometry plays a crucial role in the theory of finite and affine Lie theory. However, it is known that such a notion does not exist for the recent generalizations of finite and affine root systems such as extended affine root systems and affine reflection systems. As an alternative, we introduce the notion of a "reflectable base", a minimal subset $\Pi$ of roots such that the non-isotropic part of the root system can be recovered by reflecting roots of $\Pi$ relative to the hyperplanes determined by $\Pi$. We give a full characterization of reflectable bases for tame irreducible affine reflection systems of reduced types, excluding types $E_{6,7,8}$. As a byproduct of our results, we show that if the root system under consideration is locally finite then any reflectable base is an integral base.

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arXiv2009

Extended affine Weyl groups: Presentation by conjugation via integral collection

We give several necessary and sufficient conditions for the existence of {\it the presentation by conjugation} for a non-simply laced extended affine Weyl group. We invent a computational tool by which one can determine simply the existence of the presentation by conjugation for an extended affine Weyl group. As an application, we determine the existence of the presentation by conjugation for a large class of extended affine Weyl groups.

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