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Malihe Yousofzadeh

Publications and source records attributed to Malihe Yousofzadeh.

At least 19 recordsLinked to original sources
arXiv2026

Action of free fermions on Symmetric Functions

The Clifford algebra of the endomorphisms of the exterior algebra of a countably dimensional vector space induces natural {\em bosoni}c shadows, i.e. families of linear maps between the cohomologies of complex Grassmannians. The main result of this paper is to provide a determinantal formula expressing generating functions of such endomorphisms unifying several classical special cases. For example the action over a point recovers the Jacobi-Trudy formula in the theory of symmetric functions or the Giambelli's one in classical Schubert calculus, whereas the action of degree preserving endomorphisms take into account a finite type version of the Date-Jimbo-Kashiwara-Miwa bosonic vertex operator representation of the Lie algebra $gl(\infty)$. The fermionic actions on (finite type) bosonic spaces is described in terms of the classical theory of symmetric functions. The main guiding principle is the fact that the exterior algebra is a (non irreducible) representation of the ring of symmetric functions, which is the way we use to spell the ``finite type'' Boson-Fermion correspondence.

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arXiv2026

Quasi-Poisson Modules and Harish-Chandra AD-Modules

We introduce the notion of quasi-Poisson modules over Lie-Rinehart pairs and prove that for the Lie-Rinehart pair $(\dot A,\dot\fk)$ in which $\dot A=\bbbc[t_1^{\pm1},\ldots,t_m^{\pm1}]\ot\Lam_n$ and $\dot\fk={\rm Der}(\dot A)$, there is a one-to-one correspondence between simple cuspidal quasi-Poisson modules over $(\dot A,\dot\fk)$ and simple cuspidal Harish-Chndra $A\fk$-modules for $A:=\bbbc[t_0^{\pm1}]\ot \dot A$ and $\fk:={\rm Der}(A).$ We also classify simple cuspidal quasi-Poisson modules over the Lie-Rinehart pair $(\dot A,\dot\fk)$ and show that each such module is a tensor module $\dot A\ot Ω$ for an admissible $\frak{gl}(m+1,n)$-module $Ω$ via a prescribed action.

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arXiv2026

Quasi-integrable modules over affine Lie superalgebras (Critical level)

Representation theory of Lie (super)algebras has attracted significant research interest for many years, especially due to its applications in theoretical physics; in this regard, the representation theory of affine Lie (super)algebras is of central importance. To characterize simple modules over affine Lie (super)algebras, it is necessary to study the cases of nonzero and critical levels separately. Although a vast amount of research has been done on the representation theory of affine Lie (super)algebras $\mathcal{L}$, investigations concerning general modules at the critical level remain limited. In all existing studies, the characterization of the modules under consideration is reduced to the characterization of modules over some subalgebras of $\mathcal{L}$. Depending on the structure of the original modules, these subalgebras -- and the corresponding modules -- have different natures some of which are already known, while others need to be studied separately. In this paper, we give a complete characterization of the modules over specific subalgebras $\mathcal{G}$ of a twisted affine Lie superalgebra $\mathcal{L}$ that arise in the study of general zero level simple finite weight $\mathcal{L}$-modules. In particular, in the special case that $\\mathcal{G} = \mathcal{L}$, we obtain a complete characterization of quasi-integrable $\mathcal{L}$-modules of level zero.

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arXiv2026

Zero-level integrable modules over twisted affine Lie superalgebras

The main result of this paper is the characterization of zero-level integrable finite weight modules, over twisted affine Lie superalgebras. We prove that such a module is parabolically induced from a module which is obtained, in a prescribed way, from a module over a Lie algebra $\mathscr{L}$ which is either a $\bbbz$-graded abelian Lie algebra or a direct sum of a $\bbbz$-graded abelian Lie algebra and the so-called quadratic Lie superalgebra $\mathcal{Q}$. We give also a complete characterization of both finite dimensional simple $\mathcal{Q}$-modules as well as bounded finite weight $\bbbz$-graded simple $\mathcal{Q}$-modules.

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arXiv2024

On results of certain modules over untwisted affine Liesuperalgebras

Since 2020, finite weight modules have been studied over twisted affine Lie superalgebras. To complete the characterization of modules over affine Lie superalgebras, we need some information regarding modules over untwisted affine Lie superalgebras. There are several known results on representations of twisted affine Lie superalgebras that hold for untwisted cases, and their proofs are just a minor modification of the known ones. In this note, we gather these results for our further use.

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arXiv2022

Quasi-integrable modules, a class of non-highest weight modules over twisted affine Lie superalgebras

In this paper, we characterize quasi-integrable modules, of nonzero level, over twisted affine Lie superalgebras. We show that quasi-integrable modules are not necessarily highest weight modules. We prove that each quasi-integrable module is parabolically induced from a cuspidal module, over a finite dimensional Lie superalgebra having a Cartan subalgebra whose corresponding root system just contain real roots; in particular, the classification of quasi-integrable modules is reduced to the known classification of cuspidal modules over such Lie superalgebras.

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arXiv2021

Tight irreducible finite weight modules over twisted affine Lie superalgebras

For a twisted affine Lie superalgebra with nonzero odd part, we study {tight irreducible weight modules} with bounded weight multiplicities and show that if the action of nonzero real vectors of each affine component of the zero part is neither completely injective nor completely locally nilpotent, then these modules are parabolically induced.

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arXiv2020

Finite weight modules over twisted affine Lie superalgebras

This work provides the first step toward the classification of irreducible finite weight modules over twisted affine Lie superalgebras. We study all such modules whether the canonical central element acts as a nonzero multiple of the identity map or not. Moreover, we reduce the classification of some subclasses of irreducible finite weight modules to the classification of cuspidal modules of finite dimensional cuspidal superalgebras which is known by a work of Dimitrov, Mathieu and Penkov.

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arXiv2019

Classification of bases of twisted affine root supersystems

Following the definition of a root basis of an affine root system, we define a base of the root system of an affine Lie superalgebra to be a linearly independent subset $B$ of its root system such that each root can be written as a linear combination of elements of $B$ with integral coefficients such that all coefficients are nonnegative or all coefficients are nonpositive. Characterization and classification of bases of root systems of affine Lie algebras are known in the literature; in fact, up to $\pm 1$-multiple, each base of an affine root system is conjugate with the standard base under the Weyl group action. In the super case, the existence of those self-orthogonal roots which are not orthogonal to at least one other root, makes the situation more complicated. In this work, we give a complete characterization of bases of a twisted affine root supersystem. We precisely describe and classify them.

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arXiv2018

Extended Affine Root Supersystems

The interaction of a Lie algebra $\LL,$ having a weight space decomposition with respect to a nonzero toral subalgebra, with its corresponding root system forms a powerful tool in the study of the structure of $\LL.$ This, in particular, suggests a systematic study of the root system apart from its connection with the Lie algebra. Although there have been a lot of researches in this regard on Lie algebra level, such an approach has not been considered on Lie superalgebra level. In this work, we introduce and study extended affine root supersystems which are a generalization of both affine reflection systems and locally finite root supersystems. Extended affine root supersystems appear as the root systems of the super version of extended affine Lie algebras and invariant affine reflection algebras including affine Lie superalgebras. This work provides a framework to study the structure of this kind of Lie superalgebras refereed to as extended affine Lie superalgebras.

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arXiv2017

Current superalgebras and unitary representations

In this paper we determine the projective unitary representations of finite dimensional Lie supergroups whose underlying Lie superalgebra is $\frak{g} = A \otimes \frak{k}$, where $\frak{k}$ is a compact simple Lie superalgebra and $A$ is a supercommutative associative (super)algebra; the crucial case is when $A = Λ_s(\mathbb{R})$ is a Graßmann algebra. Since we are interested in projective representations, the first step consists of determining the cocycles defining the corresponding central extensions. Our second main result asserts that, if $\frak{k}$ is a simple compact Lie superalgebra with $\frak{k}_1\neq \{0\}$, then each (projective) unitary representation of $Λ_s(\mathbb{R})\otimes \frak{k}$ factors through a (projective) unitary representation of $\frak{k}$ itself, and these are known by Jakobsen's classification. If $\frak{k}_1 = \{0\}$, then we likewise reduce the classification problem to semidirect products of compact Lie groups $K$ with a Clifford--Lie supergroup which has been studied by Carmeli, Cassinelli, Toigo and Varadarajan.

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arXiv2015

Root Graded Lie Superalgebras

We define root graded Lie superalgebras and study their connection with centerless cores of extended affine Lie superalgebras; our definition generalizes the known notions of root graded Lie superalgebras.

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arXiv2015

Extended Affine Lie Superalgebras

We introduce the notion of extended affine Lie superalgebras and investigate the properties of their root systems. Extended affine Lie algebras, invariant affine reflection algebras, finite dimensional basic classical simple Lie superalgebras and affine Lie superalgebras are examples of extended affine Lie superalgebras.

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arXiv2014

Locally Finite Root Supersystems

We introduce the notion of locally finite root supersystems as a generalization of both locally finite root systems and generalized root systems. We classify irreducible locally finite root supersystems.

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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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