Journal of Algebraic Combinatorics (09259899)55(3)pp. 919-978
Following the definition of a root basis of an affine root system, we define a base of the root system R of an affine Lie superalgebra to be a linearly independent subset B of the linear span of R such that B⊆ R and each root can be written as a linear combination of elements of B with integral coefficients such that either all coefficients are nonnegative or all coefficients are non-positive. Characterization and classification of bases of root systems of affine Lie algebras are known in the literature; in fact, up to ± 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 the root systems of twisted affine Lie superalgerbas with nontrivial odd part. We precisely describe and classify them. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
Journal of Algebra (00218693)609pp. 407-436
Over the past three decades, representation theory of affine Lie (super)algebras has been a research spotlight in both areas Mathematics and Physics. The even parts of almost all affine Lie superalgebras L contain two affine Lie algebras; say L0(1) and L0(2). Irreducible finite weight modules (f.w.m's for short) are the first interesting class of modules over such an L. The structure of these modules strongly depends on whether of root vectors corresponding to real roots of L0(1) and L0(2) act locally nilpotently or not. We know that there is no nonzero irreducible f.w.m. of nonzero level over L for which root vectors corresponding to nonzero real roots of both L0(1) and L0(2) act locally nilpotently. This leads us to define quasi-integrable modules and study the case that root vectors corresponding to nonzero real roots of one of L0(1) and L0(2) act locally nilpotently. We give a recognition theorem for quasi-integrable modules over twisted affine Lie superalgebras. © 2022 Elsevier Inc.
Journal of Algebra (00218693)606pp. 19-29
In this paper, we study representations of a direct sum of Lie superalgebras. We deal with the question of whether the irreducible representations of a finite direct sum of Lie superalgebras are obtained in case representations of each constituent are available. We give an affirmative answer to this question for finite weight representations of a class of Lie superalgebras containing basic classical simple Lie superalgebras as well as affine Lie (super)algebras. © 2022 Elsevier Inc.
Journal of Algebra (00218693)570pp. 636-677
In 1986, Van de Leur introduced and classified affine Lie superalgebras. An affine Lie superalgebra is defined as the quotient of certain Lie superalgebra G defined by generators and relations, corresponding to a symmetrizable generalized Cartan matrix, over the so-called radical of G. Because of the interesting applications of affine Lie (super)algebras in combinatorics, number theory and physics, it is very important to recognize how far a Lie (super)algebra is to be an affine Lie (super)algebra. In this regard, we determine affine Lie superalgebras axiomatically. © 2020
Journal of Pure and Applied Algebra (00224049)225(10)
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. © 2021 Elsevier B.V.
Communications in Algebra (00927872)48(9)pp. 3673-3689
In this paper, we determine the derivations of affine Lie superalgebras. As the nature of affine Lie superalgebra of type (Formula presented.) is slightly different from the nature of the other types, we study separately the derivations for type (Formula presented.) and (Formula presented.) We precisely determine derivations up to inner derivations; in the former type, we get a four-dimensional vector space while in the latter one, we get a one-dimensional vector space. © 2020, © 2020 Taylor & Francis Group, LLC.
Journal of Algebra (00218693)564pp. 436-479
This work provides the first step toward the classification of irreducible finite weight modules over twisted affine Lie superalgebras. We divide the class of such modules into two subclasses called hybrid and tight. We reduce the classification of hybrid irreducible finite weight modules to the classification of cuspidal modules of finite dimensional cuspidal Lie superalgebras which is discussed in a work of Dimitrov, Mathieu and Penkov. © 2020 Elsevier Inc.
Journal of Pure and Applied Algebra (00224049)224(4)
We identify the universal central extension of g=A⊗k, where k is a finite dimensional perfect Lie superalgebra equipped with a nondegenerate homogeneous invariant supersymmetric bilinear form κ which is invariant under all derivations and A is a unital supercommutative associative (super)algebra. © 2019 Elsevier B.V.
Journal of Algebra (00218693)532pp. 61-79
Multiloop (super)algebras are of great importance in the literature. Affine Kac Moody Lie (super)algebras and almost all extended affine Lie algebras are realized using an affinization process on loop superalgebras and multiloop algebras respectively. An affinization process for multiloop superalgebras turns to Lie superalgebras satisfying certain properties which are the super version of extended affine Lie algebras. In this paper we systematically study this super version called extended affine Lie superalgebras. We first derive the intrinsic properties of extended affine Lie superalgebras and then study those which are simple. © 2019
Journal of Algebra (00218693)540pp. 42-62
We give an explicit description of extended affine root supersystems of type A(l, l) (l not equal 1). (C) 2019 Elsevier Inc. All rights reserved.
Journal of Pure and Applied Algebra (00224049)222(10)pp. 3303-3333
In this paper we determine the projective unitary representations of finite dimensional Lie supergroups whose underlying Lie superalgebra is g=A⊗k, where k is a compact simple Lie superalgebra and A is a supercommutative associative (super)algebra; the crucial case is when A=Λs(R) is a Graßmann algebra. Since we are interested in projective representations, the first step consists in determining the cocycles defining the corresponding central extensions. Our second main result asserts that, if k is a simple compact Lie superalgebra with k1≠{0}, then each (projective) unitary representation of Λs(R)⊗k factors through a (projective) unitary representation of k itself, and these are known by Jakobsen's classification. If k1={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. © 2017 Elsevier B.V.
Publications of the Research Institute for Mathematical Sciences (16634926)54(2)pp. 213-243
We characterize a class of root-graded Lie superalgebras containing a locally finite basic classical simple Lie superalgebra. As a by-product, we characterize the core and the core modulo the center of an extended affine Lie superalgebra whose root system is of finite type X ≠ A(m; n), B(m; n), BC(m; n), C(m; n). © 2018 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
Communications in Algebra (00927872)45(10)pp. 4292-4320
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. © 2017 Taylor & Francis.
Communications in Algebra (00927872)44(12)pp. 5426-5453
We describe the derivations of a direct limit 𝔏 of Lie superalgebras 𝔏i (i ∈ I) in an 𝔏-module 𝔲 as the inverse limit of the derivations of 𝔏i's in 𝔲. Using this, in case the first cohomology group of each 𝔏i with coefficients in 𝔲 is zero, we describe the derivations of 𝔏 in 𝔲 as the inverse limit of 𝔲/𝔲𝔏i (i ∈ I). This then allows us to compute the derivations of direct limits of finite-dimensional basic classical simple Lie superalgebras. © 2016, Copyright © Taylor & Francis Group, LLC.
Communications in Algebra (00927872)44(12)pp. 5309-5341
In this work, we consider realizations of locally extended affine Lie algebras, in the level of core modulo center. We provide a framework similar to the one for extended affine Lie algebras by "direct unions." Our approach suggests that the direct union of existing realizations of extended affine Lie algebras, in a rigorous mathematical sense, would lead to a complete realization of locally extended affine Lie algebras, in the level of core modulo center. As an application of our results, we realize centerless cores of locally extended affine Lie algebras with specific root systems of types A1, B, C, and BC. © Taylor & Francis Group, LLC.
Publications of the Research Institute for Mathematical Sciences (16634926)52(3)pp. 309-333
We introduce the notion of extended affine Lie superalgebras and investigate the properties of their root systems. Extended affine Lie algebras, invariant affine reection algebras, ffinite-dimensional basic classical simple Lie superalgebras and affine Lie superalgebras are examples of extended affine Lie superalgebras. © 2016 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
Journal of Algebra (00218693)449pp. 539-564
The interaction of a Lie algebra L, 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 L. 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. © 2015 Elsevier Inc.
Journal of Lie Theory (09495932)26(3)pp. 731-765
We define root graded Lie superalgebras and study their connec-tion with centerless cores of extended affine Lie superalgebras; our definition generalizes the known notions of root graded Lie superalgebras. © 2016 Heldermann Verlag.
Publications of the Research Institute for Mathematical Sciences (16634926)51(1)pp. 59-130
Let χ be a bi-homomorphism over an algebraically closed field of characteristic zero. Let U(χ) be a generalized quantum group, associated with χ, such that dim U+(χ) = ∞,|R+(χ) | < ∞, and R+(χ) is irreducible, where U+(χ) is the positive part of U(χ), and R+(χ) is the Kharchenko positive root system of U+(χ). In this paper, we give a list of finite-dimensional irreducible U(χ)-modules, relying on a special reduced expression of the longest element of the Weyl groupoid of R(χ) := R+(χ) ∪ (–R+(χ)). From the list, we explicitly obtain lists of finite-dimensional irreducible modules for simple Lie superalgebras g of types A-G and the (standard) quantum superalgebras Uq(g). An intrinsic gap appears between the lists for g and Uq(g), e.g, if g is B(m, n) or D(m, n). © 2015 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
Frontiers of Mathematics in China (16733452)10(3)pp. 477-509
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, the Hermitian type, the Clifford type, and the Albert type. We concretely describe Jordan G-tori of each type. © 2014, Higher Education Press and Springer-Verlag Berlin Heidelberg.
Publications of the Research Institute for Mathematical Sciences (16634926)49(4)pp. 801-829
We study central extensions of Lie algebras graded by an irreducible locally finite root system. © 2013 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
Osaka Journal of Mathematics (00306126)50(4)pp. 1039-1072
The class of invariant affine reflection algebras is the most general known extension of the class of affine Kac-Moody Lie algebras, introduced in 2008. We develop a method known as "affinization" for the class of invariant affine reflection algebras, and show that starting from an algebra belonging to this class together with a certain finite order automorphism, and applying the so called "affinization method", we obtain again an invariant affine reflection algebra. This can be considered as an important step towards the realization of invariant affine reflection algebras.
Journal of Algebra (00218693)371pp. 63-93
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. In this work, we consider the notion of a "reflectable base" for an affine reflection system R. A reflectable base for R is a minimal subset Π of roots such that the non-isotropic part of the root system can be recovered by reflecting roots of Π relative to the hyperplanes determined by Π. 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 by-product of our results, we show that if the root system under consideration is locally finite, then any reflectable base is an integral base. © 2012 Elsevier Inc.
Journal of Lie Theory (09495932)22(2)pp. 397-435
We give a complete description of the structure of Lie algebras graded by an infinite irreducible locally finite root system. © 2012 Heldermann Verlag.
Publications of the Research Institute for Mathematical Sciences (16634926)46(3)pp. 507-548
Using the well-known recognition and structural theorem(s) for root-graded Lie algebras and their universal coverings, we give a finite presentation for the universal covering algebra of a centerless Lie torus of type X not equal A, C, BC We follow a unified approach for the types under consideration
Osaka Journal of Mathematics (00306126)46(3)pp. 611-643
We study the subalgebra of fixed points of a root graded Lie algebra under a certain class of finite order automorphisms. As the centerless core of extended affine Lie algebras or equivalently irreducible centerless Lie tori are examples of root graded Lie algebras, our work is an extension of some recent result about the subalgebra of fixed points of a Lie torus under a certain finite order automorphism.
Publications of the Research Institute for Mathematical Sciences (16634926)44(1)pp. 1-14
We give a finite presentation of the universal covering algebra of a Lie torus of type Bl, l ≥ 3. © 2008 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
Communications in Algebra (00927872)35(12)pp. 4277-4302
We introduce a new class of possibly infinite dimensional Lie algebras and study their structural properties. Examples of this new class of Lie algebras are finite dimensional simple Lie algebras containing a nonzero split torus, affine and extended affine Lie algebras. Our results generalize well-known properties of these examples.
Journal of Lie Theory (09495932)15(1)pp. 145-181
We classify the BC-type extended affine root systems for nullity ≤ 3, in its most general sense. We show that these abstractly defined root systems are the root systems of a class of Lie algebras which are axiomatically defined and are closely related to the class of extended affine Lie algebras.
Journal of Algebra (00218693)287(2)pp. 351-380
It is a well-known result that the fixed point subalgebra of a finite dimensional complex simple Lie algebra under a finite order automorphism is a reductive Lie algebra so it is a direct sum of finite dimensional simple Lie subalgebras and an abelian subalgebra. We consider this for the class of extended affine Lie algebras and are able to show that the fixed point subalgebra of an extended affine Lie algebra under a finite order automorphism (which satisfies certain natural properties) is a sum of extended affine Lie algebras (up to existence of some isolated root spaces), a subspace of the center and a subspace which is contained in the centralizer of the core. Moreover, we show that the core of the fixed point subalgebra modulo its center is isomorphic to the direct sum of the cores modulo centers of the involved summands. © 2004 Elsevier Inc. All rights reserved.
