Osaka Journal of Mathematics (00306126)60(3)pp. 613-635
Extended affine Lie superalgebras are super versions of extended affine Lie algebras and, more generally, invariant affine reflection algebras. By employing a method known as “affinization”, we construct several classes of extended affine Lie superalgebras of arbitrary nullity. © 2023, Osaka University. All rights reserved.
Journal of Algebra (00218693)634pp. 1-43
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 by a type-dependent argument the existence of Chevalley involutions for centerless Lie tori. © 2023 Elsevier Inc.
Azam, S.,
Soltani, M.B.,
Tomie, M.,
Yoshii, Y. Communications in Algebra (00927872)(6)
We characterize reflectable bases for affine reflection systems (extended affine root systems) of types (Formula presented.) This completes the characterization of reflectable bases for reduced affine reflections systems. We prove that a subset of non-isotropic roots is a reflectable base if and only if it is a minimal generating set for the root lattice. Our approach also provides a new less involved proof for simply laced types different from (Formula presented.) Using our characterization, we enumerate reflectable bases for the finite type E 6. © 2022 Taylor & Francis Group, LLC.
Azam, S.,
Soltani, M.B.,
Tomie, M.,
Yoshii, Y. Communications in Algebra (00927872)50(6)pp. 2694-2718
We characterize reflectable bases for affine reflection systems (extended affine root systems) of types E-6,E-7,E-8. This completes the characterization of reflectable bases for reduced affine reflections systems. We prove that a subset of non-isotropic roots is a reflectable base if and only if it is a minimal generating set for the root lattice. Our approach also provides a new less involved proof for simply laced types different from E-6,E- 7,E- 8. Using our characterization, we enumerate reflectable bases for the finite type E-6.
Azam, S.,
Farahmand parsa, A.,
Farhadi, M.I. Journal of Algebra (00218693)597pp. 116-161
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. © 2022 Elsevier Inc.
Journal of Lie Theory (09495932)31(2)pp. 413-438
We combine the covering theory of graphs introduced by Malnic, Nedela and Skoviera, the notion of a Cayley graph and the theory of reflection systems in order to obtain a new characterization of geometric reflections in the theory of extended affine Weyl groups. As an immediate byproduct, we recover that an extended affine Weyl group of nullity greater than one is not a Coxeter group, with respect to any minimal generating set. © 2021 Heldermann Verlag.
Osaka Journal of Mathematics (00306126)58(3)pp. 563-589
We classify isomorphism and similarity classes of pointed reflection spaces of residue ≤ 2. This leads to the classification of reduced extended affine root systems whose involved pointed reflection spaces have residue ≤ 2. © 2021, Osaka University. All rights reserved.
Bulletin Of The Iranian Mathematical Society (10186301)47(SUPPL 1)pp. 1-4
Azam, S.,
Soltani, M.B.,
Tomie, M.,
Yoshii, Y. Publications of the Research Institute for Mathematical Sciences (00345318)(4)
We classify the reflectable bases of root systems of types A, B and D. We give a graph- theoretical characterization of reflectable bases and count the number of reflectable bases for each type. We also give a Dynkin-type characterization of reflectable bases of root systems of types A and D. © 2019 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
Azam, S.,
Soltani, M.B.,
Tomie, M.,
Yoshii, Y. Publications of the Research Institute for Mathematical Sciences (16634926)55(4)pp. 689-736
We classify the reflectable bases of root systems of types A, B and D. We give a graph-theoretical characterization of reflectable bases and count the number of reflectable bases for each type. We also give a Dynkin-type characterization of reflectable bases of root systems of types A and D.
Journal of Algebra and its Applications (02194988)18(3)
We establish extensions of some important features of affine theory to affine reflection systems (extended affine root systems) of type A1. We present a positivity theory which decomposes in a natural way the nonisotropic roots into positive and negative roots, then using that, we give an extended version of the well-known exchange condition for the corresponding Weyl group, and finally give an extended version of the Bruhat ordering and the Z-Lemma. Furthermore, a new presentation of the Weyl group in terms of the parity permutations is given, this in turn leads to a parity theorem which gives a characterization of the reduced words in the Weyl group. All root systems involved in this work appear as the root systems of certain well-studied Lie algebras. © 2019 World Scientific Publishing Company.
Journal of Lie Theory (09495932)29(1)pp. 247-262
We develop the notion of “extended multi-loop algebras” and determine their derivation algebras. Extended multi-loop algebras appear naturally as the core modulo center of locally extended affine Lie algebras; they are in fact an extension of n-step multi-loop algebras where the number of automorphisms are allowed to be possibly infinite and also the coordinate algebras (Laurent polynomials) are allowed to be over an infinite number of variables. © 2019 Heldermann Verlag.
Communications in Algebra (00927872)47(7)pp. 2549-2576
Extended affine Lie superalgebras are super versions of the defining axioms of extended affine Lie algebras or more generally invariant affine reflection algebras. This class includes finite dimensional basic classical simple Lie superalgebras and affine Lie superalgebras. In this paper, an affinization process is introduced for the class of extended affine Lie superalgebras, and the necessary conditions for an extended affine Lie superalgebra to be invariant under this process are presented. Moreover, new extended affine Lie superalgebras are constructed by means of the affinization process. © 2019, © 2019 Taylor & Francis Group, LLC.
Publications of the Research Institute for Mathematical Sciences (00345318)55(3)pp. 627-649
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. © 2019 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
Journal of Algebra and its Applications (02194988)16(8)
In the past two decades there has been 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 us to provide new examples of Malcev (super)algebras which extend the known Kac-Moody Malcev (super)algebras. © 2017 World Scientific Publishing Company.
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.
Algebra Colloquium (10053867)22(4)pp. 621-638
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 A1. 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. © 2015 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University.
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.
Journal of Pure and Applied Algebra (00224049)219(10)pp. 4422-4440
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 g0--module) over fields of characteristic zero into ideals. © 2015 Elsevier B.V.
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.
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.
Publications of the Research Institute for Mathematical Sciences (16634926)49(1)pp. 123-153
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 nontrivial presentation given in this generality, and can be considered as a model for other types.
Publications of the Research Institute for Mathematical Sciences (00345318)(1)
We offer a presentation for the Weyl group of an affine reection system R of type A1 as well as a presentation for the so called hyperbolic Weyl group associated with an affine reection system of type A1. 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 A1. Our presentation for the (hyperbolic) Weyl group of an affine reection system of type A1 is the first nontrivial presentation given in this generality, and can be considered as a model for other types. © 2013 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
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 Algebra and its Applications (17936829)11(3)
We study a combinatorial approach of producing new root systems from the old ones in the context of affine root systems and their new generalizations. The appearance of this approach in the literature goes back to the outstanding work of Kac in the realization of affine KacMoody Lie algebras. In recent years, this approach has been appeared in many other works, including the study of affinization of extended affine Lie algebras and invariant affine reflection algebras. © 2012 World Scientific Publishing Company.
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
Algebra Colloquium (10053867)16(3)pp. 381-396
We study the fixed point subalgebra of a centerless irreducible Lie torus under a certain finite order automorphism. We investigate which axioms of a Lie torus hold for the fixed points and which do not. We relate our study to some recent results about the fixed points of extended affine Lie algebras under a class of finite order automorphisms. © 2009 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University.
Communications in Algebra (15324125)36(3)pp. 905-927
It is known that under certain finite dimensionality condition the derivation algebra of tensor product of two algebras can be obtained in terms of the derivation algebras and the centroids of the involved algebras. We extend this theorem to infinite dimensional case and as an application, we determine the derivation algebra of the fixed point algebra of the tensor product of two algebras, with respect to the tensor product of two finite order automorphisms. These provide the framework for calculating the derivations of some infinite dimensional Lie algebras. Copyright © Taylor & Francis Group, LLC.
Journal of Algebraic Combinatorics (15729192)28(4)pp. 481-493
It is known that elliptic Weyl groups, extended affine Weyl groups of nullity 2, have a finite presentation called the generalized Coexter presentation. Similar to the finite and affine case this presentation is obtained by assigning a Dynkin diagram to the root system. Then there is a prescription to read the generators and relations from the diagram. Recently a similar presentation is given for simply laced extended affine Weyl groups of nullity 3 and rank>1. Employing a new method, we complete this work by giving a similar presentation for nullity 3 extended affine Weyl groups of type A 1. © 2007 Springer Science+Business Media, LLC.
Publications of the Research Institute for Mathematical Sciences (16634926)44(1)pp. 131-161
We give a finite presentation for reduced non-simply laced extended affine Weyl groups of arbitrary nullity. When nullity is less than or equal to 3, this presentation reduces to a very simple presentation in which the generators and relations can be easily read from a set of data attached to the root system. © 2008 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
Journal of Algebra (1090266X)319(5)pp. 1932-1953
There is a well-known presentation for finite and affine Weyl groups called the presentation by conjugation. Recently, it has been proved that this presentation holds for certain sub-classes of extended affine Weyl groups, the Weyl groups of extended affine root systems. In particular, it is shown that if nullity is ≤2, an A1-type extended affine Weyl group has the presentation by conjugation. We set up a general framework for the study of simply laced extended affine Weyl groups. As a result, we obtain certain necessary and sufficient conditions for an A1-type extended affine Weyl group of arbitrary nullity to have the presentation by conjugation. This gives an affirmative answer to a conjecture that there are extended affine Weyl groups which are not presented by "presentation by conjugation.". © 2007 Elsevier Inc. All rights reserved.
Forum Mathematicum (14355337)19(6)pp. 1029-1045
This article is about the derivation algebra of multi-loop algebras. Multi-loop algebras are algebras obtained by a generalization of a process known as twisting by automorphisms in the theory of KacMoody algebras. Multi-loop algebras are used in the realization of extended affine Lie algebras. Under certain conditions on an algebra , we determine the derivation algebra of an n-step multi-loop algebra based on as the semidirect product of a multi-loop algebra based on the derivation algebra of and the derivation algebra of the Laurent polynomials in n-variables. This in particular determines the derivation algebras of the core modulo center of (almost all) extended affine Lie algebras. © Walter de Gruyter 2007.
Publications of the Research Institute for Mathematical Sciences (00345318)43(2)pp. 403-424
Extended affine Weyl groups are the Weyl groups of extended affine root systems. Finite presentations for extended affine Weyl groups are known only for nullities ≤ 2, where for nullity 2 there is only one known such presentation. We give a finite presentation for the class of simply laced extended affine Weyl groups. Our presentation is nullity free if rank > 1 and for rank 1 it is given for nullities ≤ 3. The generators and relations are given uniformly for all types, and for a given nullity they can be read from the corresponding finite Cartan matrix and the semilattice involved in the structure of the root system. © 2007 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
Canadian Journal of Mathematics (0008414X)58(2)pp. 225-248
We investigate a class of Lie algebras which we call generalized reductive Lie algebras. These are generalizations of semi-simple, reductive, and affine Kac-Moody Lie algebras. A generalized reductive Lie algebra which has an irreducible root system is said to be irreducible and we note that this class of algebras has been under intensive investigation in recent years. They have also been called extended affine Lie algebras. The larger class of generalized reductive Lie algebras has not been so intensively investigated. We study them in this paper and note that one way they arise is as fixed point subalgebras of finite order automorphisms. We show that the core modulo the center of a generalized reductive Lie algebra is a direct sum of centerless Lie tori. Therefore one can use the results known about the classification of centerless Lie tori to classify the cores modulo centers of generalized reductive Lie algebras. ©Canadian Mathematical Society 2006.
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.
Journal of Algebra (00218693)269(2)pp. 508-527
In this paper we study the structure of the Weyl groups of nonreduced extended affine root systems. We show that similar to the case of reduced types, an extended affine Weyl group W of type BCℓ is semidirect product of a finite Weyl group W (of type Bℓ ) and a Heisenberg-like normal subgroup H which is also a characteristic subgroup of W. Moreover, H is of the form H = HnH0, where both Hn and H0 are normal subgroups of H with Hn ∩ H0 ≠{1}, Hn is naturally isomorphic to the root lattice of a finite root system of type BCℓ. Furthermore, the semidirect product of W and Hn is isomorphic to the Weyl group of a Kac-Moody affine subroot system of R of type BCℓ. © 2003 Published by Elsevier Inc.
Journal of Lie Theory (09495932)12(2)pp. 515-527
There are two notions of the extended affine root systems in the literature which both are introduced axiomatically. One, extended affine root system (SAERS for short), consists only of nonisotropic roots, while the other, extended affine root system (EARS for short), contains certain isotropic roots too. We show that there is a one to one correspondence between (reduced) SEARSs and EARSs. Namely the set of nonisotropic roots of any EARS is a (reduced) SEARS, and conversely, there is a unique way of adding certain isotropic elements to a SEARS to get an EARS. (It is known that some of extended affine root systems are not the root system of any extended affine Lie algebra.).
Communications in Algebra (00927872)28(1)pp. 465-488
In 1985 K. Saito [Sa1] introduced the concept of an extended affine Weyl group (EAWG), the Weyl group of an extended affine root system (EARS). In [A2, Section 5], we gave a presentation called "a presentation by conjugation" for the class of EAWGs of index zero, a subclass of EAWGs. In this paper we will givo a presentation which we call a "generalized presentation by conjugation" for the class of reduced EAWGs. If the extended affine Weyl group is of index zero this presentation reduces to "a presentation by conjugation". Our main result states that when the nullity of the EARS is 2, these two presentations coincide that is, EAWGs of nullity 2 have "a presentation by conjugation". In [ST] another presentation for EAWGs of nullity 2 is given. Copyright © 2000 by Marcel Dekker, Inc.
Communications in Algebra (00927872)28(6)pp. 2753-2781
We first give a characterization of the core (modulo its center) of an extended affine Lie algebra and then use this characterization to show that as in the case of affine Kac-Moody Lie algebras, many of the known examples of EALAs can be constructed from standard examples by a process known as "twisting".
Journal of Algebra (00218693)214(2)pp. 571-624
In this paper we study the Weyl groups of reduced extended affine root systems, the root systems of extended affine Lie algebras. We start by describing the extended affine Weyl group as a semidirect product of a finite Weyl group and a Heisenberg-like normal subgroup. This provides a unique expression for the Weyl group elements (in terms of some naturally arisen transformations) which is crucial in the further study of extended affine Weyl groups. We use this to give a presentation, called a presentation by conjugation, for an important subclass of extended affine Weyl groups. Using a new notion, called the index which is an invariant of the extended affine root systems, we show that one of the important features of finite and affine root systems (related to Weyl group) holds for the class of extended affine root systems. © 1999 Academic Press.
Journal of Algebra (00218693)222(1)pp. 174-189
Extended affine Weyl groups are the Weyl groups of root systems of a new class of Lie algebras called extended affine Lie algebras. In this paper we show that a (reduced) extended affine Weyl group is the homomorphic image of some indefinite Kac-Moody Weyl group where the homomorphism and its kernel are given explicitly. © 1999 Academic Press.
Allison, B.N.,
Azam, S.,
Berman, S.,
Gao, Y.,
Pianzola, A. Memoirs of the American Mathematical Society (00659266)(603)pp. 1-122
