Vertex representation of quantum N-toroidal algebra for type F4

Abstract

Recently Gao-Jing-Xia-Zhang defined the structures of quantum N-toroidal algebras uniformally, which are a kind of natural generalizations of the classical quantum toroidal algebras, just like the relation between 2-toroidal Lie algebras and N-toroidal Lie algebras. Based on this work, we construct a level-one vertex representation of quantum N-toroidal algebra for type F₄. In particular, we can also obtain a level-one vertex representation of quantum toroidal algebra for type F₄ as our special cases.     

Derivations nilpotent on subsets of prime rings

ABSTRACT

Let D be a division algebra with center F. Consider the group CK ₁(D) = D*/F*D′ where D* is the group of invertible elements of D and D′ is its commutator subgroup. In this note we shall show that, assuming a division algebra D is a product of cyclic algebras, the group CK ₁(D) is trivial if and only if D is an ordinary quaternion algebra over a real Pythagorean field F. We also characterize the cyclic central simple algebras with trivial CK ₁ and show that CK ₁ is not trivial for division algebras of index 4. Using valuation theory, the group CK ₁(D) is computed for some valued division algebras.

     

The root system and the core of an extended affine Lie algebra

Extended affine Lie algebras are higher nullity generalizations of finite dimensional simple Lie algebras and affine Kac Moody Lie algebras. In this paper we completely describe the structure of the core modulo its centre and the root system for extended affine Lie algebras of type B_l (l 3 3) B_l (l\ge 3) , C_l (l 3 2) C_l (l \ge 2), F ₄ and G ₂ .     

Constructions for Octonion and Exceptional Jordan Algebras

In this note we reverse theusual process of constructing the Lie algebras of types G ₂and F ₄ as algebras of derivations of the splitoctonions or the exceptional Jordan algebra and instead beginwith their Dynkin diagrams and then construct the algebras togetherwith an action of the Lie algebras and associated Chevalley groups.This is shown to be a variation on a general construction ofall standard modules for simple Lie algebras and it is well suitedfor use in computational algebra systems. All the structure constantswhich occur are integral and hence the construction specialisesto all fields, without restriction on the characteristic, avoidingthe usual problems with characteristics 2 and 3.     

F-Polynomials in Quantum Cluster Algebras

F-polynomials and g-vectors were defined by Fomin and Zelevinsky to give a formula which expresses cluster variables in a cluster algebra in terms of the initial cluster data. A quantum cluster algebra is a certain noncommutative deformation of a cluster algebra. In this paper, we define and prove the existence of analogous quantum F-polynomials for quantum cluster algebras. We prove some properties of quantum F-polynomials. In particular, we give a recurrence relation which can be used to compute them. Finally, we compute quantum F-polynomials and g-vectors for a certain class of cluster variables, which includes all cluster variables in type A_n\mbox{A}_{n} quantum cluster algebras.     

Varieties of almost polynomial growth: classifying their subvarieties

Let G be the infinite dimensional Grassmann algebra over a field F of characteristic zero and UT ₂ the algebra of 2 × 2 upper triangular matrices over F. The relevance of these algebras in PI-theory relies on the fact that they generate the only two varieties of almost polynomial growth, i.e., they grow exponentially but any proper subvariety grows polynomially. In this paper we completely classify, up to PI-equivalence, the associative algebras A such that A ∈ Var(G) or A ∈ Var(UT ₂).     

Radical of filters in BL‐algebras

In this paper, the notion of the radical of a filter in BL‐algebras is defined and several characterizations of the radical of a filter are given. Also we prove that A/F is an MV‐algebra if and only if D_s(A) ? F. After that we define the notion of semi maximal filter in BL‐algebras and we state and prove some theorems which determine the relationship between this notion and the other types of filters of a BL‐algebra. Moreover, we prove that A/F is a semi simple BL‐algebra if and only if F is a semi maximal filter of A. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

The decomposition numbers of Hecke algebras of typeF 4 with unequal parameters

The purpose of this paper is to calculate the decomposition numbers for Hecke algebras of typeF ₄ (andC ₃) with unequal parameters. The problem is reduced to specifying decomposition maps of the generic Hecke algebras. The concept of Kazhdan-Lusztig polynomials and left cells serves to determine their irreducible representations. We also prove a result about the minimal ring over which the irreducible representations of the generic algebras can be realized.     

Explicit irreducible representations of the¶Iwahori–Hecke algebra of Type F4

A general method for computing irreducible representations of Weyl groups and Iwahori–Hecke algebras was introduced by the first author in [10]. In that paper the representations of the algebras of types A _n , B _n , D _n and G _n were computed and it is the purpose of this paper to extend these computations to F ₄. The main goal here is to compute irreducible representations of the Iwahori–Hecke algebra of type F ₄ by only using information in the character table of the Weyl group. Received: Received: 30 July 1998     

The graded Gelfand--Kirillov dimension of verbally prime algebras

Let E be the infinite-dimensional Grassmann algebra over a field F of characteristic 0. In this article, we consider the verbally prime algebras M _n (F), M _n (E) and M _a,b (E) endowed with their gradings induced by that of Vasilovsky, and we compute their graded Gelfand--Kirillov dimensions.     

On indecomposable algebras of exponent 2

For any n ≥ 3 we give numerous examples of central division algebras of exponent 2 and index 2ⁿ over fields, which do not decompose into a tensor product of two nontrivial central division algebras, and which are sums of n + 1 quaternion algebras in the Brauer group of the field. Also, for any n ≥ 3 and any field k ₀ we construct an extension F/k ₀ and a multiquadratic extension L/F of degree 2ⁿ such that for any proper subextensions L ₁/F and L ₂/F

Split-By-Nilpotent Extensions Algebras and Stratifying Systems

Let Γ and Λ be artin algebras such that Γ is a split-by-nilpotent extension of Λ by a two sided ideal I of Γ. Consider the change of rings functors G: =_ΓΓ_Λ ?_Λ ? and F: =_ΛΛ_Γ ?_Γ ?. In this article, by assuming that I _Λ is projective, we find the necessary and sufficient conditions under which a stratifying system (Θ, ≤) in modΛ can be lifted to a stratifying system (GΘ, ≤) in mod(Γ). Furthermore, by using the functors F and G, we study the relationship between their filtered categories of modules; and some connections with their corresponding standardly stratified algebras are stated (see Theorem 5.12, Theorem 5.15 and Theorem 5.18). Finally, a sufficient condition is given for stratifying systems in mod(Γ) in such a way that they can be restricted, through the functor F, to stratifying systems in mod(Λ).     

Separable algebras over infinite fields are 2-generated and finitely presented

We prove that every separable algebra over an infinite field F admits a presentation with 2 generators and finitely many relations. In particular, this is true for finite direct sums of matrix algebras over F and for group algebras FG, where G is a finite group such that the order of G is invertible in F. We illustrate the usefulness of such presentations by using them to find a polynomial criterion to decide when 2 ordered pairs of 2 × 2 matrices (A, B) and (A′, B′) with entries in a commutative ring R are automorphically conjugate over the matrix algebra M ₂(R), under an additional assumption that both pairs generate M ₂(R) as an R-algebra.     

Representation type of endomorphism algebras of exceptional sequences of typeA n

LetF be an algebraically closed field, be a quiver of typeA _n . In this paper we prove that the endomorphism algebras of exceptional sequences over are sums of finitely many tilted algebras of typeA _m wherem≤n by using perpendicular categories, and thus the endomorphism algebras of exceptional sequences of typeA _n are representation-finite. Supported by Chinese Postdoctorate Fund and Beijing Youth Fund     

The automorphisms of Petit’s algebras

Let σ be an automorphism of a field K with fixed field F. We study the automorphisms of nonassociative unital algebras which are canonical generalizations of the associative quotient algebras K[t;σ]∕fK[t;σ] obtained when the twisted polynomial f∈K[t;σ] is invariant, and were first defined by Petit. We compute all their automorphisms if σ commutes with all automorphisms in Aut_F(K) and n≥m?1, where n is the order of σ and m the degree of f, and obtain partial results for n<m?1. In the case where K∕F is a finite Galois field extension, we obtain more detailed information on the structure of the automorphism groups of these nonassociative unital algebras over F. We also briefly investigate when two such algebras are isomorphic.     

Degrees of Quantum Function Algebras at Roots of 1

In this paper we will deal with quantum function algebrasF_q[G] in the special case when the parameterqspecializes to a root of 1. Using a combinatorial technique, we will give general formulas for the degree of such algebras and of a particular family of quotients which are fundamental objects in representation theory.     

Second cohomology group of extended W-algebras

Let \mathbbF\mathbb{F} be a field of characteristic 0, and let G be an additive subgroup of \mathbbF\mathbb{F}. We define a class of infinite-dimensional Lie algebras \mathbbF\mathbb{F}-basis {L _μ, V _μ, W _μ | μ ∈ G}, which are very closely related to W-algebras. In this paper, the second cohomology group of is determined.