Simplicity of the Lie Algebra of Skew Symmetric Elements of a Leavitt Path Algebra

For a field F of characteristic not 2 and a directed row-finite graph Γ let L(Γ) be the Leavitt Path Algebra with standard involution *. We study the lie algebra of K = K(L(Γ), *) of * ?skew-symmetric elements and find necessary and sufficient conditions for the Lie algebra [K, K] to be simple.     

On the Simplicity of Lie Algebras Associated to Leavitt Algebras

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L _𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M _d (L _𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ^?, and consider its Lie subalgebra [S ^?, S ^?]. In our main result, we show that [S ^?, S ^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L _𝕂(n)^?, L _𝕂(n)^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.     

On the Restricted Lie Algebra Structure of the Witt Lie Algebra in Finite Characteristic

The article contains an explicit formula for the restricted Lie algebra structure in the Witt Lie algebra over a field of finite characteristic. Some combinatorial lemmas can be of independent interest.     

On the Generalized Enveloping Algebra of a Color Lie Algebra

Let G be an abelian group, ε an anti-bicharacter of G and L a G-graded ε Lie algebra (color Lie algebra) over a field of characteristic zero. We prove that for all G-graded, positively filtered A such that the associated graded algebra is isomorphic to the G-graded ε-symmetric algebra S(L), there is a G- graded ε-Lie algebra L and a G-graded scalar two cocycle , such that A is isomorphic to U _ω (L) the generalized enveloping algebra of L associated with ω. We also prove there is an isomorphism of graded spaces between the Hochschild cohomology of the generalized universal enveloping algebra U(L) and the generalized cohomology of the color Lie algebra L. Supported by the EC project Liegrits MCRTN 505078.     

Stable rank of Leavitt path algebras

We characterize the values of the stable rank for Leavitt path algebras by giving concrete criteria in terms of properties of the underlying graph.

     

Derivation Algebra of Quasi Rn-filiform Lie Algebra

In this paper we explicitly determine the derivation algebra of a quasi Rn-filiform Lie algebra and prove that a quasi Pn-filiform Lie algebra is a completable nilpotent Lie algebra.     

Derivation Algebra of Quasi $R_n$-Filiform Lie Algebra

In this paper we explicitly determine the derivation algebra of a quasi $R_n$-filiform Lie algebra and prove that a quasi $R_n$-filiform Lie algebra is a completable nilpotent Lie algebra.     

On Lie Algebra Crossed Modules

Abstract

The goal of this article is to construct a crossed module representing the cocycle 〈[,],〉 generating H ³(;?) for a simple complex Lie algebra .     

Ideal Structure of Leavitt Path Algebras with Coefficients in a Unital Commutative Ring

For a (countable) graph E and a unital commutative ring R, we analyze the ideal structure of the Leavitt path algebra L _R (E) introduced by Mark Tomforde. We first modify the definition of basic ideals and then develop the ideal characterization of Mark Tomforde. We also give necessary and sufficient conditions for the primeness and the primitivity of L _R (E), and we then determine prime graded basic ideals and left (or right) primitive graded ideals of L _R (E). In particular, when E satisfies Condition (K) and R is a field, they imply that the set of prime ideals and the set of primitive ideals of L _R (E) coincide.     

*-Regular Leavitt path algebras of arbitrary graphs

If K is a field with involution and E an arbitrary graph, the involution from K naturally induces an involution of the Leavitt path algebra L _K (E). We show that the involution on L _K (E) is proper if the involution on K is positive-definite, even in the case when the graph E is not necessarily finite or row-finite. It has been shown that the Leavitt path algebra L _K (E) is regular if and only if E is acyclic. We give necessary and sufficient conditions for L _K (E) to be *-regular (i.e., regular with proper involution). This characterization of *-regularity of a Leavitt path algebra is given in terms of an algebraic property of K, not just a graph-theoretic property of E. This differs from the known characterizations of various other algebraic properties of a Leavitt path algebra in terms of graphtheoretic properties of E alone. As a corollary, we show that Handelman’s conjecture (stating that every *-regular ring is unit-regular) holds for Leavitt path algebras. Moreover, its generalized version for rings with local units also continues to hold for Leavitt path algebras over arbitrary graphs.     

The structure of Leavitt path algebras and the Invariant Basis Number property

In this paper, we provide the structure of the Leavitt path algebra of a finite graph via some step-by-step process of source eliminations, and restate Kanuni and Özaydin's nice criterion for Leavitt path algebras of finite graphs having Invariant Basis Number via matrix-theoretic language. Consequently, we give a matrix-theoretic criterion for the Leavitt path algebra of a finite graph having Invariant Basis Number in terms of a sequence of source eliminations. Using these results, we show certain classes of finite graphs for which Leavitt path algebras have Invariant Basis Number, as well as investigate the Invariant Basis Number property of Leavitt path algebras of certain Cayley graphs of finite groups.     

Simplicity of quadratic Lie conformal algebras

In this paper, simplicity of quadratic Lie conformal algebras is investigated. From the view point of the corresponding Gel’fand–Dorfman bialgebras, some su?cient conditions and necessary conditions to ensure simplicity of quadratic Lie conformal algebras are presented. By these observations, we present several new classes of infinite simple Lie conformal algebras. These results will be useful for classification purposes.     

Leavitt Path Algebras as Partial Skew Group Rings

We realize Leavitt path algebras as partial skew group rings and give new proofs, based on partial skew group ring theory of the Cuntz–Krieger uniqueness theorem and simplicity criteria for Leavitt path algebras.     

三次可解型非退化李代数及其导子李代数的结构

本文给出了非退化可解李代数的两个类型:三次可解型非退化李代数和扩充的 Heisenberg李代数,并确定三次可解型非退化李代数及其导子李代数的结构．     

一类量子环面李代数的单性(英文)

令C_q:=C_q[t₁^±1,t₂^±1,t₃^±1]为量子环面结合代数,L_q=C_q/C为量子环面李代数。本文定义了一个与q=（qij）_i,j=1³相关的指数方程体系,称之为C_q的特征方程组。通过这个特征方程组,证明了L_q是非单的当且仅当特征方程组在Z³中有非零解。对于|q|=0,证明了在C_q中存在一个极大交换子代数I,并且I严格包含中心Z（C_q）。同时文中也指出,对于|q|≠0,在C_q中不存在这样的子代数。     

FP-injective semirings,semigroup rings and Leavitt path algebras

In this paper we give characterisations of FP-injective semirings (previously termed “exact” semirings in work of the first author). We provide a basic connection between FP-injective semirings and P-injective semirings, and establish that FP-injectivity of semirings is a Morita invariant property. We show that the analogue of the Faith-Menal conjecture (relating FP-injectivity and self-injectivity for rings satisfying certain chain conditions) does not hold for semirings. We prove that the semigroup ring of a locally finite inverse monoid over an FP-injective ring is FP-injective and give a criterion for the Leavitt path algebra of a finite graph to be FP-injective.     

GICAR代数中的闭Lie理想

本文给出了GICAR代数中闭Lie理想的完全刻画.设A是GICAR代数,L是A的闭Lie理想,则存在A的闭理想J,使得[A,J]=[J,J](?) L (?) π-1(Z(A/J)),其中π是A到A/J上的商映射.反之,任意这种形式的闭子空间L是A的闭Lie理想.     

Reductivity of the Lie Algebra of a Bilinear Form

Let f: V × V → F be a totally arbitrary bilinear form defined on a finite dimensional vector space V over a field F, and let L(f) be the subalgebra of 𝔤𝔩(V) of all skew-adjoint endomorphisms relative to f. Provided F is algebraically closed of characteristic not 2, we determine all f, up to equivalence, such that L(f) is reductive. As a consequence, we find, over an arbitrary field, necessary and sufficient conditions for L(f) to be simple, semisimple or isomorphic to 𝔰𝔩(n) for some n.