Related papers: Simple Lie algebras arising from Leavitt path alge…

For any field K and directed graph E, we completely describe the elements of the Leavitt path algebra L_K(E) which lie in the commutator subspace [L_K(E),L_K(E)]. We then use this result to classify all Leavitt path algebras L_K(E) that…

For any row-finite graph $E$ and any field $K$ we construct the {\its Leavitt path algebra} $L(E)$ having coefficients in $K$. When $K$ is the field of complex numbers, then $L(E)$ is the algebraic analog of the Cuntz Krieger algebra…

For any field $K$ and for a completely arbitrary graph $E$, we characterize the Leavitt path algebras $L_K(E)$ that are indecomposable (as a direct sum of two-sided ideals) in terms of the underlying graph. When the algebra decomposes, it…

For a field $F$ and a row-finite directed graph $\Gamma$ let $L(\Gamma)$ be the Leavitt path algebra. We find necessary and sufficient conditions for the Lie algebra $[L(\Gamma),L(\Gamma)]$ to be simple.

For a field $F$ of characteristic not 2 and a directed row-finite graph $\Gamma$ let $L(\Gamma)$ be the Leavitt path algebra with the standard involution $*.$ We study the Lie algebra $K=K(L(\Gamma),*)$ of $*-$skew-symmetric elements and…

Given a directed graph E we describe a method for constructing a Leavitt path algebra $L_R(E)$ whose coefficients are in a commutative unital ring R. We prove versions of the Graded Uniqueness Theorem and Cuntz-Krieger Uniqueness Theorem…

Let $L_K(E)$ be the Leavitt path algebra of a directed graph $E$ over a field $K$. In this paper, we determine $E$ and $K$ for the Lie algebra $\mathbf{K}_{L_K(E)}$ and the Jordan algebra $\mathbf{S}_{L_K(E)}$ arising from $L_K(E)$ with…

We give necessary and sufficient conditions on a row-finite graph E so that the Leavitt path algebra L(E) is purely infinite simple. This result provides the algebraic analog to the corresponding result for the Cuntz-Krieger C$^*$-algebra…

Let $K$ be a field and $E$ be a graph. Let $L_K(E)$ be the Leavitt path algebra of $E$ over $K$ with the standard involution $^\star$. We investigate the set of skew-symmetric elements, $\mathbf{K}_{L_K(E)}=\{x\in L_K(E) : x^{\star}=-x\}$,…

Suppose that $R$ is an associative unital ring and that $E=(E^0,E^1,r,s)$ is a directed graph. Utilizing results from graded ring theory we show, that the associated Leavitt path algebra $L_R(E)$ is simple if and only if $R$ is simple,…

We classify all the pairs of a commutative associative algebra with an identity element and its finite-dimensional commutative locally-finite derivation subalgebra such that the commutative associative algebra is derivation-simple with…

We study so called regular Lie algebras, i.e. Lie algebras in which each nonzero element is regular. We make a connection with an open problem whether any element of reduced trace zero in a simple associative algebra is a commutator.

This survey of the recent developments in the investigations of a Leavitt path algebra L of an arbitrary graph E over a field K consists of two parts. In the first part describes how very often a single graph property of E implies multiple…

In this paper, we study ideal- and congruence-simpleness for the Leavitt path algebras of directed graphs with coefficients in a commutative semiring S, as well as establish some fundamental properties of those algebras. We provide a…

For any field $\K$ and integer $n\geq 2$ we consider the Leavitt algebra $L_\K(n)$; for any integer $d\geq 1$ 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…

Any finite dimensional semisimple algebra A over a field K is isomorphic to a direct sum of finite dimensional full matrix rings over suitable division rings. In this paper we will consider the special case where all division rings are…

We achieve an extremely useful description (up to isomorphism) of the Leavitt path algebra $L_K(E)$ of a finite graph $E$ with coefficients in a field $K$ as a direct sum of matrix rings over $K$, direct sum with a corner of the Leavitt…

Let $\mathbb{K}$ be a field, $R$ be an associative and commutative $\mathbb{K}$-algebra and $L$ be a Lie algebra over $\mathbb{K}$. We give some descriptions of injections from $L$ to Lie algebra of $\mathbb{K}$-derivations of $R$ in the…

In this article, we give necessary and sufficient conditions under which the Leavitt path algebra $L_K(\mathcal{G})$ of an ultragraph $\mathcal{G}$ over a field $K$ is purely infinite simple and that it is von Neumann regular. Consequently,…

Let $K$ be any field, and let $E$ be any graph. We explicitly construct the projective resolution of simple left modules over the Leavitt path algebra $L_K(E)$ associated to cycles and irreducible polynomials. Then we study the dimension of…