arXiv subjects
Publications and source records attributed to Hamed Alsulami.
Let $A$ be an associative commutative algebra with $1$ over a field of zero characteristic, $\{,\} : A \times A \to A$ is a Poisson bracket, $Z = \{ a \in A \mid \{a, A\} = (0) \}.$ We prove that if $A$ is simple as a Poisson algebra then the Lie algebra $\frac{\{A,A\}}{\{A,A\}\cap Z}$ is simple.
In this note we discuss Morita equivalence classes of arbitrary finitely presented algebras
We prove that an arbitrary countable dimensional Lie algebra over a field of characteristic $\neq 2$ that is locally of subexponential growth is embeddable in a finitely generated Lie algebra of subexponential growth.
Let $A$ be an associative algebra over a field of characteristic $\neq 2$ that is generated by a finite collection of nilpotent elements. We prove that all Lie derived powers of $A$ are finitely generated Lie algebras.
We introduce a new construction of matrix wreath products of algebras that is similar to wreath products of groups. We then use it to prove embedding theorems for Jacobson radical, nil, and primitive algebras. In §\ref{Section6}, we construct finitely generated nil algebras of arbitrary Gelfand-Kirillov dimension $\geq 8$ over a countable field which answers a question from \cite{8}.
We prove that a countable dimensional associative algebra (resp. a countable semigroup) of locally subexponential growth is $M_\infty$-embeddable as a left ideal in a finitely generated algebra (resp. semigroup) of subexponential growth. Moreover, we provide bounds for the growth of the finitely generated algebra (resp. semigroup). The proof is based on a new construction of matrix wreath product of algebras.
We prove that the property of an algebra to be finitely presented is Morita invariant.
For a finite directed graph $Γ$ we determine the center of the Cuntz-Krieger $C^*$-algebra $CK(Γ).$
In [8, 9] M. G. Corrales Garcia, D. M. Barquero, C. Martin Gonzalez, M. Siles Molina, J. F Solanilla Hernandez described the center of a Leavitt path algebra and characterized it in terms of the underlying graph. We offer a different characterization of the center. In particular, we prove that the Boolean algebra of central idempotents \ of a Leavitt path algebra of a finite graph is isomorphic to the Boolean algebra of finitary annihilator hereditary subsets of the graph.
We prove that if $A$ is finitely presented algebra with idempotent $e$ such that $A=AeA=A(1-e)A$ then the algebra $eAe$ is finitely presented.
We introduce a class of topologies on the Leavitt path algebra $L(Γ)$ of a finite directed graph and decompose a graded completion $\widehat{L}(Γ)$ as a direct sum of minimal ideals.
For a field $F$ of characteristic not 2 and a directed row-finite graph $Γ$ let $L(Γ)$ be the Leavitt path algebra with the standard involution $*.$ We study the Lie algebra $K=K(L(Γ),*)$ of $*-$skew-symmetric elements and find necessary and sufficient conditions for the Lie algebra $[K,K]$ to be simple.
We introduce ring theoretic constructions that are similar to the construction of wreath product of groups. In particular, for a given graph $Γ=(V,E)$ and an associate algebra $A,$ we construct an algebra $B=A\, wr\, L(Γ)$ with the following property: $B$ has an ideal $I$,which consists of (possibly infinite) matrices over $A$, $B/I\cong L(Γ)$, the Leavitt path algebra of the graph $Γ$. \medskip \par Let $W\subset V$ be a hereditary saturated subset of the set of vertices [1], $Γ(W)=(W,E(W,W))$ is the restriction of the graph $Γ$ to $W$, $Γ/W$ is the quotient graph [1]. Then $L(Γ)\cong L(W)$ wr $L(Γ/W)$.
Let $F$ be a field of characteristic not $2$ . An associative $F$-algebra $R$ gives rise to the commutator Lie algebra $R^{(-)}=(R,[a,b]=ab-ba).$ If the algebra $R$ is equipped with an involution $*:R\rightarrow R$ then the space of the skew-symmetric elements $K=\{a \in R \mid a^{*}=-a \}$ is a Lie subalgebra of $R^{(-)}.$ In this paper we find sufficient conditions for the Lie algebras $[R,R]$ and $[K,K]$ to be finitely generated.
We determine the structure of Leavitt path algebras of polynomial growth and discuss their automorphisms and involutions.
For a field $F$ and a row-finite directed graph $Γ$ let $L(Γ)$ be the Leavitt path algebra. We find necessary and sufficient conditions for the Lie algebra $[L(Γ),L(Γ)]$ to be simple.