Gene Abrams
Isomorphism and Morita equivalence of graph algebras.
Wed, 12/09/2009 - 16:01 — SomNambulAuthors: Gene Abrams, Mark Tomforde
Subjects: Operator Algebras
AbstractFor any countable graph $E$, we investigate the relationship between the
Leavitt path algebra $L_{\C}(E)$ and the graph C*-algebra $C^*(E)$. For graphs
$E$ and $F$, we examine ring homomorphisms, ring *-homomorphisms, algebra
homomorphisms, and algebra *-homomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$.
We prove that in certain situations isomorphisms between $L_{\C}(E)$ and
$L_{\C}(F)$ yield *-isomorphisms between the corresponding C*-algebras $C^*(E)$
and $C^*(F)$.On the simplicity of Lie algebras associated to Leavitt algebras.
Tue, 08/18/2009 - 11:53 — SomNambulAuthors: Gene Abrams, Darren Funk-Neubauer
Subjects: Rings and Algebras
AbstractFor any field $K$ and integer $n\geq 2$ we consider the Leavitt algebra $L =
L_K(n)$. $L$ is an associative algebra, but we view $L$ as a Lie algebra using
the bracket $[a,b]=ab-ba$ for $a,b \in L$. We denote this Lie algebra as $L^-$,
and consider its Lie subalgebra $[L^-,L^-]$. In our main result, we show that
$[L^-,L^-]$ is a simple Lie algebra if and only if char$(K)$ divides $n-1$. For
any positive integer $d$ we let $S = M_d(L_K(n))$ be the $d\times d$ matrix
algebra over $L_K(n)$. We give sufficient conditions for the simplicity and
non-simplicity of the Lie algebra $[S^-,S^-]$.
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