Darren Funk-Neubauer
Bidiagonal pairs, the Lie algebra sl_2, and the quantum group U_q(sl_2).
Mon, 08/08/2011 - 15:01 — SomNambulAuthors: Darren Funk-Neubauer
Subjects: Representation Theory
AbstractRoughly speaking, a bidiagonal pair is a pair of diagonalizable linear
transformations on a finite dimensional vector space, each of which acts in a
bidiagonal fashion on the eigenspaces of the other. We associate to each
bidiagonal pair a sequence of scalars called a parameter array. We present a
classification of bidiagonal pairs up to isomorphism using this concept of a
parameter array.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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