Math Dissertation: Units and Leavitt Path Algebras, March 12

The Dissertation Defense for Nick Pilewski (directed by Sergio Lopez-Permouth) is Thursday, March 12, at 2 p.m. in Morton 237.

His dissertation is  “Units and Leavitt Path Algebras.”

*Abstract*: An invertible algebra is defined to be an algebra with a basis consisting solely of units. Given a field K and a finite graph E, we give a condition on E equivalent to the Leavitt path algebra L_K(E) being an invertible K-algebra, and consequently a condition equivalent to the Cohn path algebra C_K(E) being an invertible K-algebra. Given a unital commutative ring R, sufficient conditions on E for the Leavitt path algebra L_R(E) to be an invertible R-algebra are given. As a by-product, given a field K and a graph E with finitely many vertices, we completely identify in terms of E all those right L_K(E)-modules B such that \(L_K(E) \cong L_K(E) \oplus B\) as a right L_K(E)-module. As another by-product, we show that given an arbitrary R-algebra A, matrix algebras over A and direct sums of these matrix algebras are invertible R-algebras. Additionally, we characterize the invertible semilocal algebras over a division ring, and consequently the invertible finite dimensional algebras over a division ring.