Department of Mathematics

Details

Two-sided chain condition in Leavitt path algebras

by Pinar Colak | University of Toronto
Time: 13:00  (Wednesday, Jan. 26, 2011)
Location: BA6183, Bahen Center, 40 St. George St.
Abstract:
Leavitt path algebras are a natural generalization of the Leavitt algebras, which are a class of algebras introduced by Leavitt in 1962. For a directed graph $E$, the Leavitt path algebra $L_K(E)$ of $E$ with coefficients in $K$ has received much recent attention both from algebraists and analysts over the last decade. So far, some of the algebraic properties of Leavitt path algebras have been investigated, including primitivity, simplicity and being Noetherian. First, we explicitly describe the generators of two-sided ideals in Leavitt path algebras associated to arbitrary graphs. We show that any two-sided ideal $I$ of a Leavitt path algebra associated to an arbitrary graph is generated by elements of the form $(v + \sum_{i=1}^n\lambda_i g^i)(v - \sum_{e\in S} ee^*)$, where $g$ is a cycle based at vertex $v$, and $S$ is a finite subset of $s^{-1}(v)$. Then, we use this result to describe the necessary and sufficient conditions on the arbitrary sized graph $E$, such that the Leavitt path algebra associated to $E$ satisfies two-sided chain conditions. This is joint work with Dr. Gene Abrams, Dr. Jason P. Bell and Dr. Kulumani M. Rangaswamy.

Dates in this series

· Thursday, Jan. 21, 2010: The Calderon Problem - From the Past to the Present (Leo Tzou)

· Wednesday, Feb. 03, 2010: Indecomposable characters on the group S(2^∞) (Artem Dudko)

· Wednesday, Feb. 10, 2010: Self-referencing Recursions and Labeled Binary Trees (Abraham Isgur)

· Wednesday, Feb. 17, 2010: Introduction to the Well-posedness Theory of (Dispersive) PDEs (Hiro Oh)

· Wednesday, Mar. 03, 2010: Renormalization and universality in one-dimensional dynamics: critical circle maps (Michael Yampolsky)

· Wednesday, Mar. 10, 2010: The Dimer Model on a Planar Domain: From random domino tilings to limit shapes (Alex Bloemendal)

· Wednesday, Mar. 17, 2010: The Planar Dimer Model, continued (Alex Bloemendal)

· Thursday, Sep. 16, 2010: The calculus of Euler characteristic and some of its application (Dmitry Kerner)

· Thursday, Oct. 07, 2010: Equivalences of the Axiom of Choice (Mike Pawliuk)

· Thursday, Oct. 21, 2010: Covering arrays with row limit four (Nevena Francetic)

· Thursday, Oct. 28, 2010: Which groups are Lie groups? (Yevgeniy Likuomovich)

· Thursday, Nov. 04, 2010: All started with Weyl (Victor Ivrii)

· Thursday, Nov. 11, 2010: Sparse structures in Euclidean space (Dominic Dotterrer)

· Thursday, Nov. 18, 2010: A short introduction to curve singularities (Dmitry Kerner)

· Thursday, Nov. 25, 2010: Newton diagrams and applications (Dmitry Kerner)

· Thursday, Dec. 02, 2010: Convergence and smoothing properties of Steiner symmetrizations and other simple rearrangements (Almut Burchard)

· Tuesday, Jan. 11, 2011: TeXFreak (Victor Ivrii)

· Wednesday, Jan. 26, 2011: Two-sided chain condition in Leavitt path algebras (Pinar Colak)

· Tuesday, Feb. 01, 2011: SPECIAL GRADUATE SEMINAR -- How to classify algebraic varieties (Chenyang Xu)

· Wednesday, Feb. 02, 2011: Enrico Bombieri and the Prime Number Theorem (John Friedlander)

· Wednesday, Feb. 16, 2011: Basic Financial Derivatives and the Black-Scholes PDE (Ryan Donnelly)

· Wednesday, Mar. 09, 2011: Classical mechanics and symplectic geometry (Jonathan Fisher)

· Thursday, Mar. 17, 2011: On Computability of Julia sets (Artem Dudko)

· Wednesday, Mar. 23, 2011: Variational optimization in economics (Robert McCann)

· Wednesday, Mar. 30, 2011: Exact formulas in random growth and directed polymers (Jeremy Quastel)

· Wednesday, Apr. 06, 2011: Kitaev's quantum double model for topological quantum computation -- representations, higher categories and extended TQFTs (Lucy Zhang)

· Wednesday, Apr. 13, 2011: Tropicalization and Connection to Secondary Fans (Jihyeon Jessie Yang)

· Thursday, Sep. 29, 2011: Probabilistic Cauchy Theory and Invariant Measures for Hamiltonian PDEs (Geordie Richards)

· Thursday, Oct. 13, 2011: The cohomology ring of the GKM graph of a flag manifold (Yukiko Fukukawa)

· Friday, Nov. 04, 2011: Representations of the full group of a Bratteli diagram (Artem Dudko)

· Thursday, Nov. 10, 2011: How to Solve a Cubic Equation? (Askold Khovanskii)

· Thursday, Nov. 24, 2011: Rate of escape of random walks on groups (Balint Virag)

· Thursday, Dec. 01, 2011: Some aspects of geometric measure theory (Robert Jerrard)

· Thursday, Jan. 19, 2012: Bounded generation, congruence subgroups and the splitting of primes (Kumar Murty)

· Thursday, Jan. 26, 2012: How to solve a quintic polynomial? (Yuri Burda)

· Thursday, Feb. 02, 2012: Dynamics on the circle (Elio Mazzeo)

· Thursday, Feb. 09, 2012: How to keep a can from sliding down an incline (and make it roll instead) (David Li-Bland)

· Thursday, Mar. 15, 2012: Arithmetic graphs and the sum-product phenomenon (Brandon Hanson)

· Thursday, Mar. 22, 2012: Two Inverse Problems in Medical Imaging (Nicholas Hoell)

· Thursday, Mar. 29, 2012: Enveloping algebras of nonassociative structures (Marina Tvalavadze)

· Thursday, Nov. 08, 2012: Airy's rainbow and Stokes' mist: asymptotics through the prism of geometry (Brent Pym)

· Thursday, Dec. 13, 2012: C*-algebras and the Classification Program (Danny Hay)

· Thursday, Jan. 17, 2013: Why is the Riemann Hypothesis a big deal? (Asif Zaman)