| Roozbeh Hazrat |
Classification of Leavitt path algebras in relation to symbolic dynamics |
The theory of Leavitt path algebras is intrinsically related, via graphs, to the theory of symbolic dynamics and \(C^*\)-algebras. We give an overview of how the classification of Leavitt path algebras (a major open project) is related to symbolic dynamics. We give a survey of currently open questions and new advances towards them. |
| Guillermo Cortiñas |
Bivariant algebraic K-theory and Leavitt path algebras. |
Classification problems aim to describe a class of objects (e.g. Leavitt path algebras) up to some nation of equivalence (e.g. (graded) isomorphism, Morita equivalence, etc) by means of a (usually finite) set of computable invariants. To guarantee computability one requires that the invariants satisfy certain basic properties. Bivariant algebraic K-theory kk is the universal invariant satisfying certain reasonable properties (which will be explained in the lectures). Universality means that kk is the finest invariant with those properties. We shall explain how kk is used to tackle classification problems for Leavitt path algebras.
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| Tran Giang Nam |
On automorphisms and irreducible representations of Leavitt path algebras |
We give a method to construct automorphisms of Leavitt path algebras of graphs via J. Cuntz's seminal paper [Automorphisms of certain simple \(C^*\)-algebras, Quantum fields-algebras, processes (Proc. Sympos., Univ. Bielefeld, Bielefeld, 1978), pp. 187-196, Springer, Vienna, 1980.] These automorphisms are analogous to the Anick automorphisms of a free associative algebra \(K\langle x_1, x_2,..., x_n\rangle\). For \(n= 3\) the Anick automorphism has been shown by U. U. Umirbaev [J. Reine Angew. Math. 605 (2007), 165-178]. Using these Anick type automorphisms of Leavitt path algebras we construct a new class of simple modules over the Leavitt path algebra \(L_K(R_n)\), where \(R_n\) is the rose with \(n\) etals, that is, the directed graph with one vertex and \(n\) distinct edges. There are some by now well-known classes of simple modules over a Leavitt path algebra, which after the seminal paper by X.-W. Chen [Forum Math. 27 (2015), no. 1, 549–574] are called Chen modules. The new simple modules are obtained from the Chen simple modules by twisting them by Anick type automorphisms of the Leavitt path algebra \(L_K(R_n)\). |
| Praneet Srivastava and Joshua Graham |
Higher K-theoretic tools in Leavitt path algebras |
This graduate workshop will cover two distinct areas; the first a brief introduction to algebraic K-theory, and the second an introduction to Leavitt path algebras and their algebraic K-theory.
The first part of this workshop will be devoted to establishing the definition and basic properties of higher algebraic K-theory, following the approach of Quillen. In particular, we will explore the so called \(Q\)-construction from a modern perspective. We will also introduce notions from homotopy theory and the theory of \(\infty\)-categories as required for this, building up to the group completion theorem. Finally, we will outline Quillen’s computation of all the higher K-groups of \(\mathbb{F}_p\) using
these tools.
The second part of this workshop will briefly define Leavitt path algebras and build upon the homotopy-theoretic foundations laid in this first to prove the following: if \(E\) is a finite quiver with no sources with \(d\) vertices, \(d_0\) sinks and \(k\) a field, there is a long exact sequence
\( ... \to K_n(k)^{d-d_0} \to K_n(k)^d \to K_n(L_k(E)) \to K_{n-1}(k)^{d-d_0} \to ... \)
We will also use the computations of \(K_n(\mathbb{F}_p)\) to explicitly compute the algebraic K-theory of \(L_k(E)\) where \(k\) is a finite field. |
| Elizabeth Pacheco |
Graph moves and new irreducible representations of Leavitt path algebras |
Given a directed graph, one has a set of generators and rela
tions for the Leavitt path algebra. While this completely describes
the algebra, this presentation can be cumbersome and leave a lot
of questions about the properties of the algebra and whether two
different directed graphs yield isomorphic Leavitt path algebras. In
the series of talks, we will go over various graph moves that preserve
the isomorphic classes of a Leavitt path algebras, it’s representation
theory (Morita equivalence), and the category of simple modules.
Irreducible representations of Leavitt path algebras (LPAs) were
constructed by Chen. When the Leavitt path algebra has polyno
mial growth (this corresponds to the cycles of \(\Gamma\) being mutually
disjoint, all simple modules are essentially Chen modules. Whether
irreducible representations other than the Chen modules exist or not
was raised as a natural question.
In joint work with Murad Ozaydın, we show that if an LPA (over
an arbitrary field \(F\)) has exponential growth (there are no LPAs
of intermediate growth) then it has irreducible representations that
are not Chen modules. These simple modules are submodules of
the space of step functions defined on topological Markov chains
associated to the directed graph. |
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