This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

Introduction to lie algebras and representation theory

http://atarazanas.sci.uma.es/docs/tesisuma/16611123.pdf

The Leavitt path algebra of a graph

We compute the monoid V(L
 K
 (E)) of isomorphism classes of finitely generated projective modules over certain graph algebras L
 K
 (E), and we show that this monoid satisfies the refinement property and separative cancellation. We also show that there is a natural isomorphism between the lattice of graded ideals of L
 K
 (E) and the lattice of order-ideals of V(L
 K
 (E)). When K is the field $\mathbb C$
 of complex numbers, the algebra $L_{\mathbb C}(E)$
 is a dense subalgebra of the graph C
 *-algebra C
 *(E), and we show that the inclusion map induces an isomorphism between the corresponding monoids. As a consequence, the graph C*-algebra of any row-finite graph turns out to satisfy the stable weak cancellation property.

Nonstable K -theory for Graph Algebras

We describe a special class of representations of an inverse semigroup S on Hilbert's space which we term tight. These representations are supported on a subset of the spectrum of the idempotent semilattice of S, called the tight spectrum, which is in turn shown to be precisely the closure of the space of ultra-filters, once filters are identified with semicharacters in a natural way. These representations are moreover shown to correspond to representations of the C*-algebra of the groupoid of germs for the action of S on its tight spectrum. We then treat the case of certain inverse semigroups constructed from semigroupoids, generalizing and inspired by inverse semigroups constructed from ordinary and higher rank graphs. The tight representations of this inverse semigroup are in one-to-one correspondence with representations of the semigroupoid, and consequently the semigroupoid algebra is given a groupoid model. The groupoid which arises from this construction is shown to be the same as the boundary path groupoid of Farthing, Muhly and Yeend, at least in the singly aligned, sourceless case.

Inverse semigroups and combinatorial C*-algebras

We classify the gauge-invariant ideals in the $C^*$-algebras of infinite directed graphs, and describe the quotients as graph algebras. We then use these results to identify the gauge-invariant primitive ideals in terms of the structural properties of the graph, and describe the $K$-theory of the $C^*$-algebras of arbitrary infinite graphs.

/pdf/the-ideal-structure-of-the-c-sp-algebras-of-infinite-graphs-5blefzstku.pdf

The ideal structure of the $C\sp *$-algebras of infinite graphs

We show that the Cuntz-Krieger algebras of infinite graphs and infinite {0,1}-matrices can be approximated by those of finite graphs. We then use these approximations to deduce the main uniqueness theorems for Cuntz-Krieger algebras and to compute their K-theory. Since the finite approximating graphs have sinks, we have to calculate the K-theory of Cuntz-Krieger algebras of graphs with sinks, and the direct methods we use to do this should be of independent interest.

/pdf/cuntz-krieger-algebras-of-infinite-graphs-and-matrices-43h364ehq7.pdf

Cuntz-Krieger Algebras of Infinite Graphs and Matrices

The C

 *
 -algebras of continuous functions on quantum spheres, quantum real projective spaces, and quantum complex projective spaces are realized as Cuntz-Krieger algebras corresponding to suitable directed graphs Structural results about these quantum spaces, especially about their ideals and K-theory, are then derived from the general theory of graph algebras It is shown that the quantum even and odd dimensional spheres are produced by repeated application of a quantum double suspension to two points and the circle, respectively

Quantum Spheres and Projective Spaces as Graph Algebras

For any countable directed graph E we describe the primitive ideal space of the corresponding generalized Cuntz-Krieger algebra C*E.

/pdf/the-primitive-ideal-space-of-the-c-algebras-of-infinite-1m8le8e9ta.pdf

The primitive ideal space of the $C^{*}$-algebras of infinite graphs

We recast basic topological concepts underlying differential geometry using the language and tools of noncommutative geometry. This way we characterize principal (free and proper) actions by a density condition in (multiplier) C*-algebras. We introduce the concept of piecewise triviality to adapt the standard notion of local triviality to fibre products of C*-algebras. In the context of principal actions, we study in detail an example of a non-proper free action with continuous translation map, and examples of compact principal bundles which are piecewise trivial but not locally trivial, and neither piecewise trivial nor locally trivial, respectively. We show that the module of continuous sections of a vector bundle associated to a compact principal bundle is a cotensor product of the algebra of functions defined on the total space (that are continuous along the base and polynomial along the fibres) with the vector space of the representation. On the algebraic side, we review the formalism of connections for the universal differential algebras. In the differential geometry framework, we consider smooth connections on principal bundles as equivariant splittings of the cotangent bundle, as 1-form-valued derivations of the algebra of smooth functions on the structure group, and as axiomatically given covariant differentiations of functions defined on the total space. Finally, we use the Dirac monopole connection to compute the pairing of the line bundles associated to the Hopf fibration with the cyclic cocycle of integration over S^2.

Wojciech Szymanski

Papers

The ideal structure of the $C\sp *$-algebras of infinite graphs

Cuntz-Krieger Algebras of Infinite Graphs and Matrices

Quantum Spheres and Projective Spaces as Graph Algebras

The primitive ideal space of the $C^{*}$-algebras of infinite graphs

Noncommutative Geometry Approach to Principal and Associated Bundles