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1. CJM Online first

Amini, Massoud; Elliott, George A.; Golestani, Nasser
The Category of Bratteli Diagrams
A category structure for Bratteli diagrams is proposed and a functor from the category of AF algebras to the category of Bratteli diagrams is constructed. Since isomorphism of Bratteli diagrams in this category coincides with Bratteli's notion of equivalence, we obtain in particular a functorial formulation of Bratteli's classification of AF algebras (and at the same time, of Glimm's classification of UHF~algebras). It is shown that the three approaches to classification of AF~algebras, namely, through Bratteli diagrams, K-theory, and abstract classifying categories, are essentially the same from a categorical point of view.

Keywords:C$^{*}$-algebra, category, functor, AF algebra, dimension group, Bratteli diagram
Categories:46L05, 46L35, 46M15

2. CJM 2014 (vol 67 pp. 28)

Asadollahi, Javad; Hafezi, Rasool; Vahed, Razieh
Bounded Derived Categories of Infinite Quivers: Grothendieck Duality, Reflection Functor
We study bounded derived categories of the category of representations of infinite quivers over a ring $R$. In case $R$ is a commutative noetherian ring with a dualising complex, we investigate an equivalence similar to Grothendieck duality for these categories, while a notion of dualising complex does not apply to them. The quivers we consider are left, resp. right, rooted quivers that are either noetherian or their opposite are noetherian. We also consider reflection functor and generalize a result of Happel to noetherian rings of finite global dimension, instead of fields.

Keywords:derived category, Grothendieck duality, representation of quivers, reflection functor
Categories:18E30, 16G20, 18E40, 16D90, 18A40

3. CJM 2012 (vol 65 pp. 82)

Félix, Yves; Halperin, Steve; Thomas, Jean-Claude
The Ranks of the Homotopy Groups of a Finite Dimensional Complex
Let $X$ be an $n$-dimensional, finite, simply connected CW complex and set $\alpha_X =\limsup_i \frac{\log\mbox{ rank}\, \pi_i(X)}{i}$. When $0\lt \alpha_X\lt \infty$, we give upper and lower bound for $ \sum_{i=k+2}^{k+n} \textrm{rank}\, \pi_i(X) $ for $k$ sufficiently large. We show also for any $r$ that $\alpha_X$ can be estimated from the integers rk$\,\pi_i(X)$, $i\leq nr$ with an error bound depending explicitly on $r$.

Keywords:homotopy groups, graded Lie algebra, exponential growth, LS category
Categories:55P35, 55P62, , , , 17B70

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