55 research outputs found
K-theory of locally finite graph -algebras
We calculate the K-theory of the Cuntz-Krieger algebra
associated with an infinite, locally finite graph, via the Bass-Hashimoto
operator. The formulae we get express the Grothendieck group and the Whitehead
group in purely graph theoretic terms.
We consider the category of finite (black-and-white, bi-directed) subgraphs
with certain graph homomorphisms and construct a continuous functor to abelian
groups. In this category is an inductive limit of -groups of finite
graphs, which were calculated in \cite{MM}.
In the case of an infinite graph with the finite Betti number we obtain the
formula for the Grothendieck group where is the first Betti number and
is the valency number of the graph . We note, that in the
infinite case the torsion part of , which is present in the case of a
finite graph, vanishes. The Whitehead group depends only on the first Betti
number: . These allow us to provide a
counterexample to the fact, which holds for finite graphs, that is the torsion free part of .Comment: Final version, in press at the Journal of Geometry and Physics (2013
The Golod-Shafarevich inequality for Hilbert series of quadratic algebras and the Anick conjecture
We study the question on whether the famous Golod-Shafarevich estimate, which
gives a lower bound for the Hilbert series of a (noncommutative) algebra, is
attained. This question was considered by Anick in his 1983 paper 'Generic
algebras and CW-complexes', Princeton Univ. Press., where he proved that the
estimate is attained for the number of quadratic relations and , and conjectured that this is the
case for any number of quadratic relations. The particular point where the
number of relations is equal to was addressed by Vershik.
He conjectured that a generic algebra with this number of relations is finite
dimensional.
We prove that over any infinite field, the Anick conjecture holds for
and arbitrary number of generators , and
confirm the Vershik conjecture over any field of characteristic 0. We give also
a series of related asymptotic results.Comment: 17 pages, to appear in the Proceedings of the Royal Society Edinburgh
Asymptotically optimal -step nilpotency of quadratic algebras and the Fibonacci numbers
It follows from the Golod--Shafarevich theorem that if R is an associative
algebra given by n generators and
quadratic relations, then R is not k-step nilpotent. We show that the above
estimate is asymptotically optimal, and establish number of related results.
For example, we show that for any k this estimate is attained for ifinitely
many n.Comment: to appear in Combinatoric
Quadratic automaton algebras and intermediate growth
We present an example of a quadratic algebra given by three generators and
three relations, which is automaton (the set of normal words forms a regular
language) and such that its ideal of relations does not possess a finite
Gr\"obner basis with respect to any choice of generators and any choice of a
well-ordering of monomials compatible with multiplication. This answers a
question of Ufnarovski.
Another result is a simple example (4 generators and 7 relations) of a
quadratic algebra of intermediate growth.Comment: To appear in Journal of Cobinatorial Algebr
On Koszulity for operads of Conformal Field Theory
We study two closely related operads: the Gelfand-Dorfman operad GD and the
Conformal Lie Operad CLie. The latter is the operad governing the Lie conformal
algebra structure. We prove Koszulity of the Conformal Lie operad using the
Groebner bases theory for operads and an operadic analogue of the Priddy
criterion. An example of deformation of an operad coming from the Hom
structures is considered. In particular we study possible deformations of the
Associative operad from the point of view of the confluence property. Only one
deformation, the operad which governs the identity turns out to be confluent. We introduce a new Hom structure, namely
Hom--Gelfand-Dorfman algebras and study their basic properties.Comment: 24 page
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