EVERY GRADED IDEAL OF A LEAVITT PATH ALGEBRA IS GRADED ISOMORPHIC TO A LEAVITT PATH ALGEBRA

We show that every graded ideal of a Leavitt path algebra is graded isomorphic to a Leavitt path algebra. It is known that a graded ideal I of a Leavitt path algebra is isomorphic to the Leavitt path algebra of a graph, known as the generalised hedgehog graph, which is defined based on certain sets of vertices uniquely determined by I. However, this isomorphism may not be graded. We show that replacing the short ‘spines’ of the generalised hedgehog graph with possibly fewer, but then necessarily longer spines, we obtain a graph (which we call the porcupine graph) whose Leavitt path algebra is graded isomorphic to I. Our proof can be adapted to show that, for every closed gauge-invariant ideal J of a graph $C^*$ -algebra, there is a gauge-invariant $*$ -isomorphism mapping the graph $C^*$ -algebra of the porcupine graph of J onto $J.$

On partitions of G-spaces and G-lattices

Given a [Formula: see text]-space [Formula: see text] and a non-trivial [Formula: see text]-invariant ideal [Formula: see text] of subsets of [Formula: see text], we prove that for every partition [Formula: see text] of [Formula: see text] into [Formula: see text] pieces there is a piece [Formula: see text] of the partition and a finite set [Formula: see text] of cardinality [Formula: see text] such that [Formula: see text] where [Formula: see text] is the difference set of the set [Formula: see text]. Also we investigate the growth of the sequence [Formula: see text] and show that [Formula: see text] where [Formula: see text] is the Lambert [Formula: see text]-function, defined implicitly as [Formula: see text]. This shows that [Formula: see text] grows faster than that any exponent [Formula: see text] but slower than the sequence [Formula: see text] of factorials.

TEST RANK OF THE LIE ALGEBRA F/[R′, F]

Let F be a free Lie algebra of rank n ≥ 2 and R be a fully invariant ideal of F. We show that the test rank of the Lie algebra F/[R′, F] is equal to 1 when n is even and less than or equal to 2 when n is odd.

Index maps in the K-theory of graph algebras

Let C*(E) be the graph C*-algebra associated to a graph E and let J be a gauge-invariant ideal in C*(E). We compute the cyclic six-term exact sequence in K-theory associated to the extensionin terms of the adjacency matrix associated to E. The ordered six-term exact sequence is a complete stable isomorphism invariant for several classes of graph C*-algebras, for instance those containing a unique proper nontrivial ideal. Further, in many other cases, finite collections of such sequences constitute complete invariants.Our results allow for explicit computation of the invariant, giving an exact sequence in terms of kernels and cokernels of matrices determined by the vertex matrix of E.

Torelli groups, extended Johnson homomorphisms, and new cycles on the moduli space of curves

Infinite presentations are given for all of the higher Torelli groups of once-punctured surfaces. In the case of the classical Torelli group, a finite presentation of the corresponding groupoid is also given, and finite presentations of the classical Torelli groups acting trivially on homology modulo N are derived for all N. Furthermore, the first Johnson homomorphism, which is defined from the classical Torelli group to the third exterior power of the homology of the surface, is shown to lift to an explicit canonical 1-cocycle of the Teichmüller space. The main tool for these results is the known mapping class group invariant ideal cell decomposition of the Teichmüller space.This new 1-cocycle is mapping class group equivariant, so various contractions of its powers yield various combinatorial (co)cycles of the moduli space of curves, which are also new. Our combinatorial construction can be related to former works of Kawazumi and the first-named author with the consequence that the algebra generated by the cohomology classes represented by the new cocycles is precisely the tautological algebra of the moduli space.There is finally a discussion of prospects for similarly finding cocycle lifts of the higher Johnson homomorphisms.