International Journal of Mathematics

If Γ is a graph, with distinguished vertex *, let A(Γ) denote the non-commutative path algebra on the space of semi-infinite paths in Γ beginning at *. We discuss embeddings A(Γ₁) → A(Γ₂) of AF algebras associated with graphs Γ₁ and Γ₂ from a dimension group point of view. For certain infinite T-shaped graphs, we have K₀(A(Γ)) ≅ ℤ [t], with positive cone identified with {0}∪ {P ∈ ℤ [t]: P (λ) > 0, λ ∈ (0, γ]}, where γ = γ (Γ) =||Γ||⁻² < 1/4. Hence for certain graphs there exists a unital homomorphism A(Γ₁) → A(Γ₂) if ||Γ₁|| ≤ ||Γ₂||. For certain finite T-shaped graphs K₀ (A(Γ)) ≅ ℤ [t]/<Q> where <Q> denotes the ideal generated by a polynomial Q=Q(Γ) which is essentially the characteristic polynomial of the graph Γ and positive cone identified with {0}∪ {f + <Q>: f(γ) > 0} where γ = γ(Γ) = ||Γ||^-2. Hence there exists a unital homomorphism A(Γ₁) → A(Γ₂) if ||Γ₁|| = ||Γ₂||, and Q(Γ₂) divides Q(Γ₁). The structure of K₀(A(Γ)) as an ordered ring is related to the fusion rules of rational conformal field theory.

Let M be a compact smooth manifold, let be a unital involutive subalgebra of the von Neumann algebra £ (H) of bounded linear operators of some Hilbert space H, let be the unital involutive algebra , let be an hermitian projective right -module of finite type, and let be the gauge group of unitary elements of the unital involutive algebra of right -linear endomorphisms of . We first prove that noncommutative geometry provides the suitable setting upon which a consistent theory of energy representations can be built. Three series of energy representations are constructed. The first consists of energy representations of the gauge group , being the group of unitary elements of , associated with integrable Riemannian structures of M, and the second series consists of energy representations associated with (d, ∞)-summable K-cycles over . In the case where is a von Neumann algebra of type II₁ a third series is given: we introduce the notion of regular quasi K-cycle, we prove that regular quasi K-cycles over always exist, and that each of them induces an energy representation.

We consider a version of the notion of F-inverse semigroup (studied in the algebraic theory of inverse semigroups). We point out that an action of such an inverse semigroup on a locally compact space has associated a natural groupoid construction, very similar to the one of a transformation group. We discuss examples related to Toeplitz algebras on subsemigroups of discrete groups, to Cuntz-Krieger algebras, and to crossed-products by partial automorphisms in the sense of Exel.

A theory of highest weight modules over an arbitrary finite-dimensional Lie superalgebra is constructed. A necessary and sufficient condition for the finite-dimensionality of such modules is proved. Generic finite-dimensional irreducible representations are defined and an explicit character formula for such representations is written down. It is conjectured that this formula applies to any generic finite-dimensional irreducible module over any finite-dimensional Lie superalgebra. The conjecture is proved for several classes of Lie superalgebras, in particular for all solvable ones, for all simple ones, and for certain semi-simple ones.

We determine, in terms of and , the homotopy groups of certain groups of invertibles and of certain equivalence classes in the infinite Grassmann space on a Hilbert C*--module. These results provide various interpretations of .