Introduction. In our earlier papers [2,3,4,5,6], we investigated stable operations and cooperations in elliptic cohomology and its variants, relating these to known operations on rings of modular forms. The purpose of this article is to give an introduction to these stable operation algebras, in particular explaining the connections with Hecke algebras and Morava stabilizer algebras; further de...

We introduce and study two exotic families of finite-dimensional algebras over an algebraically closed field. prove that every such algebra is derived equivalent to a higher spherical studied by Erdmann Skowroński (Arch. Math. 114, 25–39, 2020), hence it tame symmetric periodic period 4. This together with the results (Algebr. Represent. Theor. 22, 387–406, 2019; Arch. 2020) shows trivial exten...

We study the periodic cyclic homology groups of the cross-product of a finite type algebra A by a discrete group Γ. In case A is commutative and Γ is finite, our results are complete and given in terms of the singular cohomology of the sets of fixed points. These groups identify our cyclic homology groups with the “orbifold cohomology” of the underlying (algebraic) orbifold. The proof is based ...

We define and study the properties of the infinite dimensional quantized Kronecker flow. This C-dynamical system arises as a quantization of the corresponding flow on an infinite dimensional torus. We prove an ergodic theorem for a class of quantized Kronecker flows. We also study the closely related almost periodic quantum field theory of bosonic, fermionic and supersymmetric particles. We pro...

For differential operators which are invariant under the action of an abelian group Bloch theory is the tool of choice to analyze spectral properties. By shedding some new non-commutative light on this we motivate the introduction of a noncommutative Bloch theory for elliptic operators on Hilbert C-modules. It relates properties of C-algebras to spectral properties of module operators such as b...

The central result of this paper is an explicit computation of the Hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds X as their spectrum. More precisely, the Hochschild homology is identified with the space of differential forms on X, and the periodic cyclic homology with the twisted de Rham cohomology of X, thereby gener...

This work is an investigation into the structure and properties of Lie hypermatrix algebra generated by a semisimple basis. By using new algebraic tools; namely cubic hypermatrices I obtain an algebraic structure associated with the basis of a semisimple Lie algebra, and I show that the semisimple Lie basis is a generator of infinite periodic semisimple hypermatrix structures, that has a classi...

we prove that rs-bl-algebras are mv-algebras.

Let GA be an AF -algebra given by a periodic Bratteli diagram with the incidence matrix A. Depending on a polynomial p(x) ∈ Z[x], we assign to GA a finite abelian group Abp(x)(GA) = Z /p(A)Z. It is shown that for every p(x), such that p(0) = ±1, the Abp(x)(GA) is an invariant of the strong stable isomorphism class of the AF -algebra GA. Using a categorical correspondence between the elliptic cu...