One particular approach to quantum groups (matrix pseudo groups) provides the Manin quantum plane. Assuming an appropriate set of non-commuting variables spanning linearly a representation space one is able to show that the endomorphisms on that space preserving the non-commutative structure constitute a quantum group. The noncommutativity of these variables provide an example of non-commutativ...

We introduce the Arithmetic Site: an algebraic geometric space deeply related to the non-commutative geometric approach to the Riemann Hypothesis. We prove that the non-commutative space quotient of the adèle class space of the field of rational numbers by the maximal compact subgroup of the idèle class group, which we had previously shown to yield the correct counting function to obtain the co...

We develop a GL qp (2) invariant differential calculus on a two-dimensional noncommutative quantum space. Here the coordinate space for the exterior quantum plane is spanned by the differentials that are commutative (bosonic) in nature.

A family of recently discovered commutative C∗-algebras of Toeplitz operators on the unit disk can be classified as follows. Each pencil of hyperbolic straight lines determines a set of symbols consisting of functions which are constant on the corresponding cycles, the orthogonal trajectories to lines forming a pencil. The C∗-algebra generated by Toeplitz operators with such symbols turns out t...

It is well known that an (in general, non-commutative) set of non-Hermitian operators $\Lambda_j$ with real eigenvalues need not necessarily represent observables. We describe a specific class quantum models in which these plus the underlying physical Hilbert-space metric $\Theta$ are all represented terms auxiliary operator $(N+1)-$plet $Z_k$, $k=0,1,\ldots,N$. Our formalism degenerates to ${\...

We classify all the pairs of a commutative associative algebra with an identity element and its finite-dimensional commutative locally-finite derivation subalgebra such that the commutative associative algebra is derivation-simple with respect to the derivation subalgebra over an algebraically closed field with characteristic 0. Such pairs are the fundamental ingredients for constructing genera...

We have discovered an eecient algorithm for matching and uniication in associa-tive-commutative (AC) equational theories. In most cases of AC uniication our method obviates the need for solving diophantine equations, and thus avoids one of the bottlenecks of other associative-commutative uniication techniques. The algorithm eeciently utilizes powerful constraints to eliminate much of the search...

In the papers [La 5,6], we introduced the notion of (non-commutative) phase algebras (space) Phn(A), n = 0, 1, ...,∞ associated to any associative algebra A (space), defined over a field k. The purpose of this paper is to prove that this construction is useful in non-commutative deformation theory for the construction of the versal family of families of modules, see [La 4]. In particular we obt...

The cohomology groups of line bundles over complex tori (or abelian varieties) are classically studied invariants of these spaces. In this article, we compute the cohomology groups of line bundles over various holomorphic, non-commutative deformations of complex tori. Our analysis interpolates between two extreme cases. The first case is a calculation of the space of (cohomological) theta funct...