In loving memory of my mother " A mathematician is a machine for converting coffee into theorems. " —Alfréd Rényi " A comathematician, by categorical duality, is a machine for converting cotheorems into ffee. " —anonymous Preface Hopf algebras are a relatively new concept in algebra, first encountered by their namesake Heinz Hopf in 1941 in the field of algebraic topology. In the 1960s, study o...

In previous joint work with Eli Aljadeff we attached a generic Hopf Galois extension A H to each twisted algebra H obtained from a Hopf algebra H by twisting its product with the help of a cocycle α. The algebra A H is a flat deformation of H over a “big” central subalgebra B H and can be viewed as the noncommutative analogue of a versal torsor in the sense of Serre. After surveying the results...

This thesis consists of four papers. In the first paper we present methods and explicit formulas for describing simple weight modules over twisted generalized Weyl algebras. Under certain conditions we obtain a classification of a class of locally finite simple weight modules from simple modules over tensor products of noncommutative tori. As an application we describe simple weight modules ove...

Under the name prime decomposition (pd), a unique decomposition of an arbitrary N dimensional density matrix ρ into a sum of seperable density matrices with dimensions given by the coprime factors of N is introduced. For a class of density matrices a complete tensor product factorization is achieved. The construction is based on the Chinese Remainder Theorem, and the projective unitary represen...

Under the name prime decomposition (pd), a unique decomposition of an arbitrary N dimensional density matrix ρ into a sum of seperable density matrices with dimensions determined by the coprime factors of N is introduced. For a class of density matrices a complete tensor product factorization is achieved. The construction is based on the Chinese Remainder Theorem, and the projective unitary rep...

Hahn’s famous structure theorem states that totally ordered Abelian groups can be embedded in the lexicographic product of real groups. Our main theorem extends this structural description to order-dense, commutative, group-like residuated chains, which has only finitely many idempotents. It is achieved via the so-called partial-lexicographic product construction (to be introduced here) using t...