An element of a Coxeter group is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. An element of a Coxeter group is cyclically fully commutative if any of its cyclic shifts remains fully commutative. These elements were studied by Boothby et al. In particular the authors precisely identified the Coxeter groups ...

We investigate asymptotic behaviors of the strong coupling limit in the N = 2 supersymmetric non-commutative Yang-Mills theory. The strong coupling behavior is quite different from the commutative one since the non-commutative dual U(1) theory is asymptotic free, although the monodoromy is the same as that of the ordinary theory. Singularities are produced by infinitely heavy monopoles and dyon...

An r-commutative algebra is an algebra A equipped with a Yang-Baxter operator R:A ⊗ A → A ⊗ A satisfying m = mR, where m:A ⊗ A → A is the multiplication map, together with the compatibility conditions R(a⊗ 1) = 1 ⊗ a, R(1 ⊗ a) = a ⊗ 1, R(id ⊗m) = (m ⊗ id)R2R1 and R(m ⊗ id) = (id ⊗ m)R1R2. The basic notions of differential geometry extend from commutative (or supercommutative) algebras to r-comm...

After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasi-coherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, Hopf-Galois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions...

We describe the construction of non-commutative manifolds, which are the non-commutative analogs of homogeneous spaces using coherent states. In the commutative limit we obtain standard manifolds. Applications to the Fuzzy sphere and to the Fuzzy hyperboloid are discussed in more detail. *) Part of Project Nr. P8916-PHY of the \Fonds zur FF orderung der wissenschaftlichen Forschung in Osterreich".

Let Γ be a finite graph and GΓ be the corresponding free partially commutative group. In this paper we construct orthogonality theory for graphs and free partially commutative groups. The theory developed here provides tools for the study of the structure of the centraliser lattice of partially commutative groups.

Robert Lee Wilson Department of Mathematics Rutgers University The determinant of a matrix with entries in a commutative ring is a main organizing tool in commutative algebra. In these lectures, we present an analogous theory, the theory of quasideterminants, for matrices with entries in a not necessarily commutative ring. The theory of quasideterminants was originated by I. Gelfand and V. Retakh.

This paper studies the unification problem with associative, commutative, and associative-commutative functions mainly from a viewpoint of the parameterized complexity on the number of variables. It is shown that both associative and associative-commutative unification problems are W [1]-hard. A fixed-parameter algorithm and a polynomialtime algorithm are presented for special cases of commutat...

Parikh’s Theorem says that the commutative image of every context free language is the commutative image of some regular set. Pilling has shown that this theorem is essentially a statement about least solutions of polynomial inequalities. We prove the following general theorem of commutative Kleene algebra, of which Parikh’s and Pilling’s theorems are special cases: Every finite system of polyn...

1. DERIVED GEOMETRY WITH L∞ ALGEBRAS We are interested in studying formal derived moduli problems, as an orienting remark recall that particularly nice simplicial sets are those that are Kan complexes and that the nerve of a category C is Kan complex if and only if C is a groupoid. Consider the following progression. • Schemes: Functors from commutative algebras to sets; • Stacks: Functors from...