Given a collection P = {p1(x1, . . . , x2k2), . . . , pk2(x1, . . . , x2k2)} of k commutative polynomials in 2k variables, the objective is to find a condensed representation for these polynomials in terms of a single non-commutative polynomial p(X,Y ) in two k × k matrix variables X and Y . Algorithms that will generically determine whether the given family P has a non-commutative representati...

A Coxeter group element w is fully commutative if any reduced expression for w can be obtained from any other via the interchange of commuting generators. For example, in the symmetric group of degree n, the number of fully commutative elements is the nth Catalan number. The Coxeter groups with finitely many fully commutative elements can be arranged into seven infinite families An , Bn , Dn , ...

In this article, we define a non-commutative deformation of the ”symplectic invariants” (introduced in [13]) of an algebraic hyperelliptical plane curve. The necessary condition for our definition to make sense is a Bethe ansatz. The commutative limit reduces to the symplectic invariants, i.e. algebraic geometry, and thus we define non-commutative deformations of some algebraic geometry quantit...

Elliptic Quantum Planes means here non-commutative deformations of the complex projective plane P(C). We consider deformations in the realm of non-commutative (complex) algebraic geometry. As we recall in the first section, elliptic modulus parameter enters into the game. Hence the adjective “elliptic” is used. Note also that, in that world, the complex projective line P(C), namely the Riemann ...

We give characterizations of the property of being weakly reductive for finitely generated commutative semigroups. As a consequence of these characterizations we obtain an algorithm for determining from a presentation of a finitely generated commutative semigroup whether it is weakly reductive. Furthermore, we present some connections between cancellative finitely generated commutative semigrou...

We investigate SUSY of Wess-Zumino models in non(anti-)commutative Euclidean superspaces. Non(anti-)commutative deformations break 1/2 SUSY, then non(anti)commutative Wess-Zumino models do not have full SUSY in general. However, we can recover full SUSY at specific coupling constants satisfying some relations. We give a general way to construct full SUSY non(anti-)commutative Wess-Zumino models...

We say that a C∗-algebra X has the approximate n-th root property (n ≥ 2) if for every a ∈ X with ‖a‖ ≤ 1 and every ε > 0 there exits b ∈ X such that ‖b‖ ≤ 1 and ‖a − bn‖ < ε. Some properties of commutative and non-commutative C∗-algebras having the approximate nth root property are investigated. In particular, it is shown that there exists a non-commutative (resp., commutative) separable unita...

In this paper, we describe all equational theories of commutative semigroups in terms of certain well-quasi-orderings on the set of finite sequences of nonnegative integers. This description yields many old and new results on varieties of commutative semigroups. In particular, we obtain also a description of the lattice of varieties of commutative semigroups, and we give an explicit uniform sol...

A commutative local Cohen-Macaulay ring R of finite Cohen-Macaulay type is known to be an isolated singularity; that is, Spec(R) \ {m} is smooth. This paper proves a non-commutative analogue. Namely, if A is a (non-commutative) graded Artin-Schelter Cohen-Macaulay algebra which is FBN and has finite Cohen-Macaulay type, then the non-commutative projective scheme determined by A is smooth.

The maximal dimension of a commutative subalgebra of the Grassmann algebra is determined. It is shown that for any commutative subalgebra there exists an equidimensional commutative subalgebra spanned by monomials. It follows that the maximal dimension of a commutative subalgebra can be expressed in terms of the maximal size of an intersecting system of subsets of odd size in a finite set. 2010...