Two general families of new quantum deformed current algebras are proposed and identified both as infinite Hopf family of algebras, a structure which enable one to define “tensor products” of these algebras. The standard quantum affine algebras turn out to be a very special case of both algebra families, in which case the infinite Hopf family structure degenerates into standard Hopf algebras. T...

We prove a version of B ezout's theorem for non-commutative analogues of the projective spaces P n. 0. Introduction Throughout we work over an algebraically closed eld k. We establish a version of B ezout's Theorem for non-commutative projective spaces, quantum P n s for short. If Y is a quantum P n then the alternating sum of the dimension of the Ext-groups gives a bilinear form : K 0 (Y) K 0 ...

We consider Feynman’s path integral approach to quantum mechanics with a noncommutativity in position and momentum sectors of the phase space. We show that a quantum-mechanical system with this kind of noncommutativity is equivalent to the another one with usual commutative coordinates and momenta. We found connection between quadratic classical Hamiltonians, as well as Lagrangians, in their co...

We review the main features of the Weyl-Wigner formulation of noncommutative quantum mechanics. In particular, we present a ⋆-product and a Moyal bracket suitable for this theory as well as the concept of noncommutative Wigner function. The properties of these quasi-distributions are discussed as well as their relation to the sets of ordinary Wigner functions and positive Liouville probability ...

We continue the investigation of Hopf-Galois extensions with central invariants started in [30]. Our objective is not to imitate algebraic geometry using Hopf-Galois extension but to understand their geometric properties. Let H be a finite-dimensional Hopf algebra over a ground field k. Our main object of study is an H-Galois extension U ⊇ O such that O is a central subalgebra of U . Let us bri...

Recently three dimensional Einstein gravity with AdS geometry has been studied, and pointed out to be described with Chern-Simons theory by Grumiller and Jackiw. While, non-commutative Chern-Simons theory is known to be equivalent to Quantum Hall Effect, through Area Preserving Diffeomorphism by Susskind and others. Conbining these, three dimensional gravity is studied in this paper in the cont...

We initiate a unified, axiomatic study of noncommutative algebras R whose prime spectra are, in a natural way, finite unions of commutative noetherian spectra. Our results illustrate how these commutative spectra can be functorially “sewn together” to form SpecR. In particular, we construct a bimodule-determined functor ModZ → ModR, for a suitable commutative noetherian ring Z, from which there...

In this survey we present the appearance of some combinatorial notions in quantum field theory. We first focus on graph polynomials (the Tutte polynomial and its multivariate version) and their relation with the parametric representation of the commutative Φ field theory. We then generalize this to ribbon graphs and present the relation of the Bollobás–Riordan polynomial with the parametric rep...

Many important quantum algebras such as quantum symplectic space, quantum Euclidean space, quantummatrices, q-analogs of the Heisenberg algebra and the quantum Weyl algebra are semi-commutative. In addition, enveloping algebras U(L+) of even Lie color algebras are also semi-commutative. In this paper, we generalize work of Montgomery and examine the X-inner automorphisms of such algebras. The t...

Furber and Jacobs have shown in their study of quantum computation that the category of commutative C∗-algebras and PU-maps (positive linear maps which preserve the unit) is isomorphic to the Kleisli category of a comonad on the category of commutative C∗-algebras with MIU-maps (linear maps which preserve multiplication, involution and unit). [3] In this paper, we prove a non-commutative varian...