20 research outputs found
Noncommutative localization in noncommutative geometry
The aim of these notes is to collect and motivate the basic localization
toolbox for the geometric study of ``spaces'', locally described by
noncommutative rings and their categories of one-sided modules.
We present the basics of Ore localization of rings and modules in much
detail. Common practical techniques are studied as well. We also describe a
counterexample for a folklore test principle. Localization in negatively
filtered rings arising in deformation theory is presented. A new notion of the
differential Ore condition is introduced in the study of localization of
differential calculi.
To aid the geometrical viewpoint, localization is studied with emphasis on
descent formalism, flatness, abelian categories of quasicoherent sheaves and
generalizations, and natural pairs of adjoint functors for sheaf and module
categories. The key motivational theorems from the seminal works of Gabriel on
localization, abelian categories and schemes are quoted without proof, as well
as the related statements of Popescu, Watts, Deligne and Rosenberg.
The Cohn universal localization does not have good flatness properties, but
it is determined by the localization map already at the ring level. Cohn
localization is here related to the quasideterminants of Gelfand and Retakh;
and this may help understanding both subjects.Comment: 93 pages; (including index: use makeindex); introductory survey, but
with few smaller new result
Deformed Covariant Quantum Phase Spaces as Hopf Algebroids
We consider the general D=4 (10+10)-dimensional kappa-deformed quantum phase
space as given by Heisenberg double \mathcal{H} of D=4 kappa-deformed
Poincare-Hopf algebra H. The standard (4+4) -dimensional kappa - deformed
covariant quantum phase space spanned by kappa - deformed Minkowski coordinates
and commuting momenta generators ({x}_{\mu },{p}_{\mu }) is obtained as the
subalgebra of \mathcal{H}. We study further the property that Heisenberg double
defines particular quantum spaces with Hopf algebroid structure. We calculate
by using purely algebraic methods the explicite Hopf algebroid structure of
standard kappa - deformed quantum covariant phase space in Majid-Ruegg
bicrossproduct basis. The coproducts for Hopf algebroids are not unique,
determined modulo the coproduct gauge freedom. Finally we consider the
interpretation of the algebraic description of quantum phase spaces as Hopf
algebroids.Comment: 14 pages,v4; this version appeared in Physics Letters
Generalizations of Poisson structures related to rational Gaudin model
The Poisson structure arising in the Hamiltonian approach to the rational
Gaudin model looks very similar to the so-called modified Reflection Equation
Algebra. Motivated by this analogy, we realize a braiding of the mentioned
Poisson structure, i.e. we introduce a "braided Poisson" algebra associated
with an involutive solution to the quantum Yang-Baxter equation. Also, we
exhibit another generalization of the Gaudin type Poisson structure by
replacing the first derivative in the current parameter, entering the so-called
local form of this structure, by a higher order derivative. Finally, we
introduce a structure, which combines both generalizations. Some commutative
families in the corresponding braided Poisson algebra are found.Comment: LATEX, 16 p
Symmetric ordering and Weyl realizations for quantum Minkowski spaces
Symmetric ordering and Weyl realizations for non commutative quantum
Minkowski spaces are reviewed. Weyl realizations of Lie deformed spaces and
corresponding star products, as well as twist corresponding to Weyl realization
and coproduct of momenta are presented. Drinfeld twists understood in Hopf
algebroid sense are also discussed. A few examples of corresponding Weyl
realizations are given. We show that for the original Snyder space there exists
symmetric ordering, but no Weyl realization. Quadratic deformations of
Minkowski space are considered and it is demonstrated that symmetric ordering
is deformed and a generalized Weyl realization can be defined.Comment: 27 page