Zoran Škoda

Hermitian realizations of κ-Minkowski space–time

General realizations, star products and plane waves for κ-Minkowski space–time are considered. Systematic construction of general Hermitian realization is presented, with special emphasis on noncommutative plane waves and Hermitian star product. Few examples are elaborated and possible physical applications are mentioned.

Exponential Formulas and Lie Algebra Type Star Products

Given formal differential operators Fi on polynomial algebra in several variables x1;:::;xn, we discuss finding expressions Kl determined by the equation exp( P i xiFi)(exp( P j qjxj)) = exp( P l Klxl) and their applications. The expressions for Kl are related to the coproducts for deformed momenta for the noncommutative space-times of Lie algebra type and also appear in the computations with a class of star products. We find combinatorial recursions and derive formal differential equations for finding Kl. We elaborate an example for a Lie algebra su(2), related to a quantum gravity application from the literature.

HEISENBERG DOUBLE VERSUS DEFORMED DERIVATIVES

A common replacement of the tangent space to a noncommutative space whose coordinate algebra is the enveloping algebra of a Lie algebra is generated by the deformed derivatives, usually defined by procedures involving orderings among noncommutative coordinates. We show that an approach to extending the noncommutative configuration space to a phase space, based on a variant of Heisenberg double, more familiar for some other algebras, e.g. quantum groups, is in the Lie algebra case equivalent to the approach via deformed derivatives. The dependence on the ordering is now in the form of the choice of a suitable linear isomorphism between the full algebraic dual of the enveloping algebra and a space of formal differential operators of infinite order. 1. Noncommutative algebras and noncommutative geometry may play various roles in models of mathematical physics; for example describing quantum symmetry algebras. A special case of interest is when the noncommutative algebra is playing the role of the space-time of the theory, and is interpreted as a small deformation of the 1-particle configuration space. If one wants to proceed toward developing field theory on such a space, it is beneficial to introduce the extension of the deformation of configuration space to a deformation of full phase space (symplectic manifold) of the theory. Deformed momentum space for the noncommutative configuration space whose coordinate algebra is the enveloping algebra of a finite-dimensional Lie algebra (also called Lie algebra type noncommutative spaces) has been studied recently in mathematical physics literature ([1, 2, 4]), mainly in special cases, most notably variants of so-called κ-Minkowski space ([2, 9, 8, 12]). We have related several approaches to the phase space deformations in [11], for a general Lie algebra type noncommutative space. The differential forms and exterior derivative can also be extended to the same setup ([15]). The algebras in the article are over a field k of characteristic zero; both real and complex numbers appear in applications of the present formalism. We fix a Lie algebra g with basis x̂1, . . . , x̂n. 2. Lie algebra type noncommutative spaces are simply the deformation quantizations of the linear Poisson structure; given structure constants C jk linear in a deformation parameter the enveloping algebras of the Lie algebra g given in a base by [x̂j , x̂k] = C i jkx̂k is viewed as a deformation of the polynomial (symmetric) algebra S(g); by xi without hat we denote the generators ofTwo approaches to the tangent space of a noncommutative space whose coordinate algebra is the enveloping algebra of a Lie algebra are known: the Heisenberg double construction and the approach via deformed derivatives, usually defined by procedures involving orderings among noncommutative coordinates or equivalently involving realizations via formal differential operators. In an earlier work, we rephrased the deformed derivative approach introducing certain smash product algebra twisting a semicompleted Weyl algebra. We show here that the Heisenberg double in the Lie algebra case, is isomorphic to that product in a nontrivial way, involving a datum ϕ parametrizing the orderings or realizations in other approaches. This way, we show that the two different formalisms, used by different communities, for introducing the noncommutative phase space for the Lie algebra type noncommutative spaces are mathematically equivalent.

Coherent States for Hopf Algebras

Families of Perelomov coherent states are defined axiomatically in the context of unitary representations of Hopf algebras. A global geometric picture involving locally trivial noncommutative fibre bundles is involved in the construction. If, in addition, the Hopf algebra has a left Haar integral, then a formula for noncommutative resolution of identity in terms of the family of coherent states holds. Examples come from quantum groups.

Every Quantum Minor Generates an Ore Set

The subset multiplicatively generated by any given set of quantum minors and the unit element in the quantum matrix bialgebra satisfies the left and right Ore conditions. Quantum matrix groups ([9, 11, 12]) have remarkable algebraic properties and connections to several branches of mathematics and mathematical physics. A viewpoint of the noncommutative geometry may elucidate some of their properties. In the formalism in which a quantum group is described by a matrix bialgebra G, e.g. SLq(n), the shifts of the main Bruhat cells are expected also to have quantum analogues which are localizations of G. The geometry is richer and more akin to the classical case, if these localizations have good flatness properties. Ore localization is the most well-understood kind of localizations. Ore localizations are biflat (in terminology of [13]), and appear often in “quantum” situation, that is, when the noncommutative algebra is just a deformation of a commutative algebra. Thus one expects to realize the quantum main Bruhat cell and its Weyl group shifts as the (noncommutative spectra of) the localized algebras of the form G[S w ] where Sw are certain Ore subsets in G, depending on the element w in the Weyl group W . Furher support of this conjectural picture is a result of A. Joseph [5], that there is a natural family of Ore sets S̄w, w ∈ W , in the graded algebra R representing the quantum analogue of the basic affine space G/U , such that in the commutative case the spectra of the localizations R[S̄ w ] are exactly the images of the Bruhat cells in the basic affine space. However, the Ore property for the natural candidates for Sw has not been proved so far. Trying to answer the question of V. Drinfeld to find the quantum analogue of the Beilinson-Bernstein localization theorem, Y. Soibelman has shown the satisfactory localization picture for SLq(2) (unpublished), and came to the conclusion that for SLq(n), n > 2, already the Ore property of Sw is far from obvious, if not even wrong. Quantum Beilinson-Bernstein theorem has been further studied by Lunts and Rosenberg [7], Tanisaki [20] and others, but in different approaches. In his thesis, the present author has developed a direct localization approach ([15, 17]) to the construction of the coset spaces of the quantum linear groups and the locally trivial quantum principal fibrations deforming the classical fibrations of the type G → G/B and having Hopf algebras as

Some Equivariant Constructions in Noncommutative Algebraic Geometry

Abstract We here present rudiments of an approach to geometric actions in noncommutative algebraic geometry, based on geometrically admissible actions of monoidal categories. This generalizes the usual (co)module algebras over Hopf algebras which provide affine examples. We introduce a compatibility of monoidal actions and localizations which is a distributive law. There are satisfactory notions of equivariant objects, noncommutative fiber bundles and quotients in this setup.