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Zoran Škoda

Publications and source records attributed to Zoran Škoda.

32 records · Page 2Linked to original sources
arXiv2012

Exponential Formulas and Lie Algebra Type Star Products

Given formal differential operators $F_i$ on polynomial algebra in several variables $x_1,...,x_n$, we discuss finding expressions $K_l$ determined by the equation $\exp(\sum_i x_i F_i)(\exp(\sum_j q_j x_j)) = \exp(\sum_l K_l x_l)$ and their applications. The expressions for $K_l$ 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 $K_l$. We elaborate an example for a Lie algebra $su(2)$, related to a quantum gravity application from the literature.

math.QA↗
arXiv2011

Čech cocycles for quantum principal bundles

In other to study connections and gauge theories on noncommutative spaces it is useful to use the local trivializations of principal bundles. In this note we show how to use noncommutative localization theory to describe a simple version of cocycle data for the bundles on noncommutative schemes with Hopf algebras in the role of the structure group which locally look like Hopf algebraic smash products. We also show how to use these Čech cocycles for associated vector bundles. We sketch briefly some examples related to quantum groups.

math.QA↗
arXiv2011

Heisenberg double versus deformed derivatives

Two 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.

math.QA↗
arXiv2010

Categorified symmetries

Quantum field theory allows more general symmetries than groups and Lie algebras. For instance quantum groups, that is Hopf algebras, have been familiar to theoretical physicists for a while now. Nowdays many examples of symmetries of categorical flavor -- categorical groups, groupoids, Lie algebroids and their higher analogues -- appear in physically motivated constructions and faciliate constructions of geometrically sound models and quantization of field theories. Here we consider two flavours of categorified symmetries: one coming from noncommutative algebraic geometry where varieties themselves are replaced by suitable categories of sheaves; another in which the gauge groups are categorified to higher groupoids. Together with their gauge groups, also the fiber bundles themselves become categorified, and their gluing (or descent data) is given by nonabelian cocycles, generalizing group cohomology, where infinity-groupoids appear in the role both of the domain and the coefficient object. Such cocycles in particular represent higher principal bundles, gerbes, -- possibly equivariant, possibly with connection -- as well as the corresponding associated higher vector bundles. We show how the Hopf algebra known as the Drinfeld double arises in this context. This article is an expansion of a talk that the second author gave at the 5th Summer School of Modern Mathematical Physics in 2008.

math.QA↗
arXiv2009

Some equivariant constructions in noncommutative algebraic geometry

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.

math.AG↗
arXiv2009

Compatibility of (co)actions and localizations

Earlier, Lunts and Rosenberg studied a notion of compatibility of endofunctors with localization functors, with an application to the study of differential operators on noncommutative rings and schemes. Another compatibility -- of Ore localizations of an algebra with a comodule algebra structure over a given bialgebra -- introduced in my earlier work -- is here described also in categorical language, but the appropriate notion differs from that of Lunts and Rosenberg, and it involves a specific kind of distributive laws. Some basic facts about compatible localization follow from more general functoriality properties of associating comonads or even actions of monoidal categories to comodule algebras. We also introduce localization compatible pairs of entwining structures.

math.QA↗
arXiv2008

Bicategory of entwinings

We define a bicategory in which the 0-cells are the entwinings over variable rings. The 1-cells are triples of a bimodule and two maps of bimodules which satisfy an additional hexagon, two pentagons and two (co)unit triangles; and the 2-cells are the maps of bimodules satisfying two simple compatibilities. The operation of getting the "composed coring" from a given entwining, is promoted here to a canonical morphism of bicategories from a bicategory of entwinings to the Street's bicategory of corings.

math.RA↗
arXiv2007

Leibniz rules for enveloping algebras in symmetric ordering

Given a finite-dimensional Lie algebra, and a representation by derivations on the completed symmetric algebra of its dual, a number of interesting twisted constructions appear: certain twisted Weyl algebras, deformed Leibniz rules, quantized ``star'' product. We first illuminate a number of interrelations between these constructions and then proceed to study a special case in certain precise sense corresponding to the symmetric or Weyl ordering. This case has been known earlier to be related to computations with Hausdorff series, for example the expression for the star product is in such terms. For the deformed Leibniz rule, hence a coproduct, we present here a new nonsymmetric expression, which is then expanded into a sum of expressions labelled by a class of planar trees, and for a given tree evaluated by Feynman-like rules. These expressions are filtered by a bidegree and we show recursion formulas for the sums of expressions of a given bidegree, and compare the recursions to recursions for Hausdorff series, including the comparison of initial conditions. This way we show a direct corespondence between the Hausdorff series and the expression for twisted coproduct.

math.QA↗
arXiv2007

Equivariant monads and equivariant lifts versus a 2-category of distributive laws

Fix a monoidal category C. The 2-category of monads in the 2-category of C-actegories, colax C-equivarant functors, and C-equivariant natural transformations of colax functors, may be recast in terms of pairs consisting of a usual monad and a distributive law between the monad and the action of C, morphisms of monads respecting the distributive law, and transformations of monads satisfying some compatibility with the actions and distributive laws involved. The monads in this picture may be generalized to actions of monoidal categories, and actions of PRO-s in particular. If C is a PRO as well, then in special cases one gets various distributive laws of a given classical type, for example between a comonad and an endofunctor or between a monad and a comonad. The usual pentagons are in general replaced by multigons, and there are also ``mixed'' multigons involving two distinct distributive laws. Beck's bijection between the distributive laws and lifts of one monad to the Eilenberg-Moore category of another monad is here extended to an isomorphism of 2-categories. The lifts of maps of above mentioned pairs are colax C-equivariant. We finish with a short treatment of relative distributive laws between two pseudoalgebra structures which are relative with respect to the distributivity of two pseudomonads involved, what gives a hint toward the generalizations.

math.CT↗
arXiv2007

Quantum heaps, cops and heapy categories

A heap is a structure with a ternary operation which is intuitively a group with forgotten unit element. Quantum heaps are associative algebras with a ternary cooperation which are to the Hopf algebras what heaps are to groups, and, in particular, the category of copointed quantum heaps is isomorphic to the category of Hopf algebras. There is an intermediate structure of a cop in monoidal category which is in the case of vector spaces to a quantum heap about what is a coalgebra to a Hopf algebra. The representations of Hopf algebras make a rigid monoidal category. Similarly the representations of quantum heaps make a kind of category with ternary products, which we call a heapy category.

math.QA↗
arXiv2006

A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra

Given a $n$-dimensional Lie algebra $g$ over a field $k \supset \mathbb Q$, together with its vector space basis $X^0_1,..., X^0_n$, we give a formula, depending only on the structure constants, representing the infinitesimal generators, $X_i = X^0_i t$ in $g\otimes_k k [[t]]$, where $t$ is a formal variable, as a formal power series in $t$ with coefficients in the Weyl algebra $A_n$. Actually, the theorem is proved for Lie algebras over arbitrary rings $k\supset Q$. We provide three different proofs, each of which is expected to be useful for generalizations. The first proof is obtained by direct calculations with tensors. This involves a number of interesting combinatorial formulas in structure constants. The final step in calculation is a new formula involving Bernoulli numbers and arbitrary derivatives of coth(x/2). The dimensions of certain spaces of tensors are also calculated. The second method of proof is geometric and reduces to a calculation of formal right-invariant vector fields in specific coordinates, in a (new) variant of formal group scheme theory. The third proof uses coderivations and Hopf algebras.

math.RT↗
arXiv2006

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.

math.QA↗
arXiv2005

Included-row exchange principle for quantum minors

The main theorem of the paper allows to generalize a class of identities among the quantum minors for quantum linear groups to similar identities but with the row labels of the quantum minors involved permuted.

math.QA↗