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Zoran Škoda

Publications and source records attributed to Zoran Škoda.

At least 19 recordsLinked to original sources
arXiv2026

Drinfeld-Xu bialgebroid 2-cocycles twist the antipode

Ping Xu generalized Drinfeld 2-cocycles from bialgebras to associative bialgebroids over noncommutative base algebras. Any counital Drinfeld--Xu 2-cocycle twists the base algebra of the bialgebroid and a comultiplication on the total algebra, obtaining a new, twisted bialgebroid. Antipodes for bialgebroids have been considered, but finding a general way to twist the antipode, which is straightforward in the Hopf algebra case, appeared somewhat elusive. In this article, we prove that if an invertible antipode $S$ for the original bialgebroid exists, and another expression $V_F$ depending on the 2-cocycle $F$ is invertible, then the expected conjugation formula $S_F(-) = V_F^{-1} S(-) V_F$ indeed produces an invertible antipode $S_F$ for the twisted bialgebroid.

math.QA↗
arXiv2025

Enveloping algebra is a Yetter--Drinfeld module algebra over Hopf algebra of regular functions on the automorphism group of a Lie algebra

We present an elementary construction of a (highly degenerate) Hopf pairing between the universal enveloping algebra $U(\mathfrak{g})$ of a finite-dimensional Lie algebra $\mathfrak{g}$ over arbitrary field $\mathbf{k}$ and the Hopf algebra $\mathcal{O}(\mathrm{Aut}(\mathfrak{g}))$ of regular functions on the automorphism group of $\mathfrak{g}$. This pairing induces a Hopf action of $\mathcal{O}(\mathrm{Aut}(\mathfrak{g}))$ on $U(\mathfrak{g})$ which together with an explicitly given coaction makes $U(\mathfrak{g})$ into a braided commutative Yetter--Drinfeld $\mathcal{O}(\mathrm{Aut}(\mathfrak{g}))$-module algebra. From these data one constructs a Hopf algebroid structure on the smash product algebra $\mathcal{O}(\mathrm{Aut}(\mathfrak{g}))\sharp U(\mathfrak{g})$ retaining essential features from earlier constructions of a Hopf algebroid structure on infinite-dimensional versions of Heisenberg double of $U(\mathfrak{g})$, including a noncommutative phase space of Lie algebra type, while avoiding the need of completed tensor products. We prove a slightly more general result where algebra $\mathcal{O}(\mathrm{Aut}(\mathfrak{g}))$ is replaced by $\mathcal{O}(\mathrm{Aut}(\mathfrak{h}))$ and where $\mathfrak{h}$ is any finite-dimensional Leibniz algebra having $\mathfrak{g}$ as its maximal Lie algebra quotient.

math.QA↗
arXiv2024

Brane mechanics and gapped Lie n-algebroids

We draw a parallel between the BV/BRST formalism for higher-dimensional ($\ge 2$) Hamiltonian mechanics and higher notions of torsion and basic curvature tensors for generalized connections in specific Lie $n$-algebroids based on homotopy Poisson structures. The gauge systems we consider include Poisson sigma models in any dimension and ``generalised R-flux'' deformations thereof, such as models with an $(n+2)$-form-twisted R-Poisson target space. Their BV/BRST action includes interaction terms among the fields, ghosts and antifields whose coefficients acquire a geometric meaning by considering twisted Koszul multibrackets that endow the target space with a structure that we call a gapped almost Lie $n$-algebroid. Studying covariant derivatives along $n$-forms, we define suitable polytorsion and basic polycurvature tensors and identify them with the interaction coefficients in the gauge theory, thus relating models for topological $n$-branes to differential geometry on Lie $n$-algebroids.

hep-th↗
arXiv2024

Comment on "Twisted bialgebroids versus bialgebroids from Drinfeld twist"

A class of left bialgebroids whose underlying algebra $A\sharp H$ is a smash product of a bialgebra $H$ with a braided commutative Yetter--Drinfeld $H$-algebra $A$ has recently been studied in relation to models of field theories on noncommutative spaces. In [A. Borowiec, A. Pachol, ``Twisted bialgebroids versus bialgebroids from a Drinfeld twist'', J. Phys. A50 (2017) 055205] a proof has been presented that the bialgebroid $A_F\sharp H^F$ where $H^F$ and $A_F$ are the twists of $H$ and $A$ by a Drinfeld 2-cocycle $F = \sum F^1\otimes F^2$ is isomorphic to the twist of the bialgebroid $A\sharp H$ by the bialgebroid 2-cocycle $\sum 1\sharp F^1\otimes 1\sharp F^2$ induced by $F$. They assume $H$ is quasitriangular, which is reasonable for many physical applications. However the proof and the entire paper take for granted that the coaction and the prebraiding are both given by special formulas involving the R-matrix. There are counterexamples of Yetter--Drinfeld modules over quasitriangular Hopf algebras which are not of this special form. Nevertheless, the main result essentially survives. We present a proof with general coaction and the correct prebraiding, and even without the assumption of quasitriangularity.

math.QA↗
arXiv2022

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.

math-ph↗
arXiv2022

Hopf algebroids with balancing subalgebra

Recently, S. Meljanac proposed a construction of a class of examples of an algebraic structure with properties very close to the Hopf algebroids $H$ over a noncommutative base $A$ of other authors. His examples come along with a subalgebra $\mathcal{B}$ of $H\otimes H$, here called the balancing subalgebra, which contains the image of the coproduct and such that the intersection of $\mathcal{B}$ with the kernel of the projection $H\otimes H\to H\otimes_A H$ is a two-sided ideal in $\mathcal{B}$ which is moreover well behaved with respect to the antipode. We propose a set of abstract axioms covering this construction and make a detailed comparison to the Hopf algebroids of Lu. We prove that every scalar extension Hopf algebroid can be cast into this new set of axioms. We present an observation by G. Böhm that the Hopf algebroids constructed from weak Hopf algebras fit into our framework as well. At the end we discuss the change of balancing subalgebra under Drinfeld-Xu procedure of twisting of associative bialgebroids by invertible 2-cocycles.

math.QA↗
arXiv2022

Generalized Heisenberg algebra, realizations of the $\mathfrak{gl}(n)$ algebra and applications

We introduce the generalized Heisenberg algebra appropriate for realizations of the $\mathfrak{gl}(n)$ algebra. Linear realizations of the $\mathfrak{gl}(n)$ algebra are presented and the corresponding star product, coproduct of momenta and twist are constructed. The dual realization and dual $\mathfrak{gl}(n)$ algebra are considered. Finally, we present a general realization of the $\mathfrak{gl}(n)$ algebra, the corresponding coproduct of momenta and two classes of twists. These results can be applied to physical theories on noncommutative spaces of the $\mathfrak{gl}(n)$ type.

math-ph↗
arXiv2021

Why blockchain and smart contracts need semantic descriptions

We argue that there is a hierarchy of levels describing to that particular level relevant features of reality behind the content and behavior of blockchain and smart contracts in their realistic deployment. Choice, design, audit and legal control of these systems could be more informed, easier and raised to a higher level, if research on foundations of these descriptions develops and sets the formalisms, tools and standards for such descriptions.

cs.CY↗
arXiv2020

On interpolations between Jordanian twists

We consider two families of Drinfeld twists generated from a simple Jordanian twist further twisted with 1-cochains. Using combinatorial identities, they are presented as a series expansion in the dilatation and momentum generators. These twists interpolate between two simple Jordanian twists. For an expansion of a family of twists $\mathcal{F}_{L,u}$, we also show directly that the 2-cocycle condition reduces to previously proven identities.

math-ph↗
arXiv2020

Twisted exterior derivatives for universal enveloping algebras I

Consider any representation $ϕ$ of a finite-dimensional Lie algebra $g$ by derivations of the completed symmetric algebra $\hat{S}(g^*)$ of its dual. Consider the tensor product of $\hat{S}(g^*)$ and the exterior algebra $Λ(g)$. We show that the representation $ϕ$ extends canonically to the representation $\tildeϕ$ of that tensor product algebra. We construct an exterior derivative on that algebra, giving rise to a twisted version of the exterior differential calculus with the enveloping algebra in the role of the coordinate algebra. In this twisted version, the commutators between the noncommutative differentials and coordinates are formal power series in partial derivatives. The square of the corresponding exterior derivative is zero like in the classical case, but the Leibniz rule is deformed.

math.QA↗
arXiv2020

A note on symmetric orderings

Let $\hat{A}_n$ be the completion by the degree of a differential operator of the $n$-th Weyl algebra with generators $x_1,\ldots,x_n,\partial^1,\ldots,\partial^n$. Consider $n$ elements $X_1,\ldots,X_n$ in $\hat{A}_n$ of the form $$ X_i = x_i + \sum_{K = 1}^\infty \sum_{l = 1}^n\sum_{j = 1}^n x_l p_{ij}^{K-1,l}(\partial)\partial^j, $$ where $p^{K-1,l}_{ij}(\partial)$ is a degree $(K-1)$ homogeneous polynomial in $\partial^1,\ldots,\partial^n$, antisymmetric in subscripts $i,j$. Then for any natural $k$ and any function $i : \{1,\ldots,k\}\to\{1,\ldots,n\}$ we prove $$ \sum_{σ\in Σ(k)} X_{i_{σ(1)}}\cdots X_{i_{σ(k)}}\triangleright 1 = k! \,x_{i_1}\cdots x_{i_k}, $$ where $Σ(k)$ is the symmetric group on $k$ letters and $\triangleright$ denotes the Fock action of the $\hat{A}_n$ on the space of (commutative) polynomials.

math.QA↗
arXiv2020

Interpolations between Jordanian twists, the Poincaré-Weyl algebra and dispersion relations

We consider a two parameter family of Drinfeld twists generated from a simple Jordanian twist further twisted by 1-cochains. Twists from this family interpolate between two simple Jordanian twists. Relations between them are constructed and discussed. It is proved that there exists a one parameter family of twists identical to a simple Jordanian twist. The twisted coalgebra, star product and coordinate realizations of the $κ$-Minkowski noncommutative space time are presented. Real forms of Jordanian deformations are also discussed. The method of similarity transformations is applied to the Poincaré-Weyl Hopf algebra and two types of one parameter families of dispersion relations are constructed. Mathematically equivalent deformations, that are related to nonlinear changes of symmetry generators and linked with similarity maps, may lead to differences in the description of physical phenomena.

hep-th↗
arXiv2019

One Parameter Family of Jordanian Twists

We propose an explicit generalization of the Jordanian twist proposed in $r$-symmetrized form by Giaquinto and Zhang. It is proved that this generalization satisfies the 2-cocycle condition. We present explicit formulas for the corresponding star product and twisted coproduct. Finally, we show that our generalization coincides with the twist obtained from the simple Jordanian twist by twisting by a 1-cochain.

math-ph↗
arXiv2018

Hopf Algebroid Twists for Deformation Quantization of Linear Poisson Structures

In our earlier article [Lett. Math. Phys. 107 (2017), 475-503, arXiv:1409.8188], we explicitly described a topological Hopf algebroid playing the role of the noncommutative phase space of Lie algebra type. Ping Xu has shown that every deformation quantization leads to a Drinfeld twist of the associative bialgebroid of h-adic series of differential operators on a fixed Poisson manifold. In the case of linear Poisson structures, the twisted bialgebroid essentially coincides with our construction. Using our explicit description of the Hopf algebroid, we compute the corresponding Drinfeld twist explicitly as a product of two exponential expressions.

math.QA↗
arXiv2016

Lie algebra type noncommutative phase spaces are Hopf algebroids

For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way, therefore obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a completed variant of a Hopf algebroid over a noncommutative base, where the base is the enveloping algebra.

math.QA↗
arXiv2016

On Hopf algebroid structure of kappa-deformed Heisenberg algebra

The $(4+4)$-dimensional $κ$-deformed quantum phase space as well as its $(10+10)$-dimensional covariant extension by the Lorentz sector can be described as Heisenberg doubles: the $(10+10)$-dimensional quantum phase space is the double of $D=4$ $κ$-deformed Poincaré Hopf algebra $\mathbb{H}$ and the standard $(4+4)$-dimensional space is its subalgebra generated by $κ$-Minkowski coordinates $\hat{x}_μ$ and corresponding commuting momenta $\hat{p}_μ$. Every Heisenberg double appears as the total algebra of a Hopf algebroid over a base algebra which is in our case the coordinate sector. We exhibit the details of this structure, namely the corresponding right bialgebroid and the antipode map. We rely on algebraic methods of calculation in Majid-Ruegg bicrossproduct basis. The target map is derived from a formula by J-H. Lu. The coproduct takes values in the bimodule tensor product over a base, what is expressed as the presence of coproduct gauge freedom.

math-ph↗
arXiv2015

Hermitian realizations of kappa-Minkowski spacetime

General realizations, star products and plane waves for kappa-Minkowski spacetime 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.

hep-th↗