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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., Comment: v2: 18 pages, title slightly changed, presentation thoroughly revised

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., Comment: v2: 10 pages, improved presentation, in particular more structured exposition in Sec. 2 and 3. Added Concluding remarks (Sec. 4) and few references. To appear in J. Phys. A

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 $$ \sum_{\sigma \in \Sigma(k)} X_{i_{\sigma(1)}}\cdots X_{i_{\sigma(k)}}\triangleright 1 = k! \,x_{i_1}\cdots x_{i_k}, $$ where $\Sigma(k)$ is the symmetric group on $k$ letters and $\triangleright$ denotes the Fock action of the $\hat{A}_n$ on the space of (commutative) polynomials., Comment: 8 pages, v2: expositional improvements and corrections; the second half of the main theorem's proof written much more explicitly

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.

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. B\"ohm 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., Comment: v3 accepted version; v2 old title A two sided ideal in Hopf algebroid axiomatics changed to Hopf algebroids with balancing subalgebra; thoroughly revised and extended from 18 to 29 pages, entirely new sections on weak Hopf algebras and on Drinfeld-Xu twists (the first submission to a journal planned within days, but suggestions still welcome)

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.

The $(4+4)$-dimensional $\kappa$-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 $D=4$ $\kappa$-deformed Poincar\'e Hopf algebra $\mathbb{H}$ and the standard $(4+4)$-dimensional space is its subalgebra generated by $\kappa$-Minkowski coordinates $\hat{x}_\mu$ and corresponding commuting momenta $\hat{p}_\mu$. 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., Comment: 11 pages, RevTeX4, to appear in Proceedings of IX-th International Symposium "Quantum Theory and Symmetries" (QTS-9), held July 13-18, 2015, Yerevan; to be published in "Physics of Atomic Nuclei" (English Version of "Jadernaja Fizika"), ed. G. Pogosyan

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., Comment: uses kluwer.cls; v. 2: 25 pages, significant corrections, 3 authors; version 3: significant revision, 32 pages, corrections and added geometrical viewpoint and preliminaries on formal differential operators; version 4: final corrections and slightly improved readability; accepted in Letters in Mathematical Physics

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 pp

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 \v{C}ech cocycles for associated vector bundles. We sketch briefly some examples related to quantum groups.

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.

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., Comment: about 55 pages, expansion of 28-page published proceedings

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 $\phi$ 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.

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., Comment: preliminary version, posted earlier because of using and referencing some of present results in another works of mine

Consider any representation $\phi$ 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 $\Lambda(g)$. We show that the representation $\phi$ extends canonically to the representation $\tilde\phi$ 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., Comment: v2, v3: radical revisions, background more detailed, expositional corrections and, in the last section some mathematical. Slightly changed title, now part I with sequel planned. v4: 2.5 corrected/expanded to take care of completions, 15 pages. To appear in Contemporary Mathematics (2020), Proceedings of the conference "Representation Theory XVI", Dubrovnik IUC, June 19-25, 2019

For any homogeneous identity between $q$-minors, we provide an identity between $P,Q$-minors.

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., Comment: v2: changed title (previous title: Coproduct for symmetric ordering), improved exposition, now uses elsart.cls

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., Comment: 10 pages, an adaptation of an old 2001 preprint

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., Comment: 7 pages; v2: Lemma 1 corrected; part Lemma 1 (iii) added

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., Comment: v2: expositional improvements (significant in sections 5,6); v3: minor expositional improvements (including in notation, and in introduction); v4: final version, to appear in Journal of Algebra (4 minor differences from v3 due wrong uploaded file in v3)

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., Comment: 6 pages, including references

The simplicial endofunctor induced by a comonad in some category may underly a cyclic object in its category of endofunctors. The cyclic symmetry is then given by a sequence of natural transformations. We write down the commutation relations the first cyclic operator has to satisfy with the data of the comonad. If we add a version of quantum Yang Baxter relation and another relation we actually get a sufficient condition for constructing a sequence of higher cyclic operators in a canonical fashion. A degenerate case of this construction comes from so-called trivial symmetry of an additive comonad. We also consider weaker versions for paracyclic objects as well as some connections to the subject of distributive laws., Comment: 18 pages; preliminary version

Given a monoidal category C, an ordinary category M, and a monad T in M, the lifts in a strict sense of a fixed action of C on M to an action of C on the Eilenberg-Moore category of T-modules in M are in a bijective correspondence with certain families of natural transformations, indexed by the objects in the monoidal category C. These families are analogues of distributive laws between monads., Comment: 13 pages

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 results

Families of Perelomov coherent states are defined axiomatically in the context of unitary representations of Hopf algebras possessing a Haar integral. A global geometric picture involving locally trivial noncommutative fibre bundles is involved in the construction. A noncommutative resolution of identity formula is proved in that setup. Examples come from quantum groups., Comment: 19 pages, uses kluwer.cls; the exposition much improved; an example of deriving the resolution of identity via coherent states for SUq(2) added; the result differs from the proposals in literature

Viewing comodule algebras as the noncommutative analogues of affine varieties with affine group actions, we propose rudiments of a localization approach to nonaffine Hopf algebraic quotients of noncommutative affine varieties corresponding to comodule algebras. After reviewing basic background on noncommutative localizations, we introduce localizations compatible with coactions. Coinvariants of these localized coactions give local information about quotients. We define Zariski locally trivial quantum group algebraic principal and associated bundles. Compatible localizations induce localizations on the categories of Hopf modules. Their interplay with the functor of taking coinvariants and its left adjoint is stressed out. Using localization approach, we constructed a natural class of examples of quantum coset spaces, related to the quantum flag varieties of type A of other authors. Noncommutative Gauss decomposition via quasideterminants reveals a new structure in noncommutative matrix bialgebras. Particularily, in the "quantum" case, calculations with quantum minors yield the structure theorems., Comment: 34 pages bcp.sty; talk at the conference "Noncommutative geometry and quantum groups", Warszaw Sep 15-29, 2001; to appear in Banach Institute Publications