Comodule Algebra Research Articles

Differential Calculi on Quantum Principal Bundles Over Projective Bases

We propose a sheaf-theoretic approach to the theory of differential calculi on quantum principal bundles over non-affine bases. After recalling the affine case we define differential calculi on sheaves of comodule algebras as sheaves of covariant bimodules together with a morphism of sheaves -the differential- such that the Leibniz rule and surjectivity hold locally. The main class of examples is given by covariant calculi over quantum flag manifolds, which we provide via an explicit Ore extension construction. In a second step we introduce principal covariant calculi by requiring a local compatibility of the calculi on the total sheaf, base sheaf and the structure Hopf algebra in terms of exact sequences. In this case Hopf–Galois extensions of algebras lift to Hopf–Galois extensions of exterior algebras with compatible differentials. In particular, the examples of principal (covariant) calculi on the quantum principal bundles Oq(SL2(C))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}_q(\ extrm{SL}_2(\\mathbb {C}))$$\\end{document} and Oq(GL2(C))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}_q(\ extrm{GL}_2(\\mathbb {C}))$$\\end{document} over the projective space P1(C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{P}^1(\\mathbb {C})$$\\end{document} are discussed in detail.

Boundary and domain wall theories of 2d generalized quantum double model

The generalized quantum double lattice realization of 2d topological orders based on Hopf algebras is discussed in this work. Both left-module and right-module constructions are investigated. The ribbon operators and the classification of topological excitations based on the representations of the quantum double of Hopf algebras are discussed. To generalize the model to a 2d surface with boundaries and surface defects, we present a systematic construction of the boundary Hamiltonian and domain wall Hamiltonian. The algebraic data behind the gapped boundary and domain wall are comodule algebras and bicomodule algebras. The topological excitations in the boundary and domain wall are classified by bimodules over these algebras. The ribbon operator realization of boundary-bulk duality is also discussed. Finally, via the Hopf tensor network representation of the quantum many-body states, we solve the ground state of the model in the presence of the boundary and domain wall.

Separable Cowreaths Over Clifford Algebras

The fundamental notion of separability for commutative algebras was interpreted in categorical setting where also the stronger notion of heavily separability was introduced. These notions were extended to (co)algebras in monoidal categories, in particular to cowreaths. In this paper, we consider the cowreath left( Aotimes H_{4}^{op}, H_{4}, psi right) , where H_{4} is the Sweedler 4-dimensional Hopf algebra over a field k and A=Cl(alpha , beta , gamma ) is the Clifford algebra generated by two elements G, X with relations G^{2}=alpha , X^{2}=beta and XG+GX=gamma , (alpha , beta , gamma in k ) which becomes naturally an H_{4}-comodule algebra. We show that, when textrm{char}left( k right) ne 2, this cowreath is always separable and h-separable as well.

Stated skein algebras of surfaces

We study the algebraic and geometric properties of stated skein algebras of surfaces with punctured boundary. We prove that the skein algebra of the bigon is isomorphic to the quantum group $\mathcal{O}{{q^2}}(\operatorname{SL}(2))$ thus providing a topological interpretation for its structure morphisms. We also show that its stated skein algebra lifts in a suitable sense the Reshetikhin–Turaev functor, and in particular, we recover the dual $R$-matrix for $\mathcal{O}{{q^2}}(\operatorname{SL}(2)$ in a topological way. We deduce that the skein algebra of a surface with $n$ boundary components is a comodule algebra over $\mathcal{O}{{q^2}}(\operatorname{SL}(2))^{\otimes n}$ and prove that cutting along an ideal arc corresponds to Hochshild cohomology of bicomodules. We give a topological interpretation of braided tensor product of stated skein algebras of surfaces as “gluing on a triangle”; then we recover topologically some bialgebras in the category of $\mathcal{O}{{q^2}}(\operatorname{SL}(2))$-comodules, among which the “transmutation” of $\mathcal{O}\_{{q^2}}(\operatorname{SL}(2))$. We also provide an operadic interpretation of stated skein algebras as an example of a “geometric non-symmetric modular operad”. In the last part of the paper, we define a reduced version of stated skein algebras and prove that it allows to recover Bonahon–Wong's quantum trace map and interpret skein algebras in the classical limit when $q\to 1$ as regular functions over a suitable version of moduli spaces of twisted bundles.

On the second realization for the positive part of $$U_q(\widehat{sl_2})$$ of equitable type

The equitable presentation of the quantum algebra $U_q(\widehat{sl_2})$ is considered. This presentation was originally introduced by T. Ito and P. Terwilliger. In this paper, following Terwilliger's recent works the (nonstandard) positive part of $U_q(\widehat{sl_2})$ of equitable type $U_q^{IT,+}$ and its second realization (current algebra) $U_q^{T,+}$ are introduced and studied. A presentation for $U_q^{T,+}$ is given in terms of a K-operator satisfying a Freidel-Maillet type equation and a condition on its quantum determinant. Realizations of the K-operator in terms of Ding-Frenkel L-operators are considered, from which an explicit injective homomorphism from $U_q^{T,+}$ to a subalgebra of Drinfeld's second realization (current algebra) of $U_q(\widehat{sl_2})$ is derived, and the comodule algebra structure of $U_q^{T,+}$ is characterized. The central extension of $U_q^{T,+}$ and its relation with Drinfeld's second realization of $U_q(\widehat{gl_2})$ is also described using the framework of Freidel-Maillet algebras.

Abstract Goerss-Hopkins theory

We present an abstract version of Goerss-Hopkins theory in the setting of a prestable ∞-category equipped with a suitable periodicity operator. In the case of the ∞-category of synthetic spectra, this yields obstructions to realizing a comodule algebra as a homology of a commutative ring spectrum, recovering the results of Goerss and Hopkins.

Deformations of $$\mathcal {W}$$ algebras via quantum toroidal algebras

We study the uniform description of deformed \(\mathcal {W}\) algebras of type \(\textsf {A}\) including the supersymmetric case in terms of the quantum toroidal \({\mathfrak {g}}{\mathfrak {l}}_1\) algebra \({{\mathcal {E}}}\). In particular, we recover the deformed affine Cartan matrices and the deformed integrals of motion. We introduce a comodule algebra \(\mathcal {K}\) over \({{\mathcal {E}}}\) which gives a uniform construction of basic deformed \(\mathcal {W}\) currents and screening operators in types \(\textsf {B},\textsf {C},\textsf {D}\) including twisted and supersymmetric cases. We show that a completion of algebra \(\mathcal {K}\) contains three commutative subalgebras. In particular, it allows us to obtain a commutative family of integrals of motion associated with affine Dynkin diagrams of all non-exceptional types except \(\textsf {D}^{(2)}_{\ell +1}\). We also obtain in a uniform way deformed finite and affine Cartan matrices in all classical types together with a number of new examples, and discuss the corresponding screening operators.

Actions of the additive group Ga on certain noncommutative deformations of the plane

Abstract We connect the theorems of Rentschler [18] and Dixmier [10] on locally nilpotent derivations and automorphisms of the polynomial ring A 0 and of the Weyl algebra A 1, both over a field of characteristic zero, by establishing the same type of results for the family of algebras A h = 〈 x , y | y x − x y = h ( x ) 〉 , {A_h} = \left\langle {x,y|yx - xy = h\left( x \right)} \right\rangle , , where h is an arbitrary polynomial in x. In the second part of the paper we consider a field 𝔽 of prime characteristic and study 𝔽[t]-comodule algebra structures on Ah . We also compute the Makar-Limanov invariant of absolute constants of Ah over a field of arbitrary characteristic and show how this subalgebra determines the automorphism group of Ah .

The Quantum Symmetry in Nonbalanced Hopf Spin Models Determined by a Normal Coideal Subalgebra

For a finite-dimensional cocommutative semisimple Hopf C ∗ -algebra H and a normal coideal ∗ -subalgebra H 1 , we define the nonbalanced quantum double D H 1 ; H as the crossed product of H with H 1 o p ^ , with respect to the left coadjoint representation of the first algebra acting on the second one, and then construct the infinite crossed product A H 1 = ⋯ ⋊ H ⋊ H 1 ^ ⋊ H ⋊ H 1 ^ ⋊ H ⋊ ⋯ as the observable algebra of nonbalanced Hopf spin models. Under a right comodule algebra action of D H 1 ; H on A H 1 , the field algebra can be obtained as the crossed product C ∗ -algebra. Moreover, we prove there exists a duality between the nonbalanced quantum double D H 1 ; H and the observable algebra A H 1 .

A monoidal structure on the category of relative Hom-Hopf modules

We first define a Hom-Yetter–Drinfeld category with a new compatibility relation and prove that it is a pre-braided monoidal category. Secondly, let $(H,\beta )$ be a Hom-bialgebra, and $(A,\alpha )$ a left $(H,\beta )$-comodule algebra. Assume further th

Adjunctions Between Hom and Tensor as Endofunctors of (bi-)Module Category of Comodule Algebras Over a Quasi-Hopf Algebra

Adjunctions Between Hom and Tensor as Endofunctors of (bi-)Module Category of Comodule Algebras Over a Quasi-Hopf Algebra

Brauer–Clifford group of -Azumaya comodule algebras

In this paper, we extend the notion of the Brauer–Clifford group to the case of an -comodule algebra, where H is a commutative Hopf algebra, is a Lie algebra in the symmetric monoidal category of right H-comodules, and S is a commutative algebra which is an H-comodule algebra, a -module algebra and the H-coaction is compatible with the -action. This Brauer–Clifford group turns out to be an example of the Brauer group of a symmetric monoidal category.

Universal coacting Poisson Hopf algebras

We introduce the analogue of Manin’s universal coacting (bialgebra) Hopf algebra for Poisson algebras. First, for two given Poisson algebras P and U, where U is finite dimensional, we construct a Poisson algebra $$\mathcal {B}(P,\, U)$$ together with a Poisson algebra homomorphism $$\psi _{\mathcal {B}(P,\,U)} :P \rightarrow U \otimes \mathcal {B}(P,\, U)$$ satisfying a suitable universal property. $$\mathcal {B}(P,\, U)$$ is shown to admit a Poisson bialgebra structure for any pair of Poisson algebra homomorphisms subject to certain compatibility conditions. If $$P=U$$ is a finite dimensional Poisson algebra then $$\mathcal {B}(P) = \mathcal {B}(P,\, P)$$ admits a unique Poisson bialgebra structure such that $$\psi _{\mathcal {B}(P)}$$ becomes a Poisson comodule algebra and, moreover, the pair $$\bigl (\mathcal {B}(P),\, \psi _{\mathcal {B}(P)}\bigl )$$ is the universal coacting bialgebra of P. The universal coacting Poisson Hopf algebra $$\mathcal {H}(P)$$ on P is constructed as the initial object in the category of Poisson comodule algebra structures on P by using the free Poisson Hopf algebra on a Poisson bialgebra (Agore in J Math Phys 10:083502, 2014).

The gauge group of a noncommutative principal bundle and twist deformations

We study noncommutative principal bundles (Hopf–Galois extensions) in the context of coquasitriangular Hopf algebras and their monoidal category of comodule algebras. When the total space is quasi-commutative, and thus the base space subalgebra is central, we define the gauge group as the group of vertical automorphisms or equivalently as the group of equivariant algebra maps.We study Drinfeld twist (2-cocycle) deformations of Hopf–Galois extensions and show that the gauge group of the twisted extension is isomorphic to the gauge group of the initial extension. In particular, noncommutative principal bundles arising via twist deformation of commutative principal bundles have classical gauge group. We illustrate the theory with a few examples.

Cohomology of Modules Over -categories and Co--categories

AbstractLet$H$be a Hopf algebra. We consider$H$-equivariant modules over a Hopf module category${\mathcal{C}}$as modules over the smash extension${\mathcal{C}}\#H$. We construct Grothendieck spectral sequences for the cohomologies as well as the$H$-locally finite cohomologies of these objects. We also introduce relative$({\mathcal{D}},H)$-Hopf modules over a Hopf comodule category${\mathcal{D}}$. These generalize relative$(A,H)$-Hopf modules over an$H$-comodule algebra$A$. We construct Grothendieck spectral sequences for their cohomologies by using their rational$\text{Hom}$objects and higher derived functors of coinvariants.

Comodule algebras and 2-cocycles over the (Braided) Drinfeld double

We show that for dually paired bialgebras, every comodule algebra over one of the paired bialgebras gives a comodule algebra over their Drinfeld double via a crossed product construction. These constructions generalize to working with bialgebra objects in a braided monoidal category of modules over a quasitriangular Hopf algebra. Hence two ways to provide comodule algebras over the braided Drinfeld double (the double bosonization) are provided. Furthermore, a map of second Hopf algebra cohomology spaces is constructed. It takes a pair of 2-cocycles over dually paired Hopf algebras and produces a 2-cocycle over their Drinfeld double. This construction also has an analogue for braided Drinfeld doubles.

Biwreaths: A self-contained system in a 2-category that encodes different known algebraic constructions and gives rise to new ones

We introduce bimonads in a 2-category K and define biwreaths as bimonads in the 2-category bEM(K) of bimonads, in the analogous fashion as Lack and Street defined wreaths. A biwreath is then a system containing a wreath, a cowreath and their mixed versions, but also a 2-cell λ in bEM(K) governing the compatibility of the monad and the comonad structure of the biwreath. We deduce that the monad laws encode 2-(co)cycles and the comonad laws so called 3-(co)cycles, while the 2-cell conditions of the (co)monad structure 2-cells of the biwreath encode (co)actions twisted by these 2- and 3-(co)cycles. The compatibilities of λ deliver concrete expressions of the latter structure 2-cells. We concentrate on the examples of biwreaths in the 2-category induced by a braided monoidal category C and take for the distributive laws in a biwreath the braidings of the different categories of Yetter–Drinfel'd modules in C. We prove that the before-mentioned properties of a biwreath specified to the latter setting recover on the level of C different algebraic constructions known in the category MR of modules over a commutative ring R, such as Radford biproduct, Sweedler's crossed product algebra, comodule algebras over a quasi-bialgebra and the Drinfel'd twist. In this way we obtain that the known examples of (mixed) wreaths coming from MR are not merely examples, rather they are consequences of the structure of a biwreath, and that the form of their structure morphisms originates in the laws inside of a biwreath. Choosing different distributive laws and different 2-cells λ in a biwreath, leads to different and possibly new algebraic constructions.

Exotic elliptic algebras

The 4-dimensional Sklyanin algebras, over $\mathbb {C}$, $A(E,\tau )$, are constructed from an elliptic curve $E$ and a translation automorphism $\tau$ of $E$. The Klein vierergruppe $\Gamma$ acts as graded algebra automorphisms of $A(E,\tau )$. There is also an action of $\Gamma$ as automorphisms of the matrix algebra $M_2(\mathbb {C})$ making it isomorphic to the regular representation. The main object of study in this paper is the invariant subalgebra $\widetilde {A}:=\big (A(E,\tau ) \otimes M_2(\mathbb {C})\big )^{\Gamma }$. Like $A(E,\tau )$, $\widetilde {A}$ is noetherian, generated by 4 degree-one elements modulo six quadratic relations, Koszul, Artin-Schelter regular of global dimension 4, has the same Hilbert series as the polynomial ring on 4 variables, satisfies the $\chi$ condition, and so on. These results are special cases of general results proved for a triple $(A,T,H)$ consisting of a Hopf algebra $H$, an (often graded) $H$-comodule algebra $A$, and an $H$-torsor $T$. Those general results involve transferring properties between $A$, $A \otimes T$, and $(A \otimes T)^\textrm {{co} H}$. We then investigate $\widetilde {A}$ from the point of view of non-commutative projective geometry. We examine its point modules, line modules, and a certain quotient $\widetilde {B}:=\widetilde {A}/(\Theta ,\Theta ’)$ where $\Theta$ and $\Theta ’$ are homogeneous central elements of degree two. In doing this we show that $\widetilde {A}$ differs from $A$ in interesting ways. For example, the point modules for $A$ are parametrized by $E$ and 4 more points, whereas $\widetilde {A}$ has exactly 20 point modules. Although $\widetilde {B}$ is not a twisted homogeneous coordinate ring in the sense of Artin and Van den Bergh, a certain quotient of the category of graded $\widetilde {B}$-modules is equivalent to the category of quasi-coherent sheaves on the curve $E/E[2]$ where $E[2]$ is the 2-torsion subgroup. We construct line modules for $\widetilde {A}$ that are parametrized by the disjoint union $(E/\langle \xi _1\rangle ) \sqcup (E/\langle \xi _2\rangle ) \sqcup (E/\langle \xi _3\rangle )$ of the quotients of $E$ by its three subgroups of order 2.

Graded twisting of comodule algebras and module categories

Continuing our previous work on graded twisting of Hopf algebras and monoidal categories, we introduce a graded twisting construction for equivariant comodule algebras and module categories. As an example we study actions of quantum subgroups of G\subset\mathrm {SL}_{-1}(2) on K_{-1}[x,y] and show that in most cases the corresponding invariant rings K_{-1}[x,y]^G are invariant rings K[x,y]^{G'} for the action of a classical subgroup G'\subset \mathrm {SL}(2) . As another example we study Poisson boundaries of graded twisted categories and show that under the assumption of weak amenability they are graded twistings of the Poisson boundaries.

Note on the Hochschild cohomology of Hopf–Galois extension

We generalize [4, Theorem 4.3] to the case of Hopf–Galois extension, by introducing the cotensor product of a comodule algebra and its opposite algebra, and then give some applications.