We find the general solution to the twisting equation in the tensor bialgebra $T({\bf R})$ of an associative unital ring ${\bf R}$ viewed as that of fundamental representation for a universal enveloping Lie algebra and its quantum deformations. We suggest a procedure of constructing twisting cocycles belonging to a given quasitriangular subbialgebra ${\cal H}\subset T({\bf R})$. This algorithm generalizes Reshetikhin's approach, which involves cocycles fulfilling the Yang-Baxter equation. Within this framework we study a class of quantized inhomogeneous Lie algebras related to associative rings in a certain way, for which we build twisting cocycles and universal $R$-matrices. Our approach is a generalization of the methods developed for the case of commutative rings in our recent work including such well-known examples as Jordanian quantization of the Borel subalgebra of $sl(2)$ and the null-plane quantized Poincaré algebra by Ballesteros at al. We reveal the role of special group cohomologies in this process and establish the bicrossproduct structure of the examples studied.
We consider a generalisation of the Majid's mirror product of a Hopf algebra H, when one of the components of the product is replaced by a twist. This leads to a new "twisted mirror product" construction for cocycle bicrossproduct Hopf algebra where c is a 2-cocycle of "twisting" type. As an application, we obtain a canonical cocycle bicrossproduct for any Hopf algebra H.
We show that the bicrossproduct model $C[SU_2^*]{\blacktriangleright\!\!\triangleleft} U(su_2)$ quantum Poincare group in 2+1 dimensions acting on the quantum spacetime $[x_i,t]=\imathλx_i$ is related by a Drinfeld and module-algebra twist to the quantum double $U(su_2)\ltimes C[SU_2]$ acting on the quantum spacetime $[x_μ,x_ν]=\imathλε_{μνρ}x_ρ$. We obtain this twist by taking a scaling limit as $q\to 1$ of the $q$-deformed version of the above where it corresponds to a previous theory of $q$-deformed Wick rotation from $q$-Euclidean to $q$-Minkowski space. We also recover the twist result at…
Let $H$ be a Hopf algebra. Any finite-dimensional lifting of $V\in {}^{H}_{H}\mathcal{YD}$ arising as a cocycle deformation of $A=\mathfrak{B}(V)\#H$ defines a twist in the Hopf algebra $A^*$, via dualization. We follow this recipe to write down explicit examples and show that it extends known techniques for defining twists. We also contribute with a detailed survey about twists in braided categories.
A classical theorem of Scheunert on $G$-color Lie algebras, asserts in the case of finitely generated abelian groups, one can twist the algebra structure and the commutation bicharacter on $G$ by a 2-cocycle twist to a super-Lie $G$ graded, algebra. In this paper we show that this can be done for an arbitrary group.
We introduce a new class of 2-cocycles defined explicitly on the generators of certain multiparameter standard quantum groups. These allow us, through the process of twisting the familiar standard quantum groups, to generate new as well as previously known examples of non-standard quantum groups. In particular we are able to construct generalisations of both the Cremmer-Gervais deformation of SL(3) and the so called esoteric quantum groups of Fronsdal and Galindo in an explicit and straightforward manner.
This is an introduction to work on the generalisation to quantum groups of Mackey's approach to quantisation on homogeneous spaces. We recall the bicrossproduct models of the author, which generalise the quantum double. We describe the general extension theory of Hopf algebras and the nonAbelian cohomology spaces $\CH^2(H,A)$ which classify them. They form a new kind of topological quantum number in physics which is visible only in the quantum world. These same cross product quantisations can also be viewed as trivial quantum principal bundles in quantum group gauge theory. We also relate this…
In this article, we investigate the notion of a Galois object for a locally compact quantum group M. Such an object consists of a von Neumann algebra N equipped with an ergodic integrable coaction of M on N, such that the crossed product is a type I factor. We show how to construct from such a coaction a new locally compact quantum group P, which we call the reflection of M along N. By way of application, we prove the following statement: any twisting of a locally compact quantum group by a unitary 2-cocycle is again a locally compact quantum group.
We recall the abstract theory of Hopf algebra bicrossproducts and double cross products due to the author. We use it to develop some less-well known results about the quantum double as a twisting, as an extension and as $q$-Lorentz group.
We show that a semisimple Hopf algebra A is group theoretical if and only if its Drinfeld double is a twisting of the Dijkgraaf-Pasquier-Roche quasi-Hopf algebra D^{omega}(Sigma), for some finite group Sigma and some 3-cocycle omega on Sigma. We show that semisimple Hopf algebras obtained as bicrossed products from an exact factorization of a finite group Sigma are group theoretical. We also describe their Drinfeld double as a twisting of D^{omega}(Sigma), for an appropriate 3-cocycle omega coming from the Kac exact sequence.
We study deformation of tube algebra under twisting of graded monoidal categories. When a tensor category $\mathcal{C}$ is graded over a group $Γ$, a torus-valued 3-cocycle on $Γ$ can be used to deform the associator of $\mathcal{C}$. Based on a natural Fell bundle structure of the tube algebra over the action groupoid of the adjoint action of $Γ$, we show that the tube algebra of the twisted category is a 2-cocycle twisting of the original one.
We compute the quantum double, braiding and other canonical Hopf algebra constructions for the bicrossproduct Hopf algebra $H$ associated to the factorization of a finite group into two subgroups. The representations of the quantum double are described by a notion of bicrossed bimodules, generalising the cross modules of Whitehead. We also show that self-duality structures for the bicrossproduct Hopf algebras are in one-one correspondence with factor-reversing group isomorphisms. The example $Z_6Z_6$ is given in detail. We show further that the quantum double $D(H)$ is the twisting of $D(X)$ b…
Let $H$ be a Hopf algebra that is $\mathbb Z$-graded as an algebra. We provide sufficient conditions for a 2-cocycle twist of $H$ to be a Zhang twist of $H$. In particular, we introduce the notion of a twisting pair for $H$ such that the Zhang twist of $H$ by such a pair is a 2-cocycle twist. We use twisting pairs to describe twists of Manin's universal quantum groups associated to quadratic algebras and provide twisting of solutions to the quantum Yang-Baxter equation via the Faddeev-Reshetikhin-Takhtajan construction.
We re-examine all the contractions related with the ${\cal U}_q(su(2))$ deformed algebra and study the consequences that the contraction process has for their structure. We also show using ${\cal U}_q(su(2))\times{\cal U}(u(1))$ as an example that, as in the undeformed case, the contraction may generate Hopf algebra cohomology. We shall show that most of the different Hopf algebra deformations obtained have a bicrossproduct or a cocycle bicrossproduct structure, for which we shall also give their dual `group' versions. The bicovariant differential calculi on the deformed spaces associated with…
Let $A$ be a Hopf algebra over a field $K$ of characteristic 0 and suppose there is a coalgebra projection $π$ from $A$ to a sub-Hopf algebra $H$ that splits the inclusion. If the projection is $H$-bilinear, then $A$ is isomorphic to a biproduct $R #_ξH$ where $(R,ξ)$ is called a pre-bialgebra with cocycle in the category $_{H}^{H}\mathcal{YD}$. The cocycle $ξ$ maps $R \otimes R$ to $H$. Examples of this situation include the liftings of pointed Hopf algebras with abelian group of points $Γ$ as classified by Andruskiewitsch and Schneider [AS1]. One asks when such an $A$ can be twisted by a coc…
We use \cite{G} to study the algebra structure of twisted cotriangular Hopf algebras ${}_J\mathcal{O}(G)_{J}$, where $J$ is a Hopf $2$-cocycle for a connected nilpotent algebraic group $G$ over $\mathbb{C}$. In particular, we show that ${}_J\mathcal{O}(G)_{J}$ is an affine Noetherian domain with Gelfand-Kirillov dimension $\dim(G)$, and that if $G$ is unipotent and $J$ is supported on $G$, then ${}_J\mathcal{O}(G)_{J}\cong U(\g)$ as algebras, where $\g={\rm Lie}(G)$. We also determine the finite dimensional irreducible representations of ${}_J\mathcal{O}(G)_{J}$, by analyzing twisted function …
Many quantum groups and quantum spaces of interest can be obtained by cochain (but not cocycle) twist from their corresponding classical object. This failure of the cocycle condition implies a hidden nonassociativity in the noncommutative geometry already known to be visible at the level of differential forms. We extend the cochain twist framework to connections and Riemannian structures and provide examples including twist of the $S^7$ coordinate algebra to a nonassociative hyperbolic geometry in the same category as that of the octonions.
We introduce the notion of a bicocycle double cross product (resp. sum) Lie group (resp. Lie algebra), and a bicocycle double cross product bialgebra, generalizing the unified products. On the level of Lie groups the construction yields a Lie group on the product space of two pointed manifolds, none of which being necessarily a subgroup. On the level of Lie algebras, similarly, a Lie algebra is obtained on the direct sum of two vector spaces, none of which is required to be a subalgebra. Finally, on the quantum level the theory presents a bialgebra, on the tensor product of two (co)algebras th…
We continue the study of twisting of affine algebraic groups G (i.e., of Hopf 2-cocycles J for the function algebra O(G)), which was started in [EG1,EG2], and initiate the study of the associated one-sided twisted function algebras O(G)_J. We first show that J is supported on a closed subgroup H of G (defined up to conjugation), and that O(G)_J is finitely generated with center O(G/H). We then use it to study the structure of O(G)_J for connected nilpotent G. We show that in this case O(G)_J is a Noetherian domain, which is a simple algebra if and only if J is supported on G, and describe the …
We give a description of cyclic cohomology and its pairing with K-groups for 2-cocycle deformation of algebras graded over discrete groups. The proof relies on a realization of monodromy for the Gauss-Manin connection on periodic cyclic cohomology in terms of the cup product action of group cohomology.
The $\hatκ$-deformed extended Galilei Hopf group algebra, ${\rm Fun}_{\hatκ}(\tilde G_{(m)})$, is introduced. It provides an example of a cocycle bicrossproduct structure, and is shown to be the contraction limit of a pseudoextension of the $κ$-Poincaré group algebra. The possibility of obtaining another deformed extended Galilei groups is discussed, including one obtained from a non-standard Poincaré deformation.
We show that the crossed modules and bicovariant different calculi on two Hopf algebras related by a cocycle twist are in 1-1 correspondence. In particular, for quantum groups which are cocycle deformation-quantisations of classical groups the calculi are obtained as deformation-quantisation of the classical ones. As an application, we classify all bicovariant differential calculi on the Planck scale Hopf algebra $\C[x]\bicross_{\hbar,\grav}\C[p]$. This is a quantum group which has an $\hbar\to 0$ limit as the functions on a classical but non-Abelian group and a $\grav\to 0$ limit as flat spac…
We investigate the construction and properties of Clifford algebras by a similar manner as our previous construction of the octonions, namely as a twisting of group algebras of Z_2^n by a cocycle. Our approach is more general than the usual one based on generators and relations. We obtain in particular the periodicity properties and a new construction of spinors in terms of left and right multiplication in the Clifford algebra.
Non-commutative torsors (equivalently, two-cocycles) for a Hopf algebra can be used to twist comodule algebras. After surveying and extending the literature on the subject, we prove a theorem that affords a presentation by generators and relations for the algebras obtained by such twisting. We give a number of examples, including new constructions of the quantum affine spaces and the quantum tori.
We show that the octonions are a twisting of the group algebra of Z_2 x Z_2 x Z_2 in the quasitensor category of representations of a quasi-Hopf algebra associated to a group 3-cocycle. We consider general quasi-associative algebras of this type and some general constructions for them, including quasi-linear algebra and representation theory, and an automorphism quasi-Hopf algebra. Other examples include the higher 2^n-onion Cayley algebras and examples associated to Hadamard matrices.
Given a Hopf algebra $A$ graded by a discrete group together with an action of the same group preserving the grading, we define a new Hopf algebra, which we call the graded twisting of $A$. If the action is adjoint, this new Hopf algebra is a twist of $A$ by a pseudo-$2$-cocycle. Analogous construction can be carried out for monoidal categories. As examples we consider graded twistings of the Hopf algebras of nondegenerate bilinear forms, their free products, hyperoctahedral quantum groups and $q$-deformations of compact semisimple Lie groups. As applications, we show that the analogues of the…
The ribbon cocycle invariant is defined by means of a partition function using ternary cohomology of self-distributive structures (TSD) and colorings of ribbon diagrams of a framed link, following the same paradigm introduced by Carter, Jelsovsky, Kamada, Langfor and Saito in Transactions of the American Mathematical Society 2003;355(10):3947-89, for the quandle cocycle invariant. In this article we show that the ribbon cocycle invariant is a quantum invariant. We do so by constructing a ribbon category from a TSD set whose twisting and braiding morphisms entail a given TSD $2$-cocycle. Then w…
It is well known that central extensions of a group G correspond to 2-cocycles on G. Cocycles can be used to construct extensions of G-graded algebras via a version of the Drinfeld twist introduced by Majid. We show how 2-cocycles can be defined for an abstract monoidal category C, following Panaite, Staic and Van Oystaeyen. A braiding on C leads to analogues of Nichols algebras in C, and we explain how the recent work on twists of Nichols algebras by Andruskiewitsch, Fantino, Garcia and Vendramin fits in this context. Furthermore, we propose an approach to twisting the multiplication in braid…
A construction is given of a family of non-standard quantizations of the algebra of functions on a connected complex semi-simple algebraic group. For each ``disjoint'' triple in the sense of Belavin and Drinfeld, a 2-cocycle is constructed on certain multi-parameter quantum groups. The new non-standard quantum groups are the Hopf algebras obtained by twisting the known quantum groups by these 2-cocycles. In particular, the Cremmer-Gervais quantization of $SL(3)$ can be constructed in this way.
This paper begins by introducing the concept of a quasi-hom-Lie algebra which is a natural generalization of hom-Lie algebras introduced in a previous paper by the authors. Quasi-hom-Lie algebras include also as special cases (color) Lie algebras and superalgebras, and can be seen as deformations of these by homomorphisms, twisting the Jacobi identity and skew-symmetry. The natural realm for these quasi-hom-Lie algebras is as a generalization-deformation of the Witt algebra $\Witt$ of derivations on the Laurent polynomials $\C[t,t^{-1}]$. We also develop a theory of central extensions for qhl-…
To any cleft Hopf Galois object, i.e., any algebra H[t] obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle t, we attach two "universal algebras" A(H,t) and U(H,t). The algebra A(H,t) is obtained by twisting the multiplication of H with the most general two-cocycle u formally cohomologous to t. The cocycle u takes values in the field of rational functions on H. By construction, A(H,t) is a cleft H-Galois extension of a "big" commutative algebra B(H,t). Any "form" of H[t] can be obtained from A(H,t) by a specialization of B(H,t) and vice versa. If the algebra H[t] i…
We use braided groups to introduce a theory of $*$-structures on general inhomogeneous quantum groups, which we formulate as {\em quasi-$*$} Hopf algebras. This allows the construction of the tensor product of unitary representations up to a quantum cocycle isomorphism, which is a novel feature of the inhomogeneous case. Examples include $q$-Poincaré quantum group enveloping algebras in $R$-matrix form appropriate to the previous $q$-Euclidean and $q$-Minkowski spacetime algebras $R_{21}x_1x_2=x_2x_1R$ and $R_{21}u_1Ru_2=u_2R_{21}u_1R$. We obtain unitarity of the fundamental differential repre…
In 1997 we proved that any triangular semisimple Hopf algebra over an algebraically closed field k of characteristic 0 is obtained from the group algebra k[G] of a finite group G, by twisting its comultiplication by a twist in the sense of Drinfeld. In this paper, we generalize this result to not necessarily finite-dimensional cotriangular Hopf algebras. Namely, our main result says that a cotriangular Hopf algebra A over k is obtained from a function algebra of a pro-algebraic group by twisting by a Hopf 2-cocycle, and possibly changing its R-form by a central grouplike element of A^* of orde…
Adapting the idea of twisted tensor products to the category of finitely generated algebras, we define on its opposite, the category QLS of quantum linear spaces, a family of objects hom(B,A)^{op}, one for each pair A^{op},B^{op} there, with analogous properties to its internal Hom ones, but representing spaces of transformations whose coordinate rings hom(B,A) and the ones of their respective domains B^{op} do not commute among themselves. The mentioned non commutativity is controlled by a collection of twisting maps τ_{A,B}. We show that the (bi)algebras end(A)=hom(A,A), under certain circum…
Reflection equation algebras and related U_q(g)-comodule algebras appear in various constructions of quantum homogeneous spaces and can be obtained via transmutation or equivalently via twisting by a cocycle. In this paper we investigate algebraic and representation theoretic properties of such so called `covariantized' algebras, in particular concerning their centres, invariants, and characters. Generalising M. Noumi's construction of quantum symmetric pairs we define a coideal subalgebra B_f of U_q(g) for each character f of a covariantized algebra. The locally finite part F_l(U_q(g)) of U_q…
The Drinfeld twist for the opposite quasi-Hopf algebra is determined and is shown to be related to the (second) Drinfeld twist. The twisted Drinfeld twist is investigated. In the quasi-triangular case it is shown that the Drinfeld u operator arises from the equivalence of the opposite quasi-Hopf algebra to the quasi-Hopf algebra induced by twisting with the R-matrix. The Altschuler-Coste u operator arises in a similar way and is shown to be closely related to the Drinfeld u operator. The quasi-cocycle condition is introduced, and is shown to play a central role in the uniqueness of twisted str…
In previous joint work with Eli Aljadeff we attached a generic Hopf Galois extension A(H,c) to each twisted algebra H(c) obtained from a Hopf algebra H by twisting its product with the help of a cocycle c. The algebra A(H,c) is a flat deformation of H(c) over a "big" central subalgebra B(H,c) and can be viewed as the noncommutative analogue of a versal torsor in the sense of Serre. After surveying the results on A(H,c) obtained with Aljadeff, we establish three new results: we present a systematic method to construct elements of the commutative algebra B(H,c), we show that a certain important …
The Yang-Baxter equation admits two classes of elliptic solutions, the vertex type and the face type. On the basis of these solutions, two types of elliptic quantum groups have been introduced (Foda et al., Felder). Fronsdal made a penetrating observation that both of them are quasi-Hopf algebras, obtained by twisting the standard quantum affine algebra U_q(g). In this paper we present an explicit formula for the twistors in the form of an infinite product of the universal R matrix of U_q(g). We also prove the shifted cocycle condition for the twistors, thereby completing Fronsdal's findings. …
Bialgebroids (resp. Hopf algebroids) are bialgebras (Hopf algebras) over noncommutative rings. Drinfeld twist techniques are particularly useful in the (deformation) quantization of Lie algebras as well as underlying module algebras (=quantum spaces). Smash product construction combines these two into the new algebra which, in fact, does not depend on the twist. However, we can turn it into bialgebroid in the twist dependent way. Alternatively, one can use Drinfeld twist techniques in a category of bialgebroids. We show that both techniques indicated in the title: twisting of a bialgebroid or …
