Twist deformation U_F(g) is equivalent to the quantum group Fun_d(G#) and has two preferred bases: the one originating from U(g) and that of the coordinate functions on the dual Lie group G#. The costructure of the Hopf algebra U_F(g) is analized in terms of group G#. The weight diagram of the adjoint representation of the algebra g# is constructed in terms of the root system L(g). The explicit form of the g --> g# transformation can be obtained for any simple Lie algebra g and the factorizable chain F of extended Jordanian twists. The dual group approach is used to find new solutions of the twist equations. The parametrized family of extended Jordanian deformations for U(sl(3)) is constructed and studied in terms of SL(3)#. New realizations of the parabolic twist are found.
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.
It is demonstrated how chains of twists for classical Lie algebras induce the new twist deformations (the deformed Yangians) that quantize the generalized rational solutions of the classical Yang-Baxter equation. For the case of Y(g) with g=so(2N+1) the explicit expression for the corresponding R-matrix is given in the defining representation.
Drinfeld twist is applied to the Lie algebra gl(2) so that a two-parametric deformation of it is obtained, which is identical to the Jordanian deformation of the gl(2) obtained by Aneva et al. The same twist element is applied to deform the Lie superalgebra sl(1/2), since the gl(2) is embedded into the sl(1/2). By making use of the FRT-formalism, we construct a deformation of the Lie supergroup $ SL(1/2). $
Multidimensional Heisenberg algebras, whose creation and annihilation operators are the N-dimensional vectors, can be injected into simple Lie algebras g. It is demonstrated that the spectrum of their deformations can be investigated using chains of extended Jordanian twists applied to U(g). In the case of U(sl(N)) (for N>5) the two-dimensional Heisenberg subalgebras have nine deformed costructures connected by four "internal" and "external" twists composing the commutative diagram.
In this paper we provide an explicit construction of star products on U(g)-module algebras by using the Fedosov approach. This construction allows us to give a constructive proof to Drinfel'd theorem and to obtain a concrete formula for Drinfel'd twist. We prove that the equivalence classes of twists are in one-to-one correspondence with the second Chevalley-Eilenberg cohomology of the Lie algebra g. Finally, we show that for Lie algebras with Kähler structure we obtain a strongly positive universal deformation of *-algebras by using a Wick-type deformation. This results in a positive Drinfel'…
The minimal twist map introduced by B. Abdesselam, A. Chakrabarti, R. Chakrabarti and J. Segar (Mod. Phys. Lett. A 14 (1999) 765) for the non-standard (Jordanian) quantum sl(2,R) algebra is used to construct the twist maps for two different non-standard quantum deformations of the (1+1) Schrodinger algebra. Such deformations are, respectively, the symmetry algebras of a space and a time uniform lattice discretization of the (1+1) free Schrodinger equation. It is shown that the corresponding twist maps connect the usual Lie symmetry approach to these discrete equations with non-standard quantum…
We show that certain twisting deformations of a family of supersolvable groups are simple as Hopf algebras. These groups are direct products of two generalized dihedral groups. Examples of this construction arise in dimensions 60 and p^2q^2, for prime numbers p, q with q dividing p-1. We also show that certain twisting deformation of the symmetric group is simple as a Hopf algebra. On the other hand, we prove that every twisting deformation of a nilpotent group is semisolvable. We conclude that the notions of simplicity and (semi)solvability of a semisimple Hopf algebra are not determined by i…
We study new coalgebra structures on the tensor product of two coalgebras $C$ and $D$ by twisting the tensor product coalgebra via a twist map $Ψ: C \otimes D \rightarrow D \otimes C$. We deal with the general case in which the counit of the tensor product coalgebra is deformed as well. Some classes of such deformations are analyzed and a notion of equivalence of twists is discussed. We also present the dual deformation of tensor product algebras and provide examples.
We consider the quantum-group self-duality equation in the framework of the gauge theory on a deformed twistor space. Quantum deformation of the Atiyah-Drinfel'd-Hitchin-Manin and t'Hooft multi-instanton solutions are constructed.
A new non-standard deformation of all types of classical Lie algebras is constructed by means of Drinfel'd twist based on a six dimensional subalgebra. This is an extension of extended twists introduced by Kulish et al. For the algebra M_3 = so(3,2), a relation to a known non-standard deformation is discussed.
This paper together with the previous one (arXiv:hep-th/0604146) presents the detailed description of all quantum deformations of D=4 Lorentz algebra as Hopf algebra in terms of complex and real generators. We describe here in detail two quantum deformations of the D=4 Lorentz algebra o(3,1) obtained by twisting of the standard q-deformation U_{q}(o(3,1)). For the first twisted q-deformation an Abelian twist depending on Cartan generators of o(3,1) is used. The second example of twisting provides a quantum deformation of Cremmer-Gervais type for the Lorentz algebra. For completeness we describ…
A noncommutative algebra of the complex $q$-twistors and their differentials is considered on the basis of the quantum $GL_q (4)\times SL_q (2)$ group. Real and pseudoreal $q$-twistors are discussed too. We consider the quantum-group self-duality equation in the framework of the local gauge algebra of differential forms on $q$-twistor spaces. Quantum deformations of the general multi-instanton solutions are constructed. The corresponding noncommutative algebras of moduli are introduced. The general $q$-instanton connection is a function of the $q$-twistors and the $q$-moduli .
We construct explicitly the symmetries of the isospectral deformations as twists of Lie algebras and demonstrate that they are isometries of the deformed spectral triples.
We construct twisting elements for module algebras of restricted two-parameter quantum groups from factors of their R-matrices. We generalize the theory of Giaquinto and Zhang to universal deformation formulas for categories of module algebras and give examples arising from R-matrices of two-parameter quantum groups.
The Drinfeld twist is applyed to deforme the rank one orthosymplectic Lie superalgebra $osp(1|2)$. The twist element is the same as for the $sl(2)$ Lie algebra due to the embedding of the $sl(2)$ into the superalgebra $osp(1|2)$. The R-matrix has the direct sum structure in the irreducible representations of $osp(1|2)$. The dual quantum group is defined using the FRT-formalism. It includes the Jordanian quantum group $SL_ξ(2)$ as subalgebra and Grassmann generators as well.
We construct noncommutative principal bundles deforming principal bundles with a Drinfeld twist (2-cocycle). If the twist is associated with the structure group then we have a deformation of the fibers. If the twist is associated with the automorphism group of the principal bundle, then we obtain noncommutative deformations of the base space as well. Combining the two twist deformations we obtain noncommutative principal bundles with both noncommutative fibers and base space. More in general, the natural isomorphisms proving the equivalence of a closed monoidal category of modules and its twis…
Covariance of a quantum space with respect to a quantum enveloping algebra ties the deformation of the multiplication of the space algebra to the deformation of the coproduct of the enveloping algebra. Since the deformation of the coproduct is governed by a Drinfeld twist, the same twist naturally defines a covariant star product on the commutative space. However, this product is in general not associative and does not yield the quantum space. It is shown that there are certain Drinfeld twists which realize the associative product of the quantum plane, quantum Euclidean 4-space, and quantum Mi…
In this thesis we give obstructions for Drinfel'd twist deformation quantization on several classes of symplectic manifolds. Motivated from this quantization procedure, we further construct a noncommutative Cartan calculus on any braided commutative algebra, as well as an equivariant Levi-Civita covariant derivative for any non-degenerate equivariant metric. This generalizes and unifies the Cartan calculus on a smooth manifold and the Cartan calculus on twist star product algebras. We prove that the Drinfel'd functor leads to equivalence classes in braided commutative geometry and commutes wit…
In this paper we apply a method devised in \cite{HartLarsSilv1D,LarsSilv1D} to the three-dimensional simple Lie algebra $\sll$. One of the main points of this deformation method is that the deformed algebra comes endowed with a canonical twisted Jacobi identity. We show in the present paper that when our deformation scheme is applied to $\sll$ we can, by choosing parameters suitably, deform $\sll$ into the Heisenberg Lie algebra and some other three-dimensional Lie algebras in addition to more exotic types of algebras, this being in stark contrast to the classical deformation schemes where $\s…
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 p…
We discussed twisted quantum deformations of D=4 Lorentz and Poincare algebras. In the case of Poincare algebra it is shown that almost all classical r-matrices of S.Zakrzewski classification can be presented as a sum of subordinated r-matrices of Abelian and Jordanian types. Corresponding twists describing quantum deformations are obtained in explicit form. This work is an extended version of the paper \url{arXiv:0704.0081v1 [math.QA]}.
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 ex…
We introduce a general theory of twisting algebraic structures based on actions of a bialgebra. These twists are closely related to algebraic deformations and also to the theory of quasi-triangular bialgebras. In particular, a deformation produced from a universal deformation formula (UDF) is a special case of a twist. The most familiar example of a deformation produced from a UDF is perhaps the "Moyal product" which (locally) is the canonical quantization of the algebra of functions on a symplectic manifold in the direction of the Poisson bracket. In this case, the derivations comprising the …
We show how some classical r-matrices for the D=4 Poincare algebra can be supersymmetrized by an addition of part depending on odd supercharges. These r-matrices for D=4 super-Poincare algebra can be presented as a sum of the so-called subordinated r-matrices of super-Abelian and super-Jordanian type. Corresponding twists describing quantum deformations are obtained in an explicit form. These twists are the super-extensions of twists obtained in the paper arXiv:0712.3962.
Covariance ties the noncommutative deformation of a space into a quantum space closely to the deformation of the symmetry into a quantum symmetry. Quantum deformations of enveloping algebras are governed by Drinfeld twists, inner automorphisms which relate the deformed to the undeformed coproduct. While Drinfeld twists naturally define a covariant star product on the space algebra, this product is in general not associative and does not yield a quantum space. It is reported that, nevertheless, there are certain Drinfeld twists which realize the quantum plane, quantum Euclidean 4-space, and qua…
The twisting function describing a nonstandard (super-Jordanian) quantum deformation of $osp(1|2)$ is given in explicite closed form. The quantum coproducts and universal R-matrix are presented. The non-uniqueness of the twisting function as well as two real forms of the deformed $osp(1|2)$ superalgebras are considered. One real quantum $osp(1|2)$ superalgebra is interpreted as describing the $κ$-deformation of D=1, N=1 superconformal algebra, which can be applied as a symmetry algebra of N=1 superconformal mechanics.
A new twisted deformation, U_z(so(4,2)), of the conformal algebra of the (3+1)-dimensional Minkowskian spacetime is presented. This construction is provided by a classical r-matrix spanned by ten Weyl-Poincare generators, which generalizes non-standard quantum deformations previously obtained for so(2,2) and so(3,2). However, by introducing a conformal null-plane basis it is found that the twist can indeed be supported by an eight-dimensional carrier subalgebra. By construction the Weyl-Poincare subalgebra remains as a Hopf subalgebra after deformation. Non-relativistic limits of U_z(so(4,2)) …
We use the decomposition of o(3,1)=sl(2;C)_1\oplus sl(2;C)_2 in order to describe nonstandard quantum deformation of o(3,1) linked with Jordanian deformation of sl(2;C}. Using twist quantization technique we obtain the deformed coproducts and antipodes which can be expressed in terms of real physical Lorentz generators. We describe the extension of the considered deformation of D=4 Lorentz algebra to the twist deformation of D=4 Poincare algebra with dimensionless deformation parameter.
In this paper we consider deformations of an algebroid stack on an etale groupoid. We construct a differential graded Lie algebra (DGLA) which controls this deformation theory. In the case when the algebroid is a twisted form of functions we show that this DGLA is quasiisomorphic to the twist of the DGLA of Hochschild cochains on the algebra of functions on the groupoid by the characteristic class of the corresponding gerbe.
We describe nonassociative deformations of geometry probed by closed strings in non-geometric flux compactifications of string theory. We show that these non-geometric backgrounds can be geometrised through the dynamics of open membranes whose boundaries propagate in the phase space of the target space compactification, equiped with a twisted Poisson structure. The effective membrane target space is determined by the standard Courant algebroid over the target space twisted by an abelian gerbe in momentum space. Quantization of the membrane sigma-model leads to a proper quantization of the non-…
Given a Hopf algebra H, we study modules and bimodules over an algebra A that carry an H-action, as well as their morphisms and connections. Bimodules naturally arise when considering noncommutative analogues of tensor bundles. For quasitriangular Hopf algebras and bimodules with an extra quasi-commutativity property we induce connections on the tensor product over A of two bimodules from connections on the individual bimodules. This construction applies to arbitrary connections, i.e. not necessarily H-equivariant ones, and further extends to the tensor algebra generated by a bimodule and its …
We outline how Drinfeld twist deformation techniques can be applied to the deformation quantization of principal bundles into noncommutative principal bundles, and more in general to the deformation of Hopf-Galois extensions. First we twist deform the structure group in a quantum group, and this leads to a deformation of the fibers of the principal bundle. Next we twist deform a subgroup of the group of authomorphisms of the principal bundle, and this leads to a noncommutative base space. Considering both deformations we obtain noncommutative principal bundles with noncommutative fiber and bas…
We discuss quantum deformations of Jordanian type for Lie superalgebras. These deformations are described by twisting functions with support from Borel subalgebras and they are multiparameter in the general case. The total twists are presented in explicit form for the Lie superalgebras sl(m|n) and osp(1|2n). We show also that the classical $r$-matrix for a light-cone deformation of D=4 super-Poincare algebra is of Jordanian type and a corresponding twist is given in explicit form.
Let H be a Hopf algebra, A a left H-module algebra and V a left H-module A-bimodule. We study the behavior of the right A-linear endomorphisms of V under twist deformation. We in particular construct a bijective quantization map to the right A_\star-linear endomorphisms of V_\star, with A_\star,V_\star denoting the usual twist deformations of A,V. The quantization map is extended to right A-linear homomorphisms between left H-module A-bimodules and to right connections on V. We then investigate the tensor product of linear maps between left H-modules. Given a quasitriangular Hopf algebra we ca…
We discussed quantum deformations of D=4 Lorentz and Poincare algebras. In the case of Poincare algebra it is shown that almost all classical r-matrices of S. Zakrzewski classification correspond to twisted deformations of Abelian and Jordanian types. A part of twists corresponding to the r-matrices of Zakrzewski classification are given in explicit form.
We analyse the symmetries underlying nonassociative deformations of geometry in non-geometric R-flux compactifications which arise via T-duality from closed strings with constant geometric fluxes. Starting from the non-abelian Lie algebra of translations and Bopp shifts in phase space, together with a suitable cochain twist, we construct the quasi-Hopf algebra of symmetries that deforms the algebra of functions and the exterior differential calculus in the phase space description of nonassociative R-space. In this setting nonassociativity is characterised by the associator 3-cocycle which cont…
In this article we develop an approach to deformations of the Witt and Virasoro algebras based on $σ$-derivations. We show that $σ$-twisted Jacobi type identity holds for generators of such deformations. For the $σ$-twisted generalization of Lie algebras modeled by this construction, we develop a theory of central extensions. We show that our approach can be used to construct new deformations of Lie algebras and their central extensions, which in particular include naturally the $q$-deformations of the Witt and Virasoro algebras associated to $q$-difference operators, providing also correspond…
