Salih Celik
The two-parametric quantum deformation of the algebra of coordinate functions on the supergroup GL$(1| 1)$ via a contraction of GL$_{p,q}(1| 1)$ is presented. Related differential calculus on the quantum superplane is introduced.
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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). $
We constructed a multi-parametric deformation of the Brauer algebra representation related with the symplectic Lie algebras. The notion of Manin matrix of type C was generalised to the case of the multi-parametric deformation by using this representation and corresponding quadratic algebras. We derived pairing operators for these quadratic algebras and minors for the considered Manin matrices. The rank of pairing operators and dimensions of components of quadratic algebras were calculated.
A three-parametric $R$-matrix satisfying a graded Yang-Baxter equation is introduced.This $R$-matrix allows us to construct new quantum supergroups which are deformations of the supergroup $GL(1/1)$ and the universal enveloping algebra $U[gl(1/1)]$.
A two-parametric deformation of U[sl(2)] and its representations are considered. This newly introduced two-parametric quantum group denoted as $U_{pq}[sl(2)]$ admits a class of infinite-dimensional representations which have no classical (non-deformed) and one-parametric deformation analogues, even at generic deformation parameters. Interestingly that finite-dimensional representations of $U_{pq}[sl(2)]$ allow arbitrary complex "spins" (i.e., not necessary they to be integral or half-integral numbers), unlike those in the classical and one-parametric deformation cases.
The two-parametric quantum superalgebra $U_{p,q}[gl(2/2)]$ and its induced representations are considered. A method for constructing all finite-dimensional irreducible representations of this quantum superalgebra is also described in detail. It turns out that finite-dimensional representations of the two-parametric $U_{p,q}[gl(2/2)]$, even at generic deformation parameters, are not simply trivial deformations from those of the classical superalgebra $gl(2/2)$, unlike the one-parametric cases.
The two-parametric quantum superalgebra $U_{p,q}[gl(2/1)]$ is consistently defined. A construction procedure for induced representations of $U_{p,q}[gl(2/1)]$ is described and allows us to construct explicitly all (typical and nontypical) finite-dimensional representations of this quantum superalgebra. In spite of some specific features, the present approach is similar to a previously developed method [1] which, as shown here, is applicable not only to the one-parametric quantum deformations but also to the multi-parametric ones.
Some one- and two-parametric deformations of U[sl(2)] and their representations are considered. Interestingly, a newly introduced two-parametric deformation admits a class of infinite - dimensional representations which have no classical (non-deformed) and one-parametric deformation analogues, even at generic deformation parameters.
The purpose of this paper is to study Virasoro extensions of the q-deformed Witt Hom-Lie superalgebra. Moreover, we provide the cohomology and deformations of the Ramond Hom-superalgebra and special Ramond Hom-superalgebra.
We examine PBW deformations of finite group extensions of skew polynomial rings, in particular the quantum Drinfeld orbifold algebras defined by the first author. We give a homological interpretation, in terms of Gerstenhaber brackets, of the necessary and sufficient conditions on parameter functions to define a quantum Drinfeld orbifold algebra, thus clarifying the conditions. In case the acting group is trivial, we determine conditions under which such a PBW deformation is a generalized enveloping algebra of a color Lie algebra; our PBW deformations include these algebras as a special case.
This paper suveys some recent algebraic developments in two parameter Quantum deformations and their Nonstandard (or Jordanian) counterparts. In particular, we discuss the contraction procedure and the quantum group homomorphisms associated to these deformations. The scheme is then set in the wider context of the coloured extensions of these deformations, namely, the so-called Coloured Quantum Groups.
Cohomology, Extensions and Deformations of restricted Lie triple systems
本文构建了受限李三系的低阶上同调理论，定义了直到5阶的受限上链与上边界算子，并阐释了低阶上同调群在导子、模扩张、形式形变及系统扩张中的代数意义。
Some introductory concepts and basic definitions of the Lie superalgebras and their quantum deformations are exposed. Especially the induced representation methods in both cases are described. Based on the Kac representation theory we have succeeded in constructing representations of several higher rank superalgebras. When representations of quantum superalgebras are concerned, we have developed a method which can be applied not only to the one-parametric quantum deformations but also to the multi-parametric ones. As an intermediate step the Gel'fand-Zetlin basis description is extended to the…
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…
Using the multi-parametric deformation of the algebra of functions on $ \GL{n+1} $ and the universal enveloping algebra $ \U{\igl{n+1}} $, we construct the multi-parametric quantum groups $ \IGLq{n} $ and $ \Uq{\igl{n}} $.
$G_\infty$-structure is shown to exist on the deformation complex of a morphism of associative algebras. The main step of the construction is extension of a $B_\infty$-algebra by an associative algebra. Actions of $B_\infty$-algebras on associative and $B_\infty$-algebras are analyzed, extensions of $B_\infty$-algebras by associative and $B_\infty$-algebras, that they act upon, are constructed. The resulting $G_\infty$-algebra on the deformation complex of a morphism is shown to be quasi-isomorphic to the $G_\infty$-algebra on deformation complex of the corresponding diagram algebra.
We study quantum deformed $gl(n)$ and $igl(n)$ algebras on a quantum space discussing multi-parametric extension. We realize elements of deformed $gl(n)$ and $igl(n)$ algebras by a quantum fermionic space. We investigate a map between deformed $igl(2)$ algebras of our basis and other basis.
Let $A$ be a star product on a symplectic manifold $(M,ω_0)$, $\frac{1}{t}[ω]$ its Fedosov class, where $ω$ is a deformation of $ω_0$. We prove that for a complex polarization of $ω$ there exists a commutative subalgebra, $O$, in $A$ that is isomorphic to the algebra of functions constant along the polarization. Let $F(A)$ consists of elements of $A$ whose commutator with $O$ belongs to $O$. Then, $F(A)$ is a Lie algebra which is an $O$-extension of the Lie algebra of derivations of $O$. We prove a formula which relates the class of this extension, the Fedosov class, and the Chern class of $P$…
We study a noncommutative deformation of the commutative Hopf algebra of rooted trees which was shown by Connes and Kreimer to be related to the mathematical structure of renormalization in quantum field theories. The requirement of the existence of an antipode for the noncommutative deformation leads to a natural extension of the algebra. Noncommutative deformations of the Connes-Kreimer algebra might be relevant for renormalization of field theories on noncommutative spaces and there are indications that in this case the extension of the algebra might be linked to a mixing of infrared and ul…
Deformation cohomology of Nijenhuis algebras and applications to extensions, inducibility of automorphisms and homotopy algebras
本文引入并研究了Nijenhuis算子和Nijenhuis代数的上同调，证明其控制结合结构与算子的同时形变，并应用于阿贝尔扩张、自同构可诱导性及同伦Nijenhuis代数的刻画。
The two-parametric quantum superalgebra $U_{pq}[gl(2/2)]$ and its representations are considered. All finite-dimensional irreducible representations of this quantum superalgebra can be constructed and classified into typical and nontypical ones according to a proposition proved in the present paper. This proposition is a nontrivial deformation from the one for the classical superalgebra gl(2/2), unlike the case of one-parametric deformations.
In order to unify the methods which have been applied to various topics such as BRST theory of constraints, Poisson brackets of local functionals, and certain developments in deformation theory, we formulate a new concept which we call the {\it chain extension} of a $D$-algebra. We develop those aspects of this new idea which are central to applications to algebra and physics. Chain extensions may be regarded as generalizations of ordinary algebraic extensions of Lie algebras. Applications of our theory provide a new constructive approach to BRST theories which only contains three terms; in pa…
We study deformation of Courant pairs with a commutative algebra base. We consider the deformation cohomology bi-complex and describe a universal infinitesimal deformation. In a sequel, we formulate an extension of a given deformation of a Courant pair to another with extended base. This leads to describe the obstruction in extending a given deformation. We also discuss about the construction of versal deformation of Courant pairs. As an application, we explicitly compute universal infinitesimal deformation of Poisson algebra structures
Deformations of the Lie algebras so(4), so(3,1), and e(3) that leave their so(3) subalgebra undeformed and preserve their coset structure are considered. It is shown that such deformed algebras are associative for any choice of the deformation parameters. Their Casimir operators are obtained and some of their unitary irreducible representations are constructed. For vanishing deformation, the latter go over into those of the corresponding Lie algebras that contain each of the so(3) unitary irreducible representations at most once. It is also proved that similar deformations of the Lie algebras …
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…
We construct highest-weight modules and a Yangian extension of the centrally extended sl(1|1)^2 superalgebra, that is a symmetry of the worldsheet scattering associated with the AdS3/CFT3 duality. We demonstrate that the R-matrix intertwining atypical modules is of a trigonometric type. We also consider a quantum deformation of this superalgebra, its modules, and obtain a quantum affine extension of the Drinfeld-Jimbo type that describes a deformed worldsheet scattering.
A quantization of a non-standard rational solution of CYBE for $sl_2$ is given explicitly. We obtain the quantization with the help of a twisting of the usual Yangian $Y(sl_2$. This quantum object (deformed Yangian $Y_{η,ξ}(sl_2))$ is a two-parametric deformation of the universal enveloping algebra $U(sl_2[u])$ of the positive current algebra $sl_2[u]$. We consider the pseudotriangular structure on $Y_{η,ξ}(sl_2)$, the quantum double $DY_{η,ξ}(sl_2)$ its the universal R-matrix and also the RTT-realization of $Y_{η,ξ}(sl_2)$.
As a generalization of the linear Poisson bracket on the dual space of a Lie algebra, we introduce certain non-linear Poisson brackets which are ``cocycle perturbations'' of the linear Poisson bracket. We show that these special Poisson brackets are equivalent to Poisson brackets of central extension type, which resemble the central extensions of an ordinary Lie bracket via Lie algebra cocycles. We are able to formulate (strict) deformation quantizations of these Poisson brackets by means of twisted group C*-algebras. We also indicate that these deformation quantizations can be used to constru…
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-…
The centrally extended superalgebra psu(2|2)xR^3 was shown to play an important role for the integrable structures of the one-dimensional Hubbard model and of the planar AdS/CFT correspondence. Here we consider its quantum deformation U_q(psu(2|2)xR^3) and derive the fundamental R-matrix. From the latter we deduce an integrable spin chain Hamiltonian with three independent parameters and the corresponding Bethe equations to describe the spectrum on periodic chains. We relate our Hamiltonian to a two-parametric Hamiltonian proposed by Alcaraz and Bariev which can be considered a quantum deforma…
Consider a group $Γ$ acting on a formal (Fedosov) deformation quantization $\mathbb{A}_\hbar(M)$ of a symplectic manifold $(M,ω)$. This canonically induces an action of $Γ$ by symplectomorphisms on $M$. We examine the reverse problem of extending group actions by symplectomorphisms to the deformation quantization. To do this we first define a notion of extension that does not impose restrictions on the Fedosov connection realizing $\mathbb{A}_\hbar(M)$ in its gauge equivalence class by considering composition of pull-back with certain inner automorphisms of sections of the Weyl bundle. We refo…
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