R. Chakrabarti, J. Segar
We introduce and investigate a one parameter family of quantum spaces invariant under the left (right) coactions of the group-like element ${\cal T}_{h}^{(j=1)}$ of the Jordanian function algebra $Fun_{h}(SL(2))$. These spaces may be regarded as Jordanian quantization of the two-dimensional spheres.
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The quantum deformation of the Jordanian twist F_qJ for the standard quantum Borel algebra U_q(B) is constructed. It gives the family U_qJ(B) of quantum algebras depending on parameters x and h. In a generic point these algebras represent the hybrid (standard-nonstandard) quantization. The quantum Jordanian twist can be applied to the standard quantization of any Kac-Moody algebra. The corresponding classical r-matrix is a linear combination of the Drinfeld- Jimbo and the Jordanian ones. The obtained two-parametric families of Hopf algebras are smooth and for the limit values of the parameters…
Using the contraction procedure introduced by us in Ref. \cite{ACC2}, we construct, in the first part of the present letter, the Jordanian quantum Hopf algebra ${\cal U}_{\sf h}(sl(3))$ which has a remarkably simple coalgebraic structure and contains the Jordanian Hopf algebra ${\cal U}_{\sf h}(sl(2))$, obtained by Ohn, as a subalgebra. A nonlinear map between ${\cal U}_{\sf h}(sl(3))$ and the classical $sl(3)$ algebra is then established. In the second part, we give the higher dimensional Jordanian algebras ${\cal U}_{\sf h}(sl(N))$ for all $N$. The Universal ${\cal R}_{\sf h}$-matrix of ${\c…
Extending a recently proposed procedure of construction of various elements of diffential geometry on noncommutative algebras, we obtain these structures on noncommutative superalgebras. As an example, a quantum superspace covariant under the action of super-Jordanian $OSp_h(2/1)$ is studied. It is shown that there exist a two paramater family of torsionless connections, and the curvature computed from this family of connections is bilinear. It is also shown that the connections are not compatible with the metric.
The explicit expressions of the representation functions (D-functions) for Jordanian quantum group SL_h(2) are obtained by combination of tensor operator technique and Drinfeld twist. It is shown that the D-functions can be expressed in terms of Jacobi polynomials as the undeformed D-functions can. Some of the important properties of the D-functions for SL_h(2) such as Winger's product law, recurrence relations, RTT type relations are also presented.
A triangular quantum deformation of $ osp(2/1) $ from the classical $r$-matrix including an odd generator is presented with its full Hopf algebra structure. The deformation maps, twisting element and tensor operators are considered for the deformed $ osp(2/1)$. It is also shown that its subalgebra generated by the Borel subalgebra is self-dual.
We classify all 4-dimensional first order bicovariant calculi on the Jordanian quantum group GL_{h,g}(2) and all 3-dimensional first order bicovariant calculi on the Jordanian quantum group SL_{h}(2). In both cases we assume that the bicovariant bimodules are generated as left modules by the differentials of the quantum group generators. It is found that there are 3 1-parameter families of 4-dimensional bicovariant first order calculi on GL_{h,g}(2) and that there is a single, unique, 3-dimensional bicovariant calculus on SL_{h}(2). This 3-dimensional calculus may be obtained through a classic…
The limiting transitions between different types of quantizations are studied by the deformation theory methods. We prove that for the first order coboundary deformation (g,g*_1 + x g*_2) of a Lie bialgebra (g,g*) one can always get the quantized Lie bialgebra (g,g*_2) as a limit of the sequence of quantizations of the type A(g,g*_1). The obtained results are illustrated by some low-dimensional examples of quantized Lie algebras and superalgebras.
Tensor products of irreducible representations of the Jordanian quantum algebras U_h(sl(2)) and U_h(su(1,1)) are considered. For both the highest weight finite dimensional representations of U_h(sl(2)) and lowest weight infinite dimensional ones of U_h(su(1,1)), it is shown that tensor product representations are reducible and that the decomposition rules to irreducible representations are exactly the same as those of corresponding Lie algebras.
The generators of the Jordanian quantum algebra ${\cal U}_h(sl(2))$ are expressed as nonlinear invertible functions of the classical $sl(2)$ generators. This permits immediate explicit construction of the finite dimensional irreducible representations of the algebra ${\cal U}_h(sl(2))$. Using this construction, new finite dimensional solutions of the Yang-Baxter equation may be obtained.
It is known that the inhomogeneous quantum group IGL_{q,r}(2) can be constructed as a quotient of the multiparameter q-deformation of GL(3). We show that a similar result holds for the inhomogeneous Jordanian deformation and exhibit its Hopf structure.
Recently, a class of transformations of $R_q$-matrices was introduced such that the $q \to 1$ limit gives explicit nonstandard $R_h$-matrices. The transformation matrix is singular as $q \to 1$. For the transformed matrix, the singularities, however, cancel yielding a well-defined construction. We have shown that our method can be implemented systematically on $R_q$ matrices of all dimensions of $U_q(sl(N)), U_Q(osp(1|2))$ and $U_q(sl(2|1))$ algebras. Explicit constructions are presented for $U_q(sl(2)), U_q(sl(3)), U_q(osp(1|2))$ and $U_q(sl(2|1))$ algebras, while choosing $R_q$ matrix for (f…
The quantum group GL_p,q(2) is known to be related to the Jordanian GL_h,h'(2) via a contraction procedure. It can also be realised using the generators of the Hopf algebra G_r,s. We contract the G_r,s quantum group to obtain its Jordanian analogue G_m,k, which provides a realisation of GL_h,h'(2) in a manner similar to the q-deformed 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.
Representation theory for the Jordanian quantum algebra $U=U_h(sl(2))$ is developed. Closed form expressions are given for the action of the generators of U on the basis vectors of finite dimensional irreducible representations. It is shown how representation theory of U has a close connection to combinatorial identities involving summation formulas. A general formula is obtained for the Clebsch-Gordan coefficients $C^{j_1,j_2,j}_{n_1,n_2,m}(h)$ of U. These Clebsch-Gordan coefficients are shown to coincide with those of su(2) for $n_1+n_2 \leq m$, but for $n_1+n_2 > m$ they are in general a no…
We consider a superextension of the extended Jordanian twist, describing nonstandard quantization of anti-de-Sitter ($AdS$) superalgebra $osp(1|4)$ in the form of Hopf superalgebra. The super-Jordanian twisting function and corresponding basic coproduct formulae for the generators of $osp(1|4)$ are given in explicit form. The nonlinear transformation of the classical superalgebra basis not modifying the defining algebraic relations but simplifying coproducts and antipodes is proposed. Our physical application is to interpret the new super-Jordanian deformation of $osp(1|4)$ superalgebra as def…
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.
A non-standard quantum deformation of the two-photon algebra $h_6$ is constructed, and its quantum universal R-matrix is given. Representations of this new quantum algebra are studied on the Fock space and translated into Fock-Bargmann realizations that provide a direct formalism for the definition of deformed states of light. Finally, the isomorphism between $h_6$ and the (1+1) Schrödinger algebra is used to introduce a new (non-standard) Hopf algebra deformation of this latter symmetry algebra.
We find the Hopf algebra $U_{g,h}$ dual to the Jordanian matrix quantum group $GL_{g,h}(2)$. As an algebra it depends only on the sum of the two parameters and is split in two subalgebras: $U'_{g,h}$ (with three generators) and $U(Z)$ (with one generator). The subalgebra $U(Z)$ is a central Hopf subalgebra of $U_{g,h}$. The subalgebra $U'_{g,h}$ is not a Hopf subalgebra and its coalgebra structure depends on both parameters. We discuss also two one-parameter special cases: $g =h$ and $g=-h$. The subalgebra $U'_{h,h}$ is a Hopf algebra and coincides with the algebra introduced by Ohn as the dua…
The quantum group G_r,s provides a realisation of the two parameter quantum GL_p,q(2) which is known to be related to the two parameter nonstandard GL_hh'(2) group via a contraction method. We apply the contraction procedure to G_r,s and obtain a new Jordanian quantum group G_m,k. Furthermore, we provide a realisation of GL_h,h'(2) in terms of G_m,k. The contraction procedure is then extended to the coloured quantum group GL_r{λ,μ}(2) to yield a new Jordanian quantum group GL_m{λ,μ}(2). Both G_r,s and G_m,k are then generalised to their coloured versions which inturn provide similar realisatio…
Tensor operators for the Jordanian quantum algebra Uh(sl(2)) are considered. Some explicit examples of them, which are obtained in the boson or fermion realization, are given and their properties are studied. It is also shown that the Wigner-Eckart's theorem can be extended to Uh(sl(2)).
A universal $R$-matrix for the non-standard (Jordanian) quantum deformation of $sl(2,\R)$ is presented. A family of solutions of the quantum Yang--Baxter equation is obtained from some finite dimensional representations of this Lie bialgebra quantization of $sl(2,\R)$.
Starting from the classical r-matrix of the non-standard (or Jordanian) quantum deformation of the sl(2,R) algebra, new triangular quantum deformations for the real Lie algebras so(2,2), so(3,1) and iso(2,1) are simultaneously constructed by using a graded contraction scheme; these are realized as deformations of conformal algebras of (1+1)-dimensional spacetimes. Time-type and space-type quantum algebras are considered according to the generator that remains primitive after deformation: either the time or the space translation, respectively. Furthermore by introducing differential-difference …
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 investigate the algebro-geometric structure of a novel two-parameter quantum deformation which exhibits the nature of a semidirect or cross-product algebra built upon GL(2) x GL(1), and is related to several other known examples of quantum groups. Following the R-matrix framework, we construct the L+/- functionals and address the problem of duality for this quantum group. This naturally leads to the construction of a bicovariant differential calculus that depends only on one deformation parameter, respects the cross-product structure and has interesting applications. The corresponding Jorda…
Quantum sphere is introduced as a quotient of the so-called Reflection Equation Algebra. This enables us to construct some line bundles on it by means of the Cayley-Hamilton identity whose a quantum version was discovered in \cite{PS}, \cite{GPS}. A new way to introduce some elements of "braided geometry" on the quantum sphere is discussed.
Weighted circle actions on the quantum Heeqaard 3-sphere are considered. The fixed point algebras, termed quantum weighted Heegaard spheres, and their representations are classified and described on algebraic and topological levels. On the algebraic side, coordinate algebras of quantum weighted Heegaard spheres are interpreted as generalised Weyl algebras, quantum principal circle bundles and Fredholm modules over them are constructed, and the associated line bundles are shown to be non-trivial by an explicit calculation of their Chern numbers. On the topological side, the C*-algebras of conti…
The quantum flag manifold ${SU_q(3)/\mathbb{T}^2}$ is interpreted as a noncommutative bundle over the quantum complex projective plane with the quantum or Podleś sphere as a fibre. A connection arising from the (associated) quantum principal $U_q(2)$-bundle is described.
The dual coalgebra of Podleś' quantum sphere O_q(S^2_c) is determined explicitly. This result is used to classify all finite dimensional covariant first order differential calculi over O_q(S^2_c) for all but exceptional values of the parameter c.
We quantize homogeneous vector bundles over an even complex sphere $\mathbb{S}^{2n}$ as one-sided projective modules over its quantized coordinate ring. We realize them in two different ways: as locally finite $\mathbb{C}$-homs between pseudo-parabolic Verma modules and as induced modules of the quantum orthogonal group. Based on this alternative, we study representations of a quantum symmetric pair related to $\mathbb{S}^{2n}_q$ and prove their complete reducibility.
We study covariant differential calculus on the quantum spheres S_q^2N-1. Two classification results for covariant first order differential calculi are proved. As an important step towards a description of the noncommutative geometry of the quantum spheres, a framework of covariant differential calculus is established, including a particular first order calculus obtained by factorization, higher order calculi and a symmetry concept.
Noncommutative (NC) sphere is introduced as a quotient of the enveloping algebra of the Lie algebra su(2). Using the Cayley-Hamilton identities we introduce projective modules which are analogues of line bundles on the usual sphere (we call them quantum line bundles) and define a multiplicative structure in their family. Also, we compute a pairing between certain quantum line bundles and finite dimensional representations of the NC sphere in the spirit of the NC index theorem. A new approach to constructing the differential calculus on a NC sphere is suggested. The approach makes use of the pr…
It is shown that the isospectral bi-equivariant spectral triple on quantum SU(2) and the isospectral equivariant spectral triples on the Podles spheres are related by restriction. In this approach, the equatorial Podles sphere is distinguished because only in this case the restricted spectral triple admits an equivariant grading operator together with a real structure (up to infinitesimals of arbitrary high order). The real structure is expressed by the Tomita operator on quantum SU(2) and it is shown that the failure of the real structure to satisfy the commutant property is related to the fa…
We construct a quantum version of the SU(2) Hopf bundle $S^7 \to S^4$. The quantum sphere $S^7_q$ arises from the symplectic group $Sp_q(2)$ and a quantum 4-sphere $S^4_q$ is obtained via a suitable self-adjoint idempotent $p$ whose entries generate the algebra $A(S^4_q)$ of polynomial functions over it. This projection determines a deformation of an (anti-)instanton bundle over the classical sphere $S^4$. We compute the fundamental $K$-homology class of $S^4_q$ and pair it with the class of $p$ in the $K$-theory getting the value -1 for the topological charge. There is a right coaction of $SU…
We study irreducible representations of a class of quantum spheres, quotients of quantum symplectic spheres.
We propose in this paper the construction of non-commutative Chern characters of the C*-algebras of spheres and quantum spheres. The final computation gives us a clear relation with the ordinary Z/(2)-graded Chern characters of tori or their normalizers.
We describe the quantum sphere of Podleś for $c=0$ by means of a stereographic projection which is analogous to that which exhibits the classical sphere as a complex manifold. We show that the algebra of functions and the differential calculus on the sphere are covariant under the coaction of fractional transformations with \SU coefficients as well as under the action of \SU vector fields. Going to the classical limit we obtain the Poisson sphere. Finally, we study the invariant integration of functions on the sphere and find its relation with the translationally invariant integration on the c…
We introduce and study two new examples of noncommutative spheres: the half-liberated sphere, and the free sphere. Together with the usual sphere, these two spheres have the property that the corresponding quantum isometry group is "easy", in the representation theory sense. We present as well some general comments on the axiomatization problem, and on the "untwisted" and "non-easy" case.
Definition of Dirac operators on the quantum group $SU_{q}(2)$ and the quantum sphere $S^{2}_{q μ}$ are discussed. In both cases similar $SU_{q}(2)$-invariant form is obtained. It is connected with corresponding Laplace operators.
The Noncommutative Index Theorem is used to prove that the Chern character of quantum Hopf line bundles over the standard Podles quantum sphere equals the winding number of the representations defining these bundles. This result gives an estimate of the positive cone of the algebraic $K_0$ of the standard quantum sphere.
The cup and cap product in twisted Hochschild (co)homology is computed for the standard quantum 2-sphere and used to construct a cyclic 2-cocycle that represents the fundamental Hochschild class.
We study relations between different kinds of non-commutative spheres which have appeared in the context of ADS/CFT correspondences recently, emphasizing the connections between spaces that have manifest quantum group symmetry and spaces that have manifest classical symmetry. In particular we consider the quotient $SU_q(2)/U(1)$ at roots of unity, and find its relations with the fuzzy sphere with manifest classical SU(2) symmetry. Deformation maps between classical and quantum symmetry, the $U_q(SU(2))$ module structure of quantum spheres and the structure of indecomposable representations of …
We show that the C*-algebra of a quantum sphere $C(S_{q}^{2n+1})$ can be realized as a groupoid C*-algebra of a groupoid which is explicitly identified and is independent of the parameter $q$.
We show that the noncommutative spheres of Connes and Landi are quantum homogeneous spaces for certain compact quantum groups. We give a general construction of homogeneous spaces which support noncommutative spin geometries.
We investigate the spin $1/2$ fermions on quantum two spheres. It is shown that the wave functions of fermions and a Dirac Operator on quantum two spheres can be constructed in a manifestly covariant way under the quantum group $SU(2)_q$. The concept of total angular momentum and chirality can be expressed by using $q$-analog of Pauli-matrices and appropriate commutation relations.
We introduce the notion of noncommutative complex spheres with partial commutation relations for the coordinates. We compute the corresponding quantum symmetry groups of these spheres, and this yields new quantum unitary groups with partial commutation relations. We also discuss some geometric aspects of the quantum orthogonal groups associated with the mixture of classical and free independence discovered by Speicher and Weber. We show that these quantum groups are quantum symmetry groups on some quantum spaces of spherical vectors with partial commutation relations.
The sphere $S^{N-1}_\mathbb R$ has a half-liberated analogue $S^{N-1}_{\mathbb R,*}$, and a free analogue $S^{N-1}_{\mathbb R,+}$. This is a presentation of the construction and main properties of these noncommutative spheres, $S^{N-1}_\mathbb R\subset S^{N-1}_{\mathbb R,*}\subset S^{N-1}_{\mathbb R,+}$, and of their quantum isometry groups.
We construct a Dirac operator on the quantum sphere $S^2_q$ which is covariant under the action of $SU_q(2)$. It reduces to Watamuras' Dirac operator on the fuzzy sphere when $q\to 1$. We argue that our Dirac operator may be useful in constructing $SU_q(2)$ invariant field theories on $S^2_q$ following the Connes-Lott approach to noncommutative geometry.
Hilbert space representations of the cross product *-algebra of the Hopf *-algebra U_q(su_2) and its module *-algebras O(S^2_{qc}) of Podles' spheres are studied. Two classes of representations are described by explicit formulas for the actions of the generators.
We discuss the left-covariant 3-dimensional differential calculus on the quantum sphere $SU_q (2)/U(1) $. The $SU_q (2)$-spinor harmonics are treated as coordinates of the quantum sphere. We consider the gauge theory for the quantum group $SU_q (2)\times U(1) $ on the deformed Euclidean space $E_q (4)$. A $q$-generalization of the harmonic-gauge-field formalism is suggested . This formalism is applied for the harmonic (twistor) interpretation of the quantum-group self-duality equation (QGSDE). We consider the zero-curvature representation and the general construction of QGSDE-solutions in term…
We construct new families of spectral triples over quantum spheres, with a particular attention focused on the standard Podles quantum sphere and twisted Dirac operators.
Noncommutative analogues of n-dimensional balls are defined by repeated application of the quantum double suspension to the classical low-dimensional spaces. In the `even-dimensional' case they correspond to the Twisted Canonical Commutation Relations of Pusz and Woronowicz. Then quantum spheres are constructed as double manifolds of noncommutative balls. Both C*-algebras and polynomial algebras of the objects in question are defined and analyzed, and their relations with previously known examples are presented. Our construction generalizes that of Hajac, Matthes and Szymanski for `dimension 2…
Quantum super 2-shpheres and the corresponding quantum super transformation group are introduced in analogy to the well-known quantum 2-shpheres and quantum SL(2), connection between little $t$-Jacobi polynomials and the finite dimensional representations of the quantum super group is formulated, and the Peter-Weyl theorem is obtained.
For $μ\in (0,1), c> 0,$ we identify the quantum group $SO_μ(3)$ as the universal object in the category of compact quantum groups acting by `orientation and volume preserving isometries' in the sense of \cite{goswami2} on the natural spectral triple on the Podles sphere $S^2_{μ, c}$ constructed by Dabrowski, D'Andrea, Landi and Wagner in \cite{Dabrowski_et_al}.
We construct spectral triples on all Podles quantum spheres. These noncommutative geometries are equivariant for a left action of $U_q(su(2))$ and are regular, even and of metric dimension 2. They are all isospectral to the undeformed round geometry of the 2-sphere. There is also an equivariant real structure for which both the commutant property and the first order condition for the Dirac operators are valid up to infinitesimals of arbitrary order.
We present a candidate of a vector space basis for the algebra $\mathcal{O}(S_q^{4n-1})$ of the quantum symplectic sphere for every $n\geq 1$. The algebra $\mathcal{O}(S_q^{4n-1})$ is defined as a certain subalgebra of the quantum symplectic group $\mathcal{O}(SP_q(2n))$. A non-trivial application of the Diamond Lemma is used to construct the vector space basis and the conjecture is supported by computer experiments for $n=1,2,...,8$.
A list of known quantum spheres of dimension one, two and three is presented.
In this thesis, we give a unification of the quantum WRT invariants. Given a rational homology 3-sphere M and a link L inside, we define the unified invariants, such that the evaluation of these invariants at a root of unity equals the corresponding quantum WRT invariant. In the SU(2) case, we assume the order of the first homology group of the manifold to be odd. Therefore, for rational homology 3-spheres, our invariants dominate the whole set of SO(3) quantum WRT invariants and, for manifolds with the order of the first homology group odd, the whole set of SU(2) quantum WRT invariants. We fu…
A Dirac operator D on the standard Podles sphere is defined and investigated. It yields a spectral triple such that |D|^{-z} is of trace class for Re z>0. Commutators with the Dirac operator give the distinguished 2-dimensional covariant differential calculus on the standard Podles sphere. The twisted cyclic cocycle associated with the volume form of the differential calculus is expressed by means of the Dirac operator.
The real sphere $S^{N-1}_\mathbb R$ appears as increasing union, over $d\in\{1,...,N\}$, of its "polygonal" versions $S^{N-1,d-1}_\mathbb R=\{x\in S^{N-1}_\mathbb R|x_{i_0}... x_{i_d}=0,\forall i_0,...,i_d\ {\rm distinct}\}$. Motivated by general classification questions for the undeformed noncommutative spheres, smooth or not, we study here the quantum isometries of $S^{N-1,d-1}_\mathbb R$, and of its various noncommutative analogues, obtained via liberation and twisting. We discuss as well a complex version of these results, with $S^{N-1}_\mathbb R$ replaced by the complex sphere $S^{N-1}_\m…
Spectral triples on the q-deformed spheres of dimension two and three are reviewed.
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