We give a complete classification of bicovariant first order differential calculi on the quantum enveloping algebra U_q(b+) which we view as the quantum function algebra C_q(B+). Here, b+ is the Borel subalgebra of sl_2. We do the same in the classical limit q->1 and obtain a one-to-one correspondence in the finite dimensional case. It turns out that the classification is essentially given by finite subsets of the positive integers. We proceed to investigate the classical limit from the dual point of view, i.e. with ``function algebra'' U(b+) and ``enveloping algebra'' C(B+). In this case there are many more differential calculi than coming from the q-deformed setting. As an application, we give the natural intrinsic 4-dimensional calculus of kappa-Minkowski space and the associated formal integral.
If the bimodule of 1-forms of a differential calculus over an associative algebra is the direct sum of 1-dimensional bimodules, a relation with automorphisms of the algebra shows up. This happens for some familiar quantum space calculi.
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 …
We equip a family of algebras whose noncommutativity is of Lie type with a derivation based differential calculus obtained, upon suitably using both inner and outer derivations, as a reduction of a redundant calculus over the Moyal four dimensional space.
For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace-Beltrami operator for arbitrary k-forms is given. Under some technical assumptions it is proved that Woronowicz' external algebra of left-invariant differential forms either contains a unique form of maximal degree or it is infinite dimensional. Using Jucys-Murphy elements of the Hecke algebra the eigenvalues of the Laplace-Beltrami operator for the Hopf algebra O(SLq(N)) are computed.
We study covariant derivatives on a class of centered bimodules $\mathcal{E}$ over an algebra A. We begin by identifying a $\mathbb{Z} ( A ) $-submodule $ \mathcal{X} ( A ) $ which can be viewed as the analogue of vector fields in this context; $ \mathcal{X} ( A ) $ is proven to be a Lie algebra. Connections on $\mathcal{E}$ are in one to one correspondence with covariant derivatives on $ \mathcal{X} ( A ). $ We recover the classical formulas of torsion and metric compatibility of a connection in the covariant derivative form. As a result, a Koszul formula for the Levi-Civita connection is als…
We show that if two Hopf algebras are monoidally equivalent, then their categories of bicovariant differential calculi are equivalent. We then classify, for $q \in \mathbb{C}^*$ not a root of unity, the finite dimensional bicovariant differential calculi over the Hopf algebra $\mathcal{O}_q(SL_2)$. Using a monoidal equivalence between free orthogonal Hopf algebras and $\mathcal{O}_q(SL_2)$ for a given $q$, this leads us to the classification of finite dimensional bicovariant differential calculi over free orthogonal Hopf algebras.
A brief review of bicovariant differential calculi on finite groups is given, with some new developments on diffeomorphisms and integration. We illustrate the general theory with the example of the nonabelian finite group S_3.
We study some aspects of the theory of non-commutative differential calculi over complex algebras, especially over the Hopf algebras associated to compact quantum groups in the sense of S.L. Woronowicz. Our principal emphasis is on the theory of twisted graded traces and their associated twisted cyclic cocycles. One of our principal results is a new method of constructing differential calculi, using twisted graded traces.
For transcendental values of q the quantum tangent spaces of all left-covariant first order differential calculi of dimension less than four on the quantum group $\SLq 2$ are given. All such differential calculi $Γ$ are determined and investigated for which the left-invariant differential one-forms $ω(u^1_2)$, $ω(u^2_1)$ and $ω(u^1_1-u^2_2)$ generate $Γ$ as a bimodule and the universal higher order differential calculus has the same dimension as in the classical case. Important properties (cohomology spaces, *-structures, braidings, generalized Lie brackets) of these calculi are examined as we…
A unitary orthosymplectic quantum supergroup is introduced. Two covariant differential calculi on the quantum superspace $SP_q^{1|2}$ are presented. The $h$-deformed symplectic superspaces via a contraction of the $q$-deformed symplectic superspaces are obtained. A new $h$-deformation of the Heisenberg superalgebra is given.
For two covariant differential *-calculi, the twisted cyclic cocycle associated with the volume form is represented in terms of commutators [F,ρ(x)] for some self-adjoint operator F and some *-representation $ρ$ of the underlying *-algebra.
We consider Hopf bimodules and crossed modules over a Hopf algebra $H$ in a braided category. They are the key-stones for braided bicovariant differential calculi and their invariant vector fields respectively, as well as for the construction of braided Hopf algebra cross products. We show that the notions of Hopf bimodules and crossed modules are equivalent. A generalization of the Radford-Majid criterion to the braided case is given and it is seen that bialgebra cross products over the Hopf algebra $H$ are precisely described by $H$-crossed module bialgebras. We study the theory of (bicovari…
We study *-differential calculi over compact quantum groups in the sense of S.L. Woronowicz. Our principal results are the construction of a Hodge operator commuting with the Laplacian, the derivation of a corresponding Hodge decomposition of the calculus of forms, and, for Woronowicz' first calculus, the calculation of the eigenvalues of the Laplacian
We study $N^2-1$ dimensional left-covariant differential calculi on the quantum group $SL_q(N)$ for which the generators of the quantum Lie algebras annihilate the quantum trace. In this way we obtain one distinguished calculus on $SL_q(2)$ (which corresponds to Woronowicz' 3D-calculus on $SU_q(2)$) and two distinguished calculi on $SL_q(3)$ such that the higher order calculi give the ordinary differential calculus on $SL(2)$ and $SL(3)$, respectively, in the limit $q\rightarrow 1$. Two new differential calculi on $SL_q(3)$ are introduced and developed in detail.
The differential caluli $(Gamma,d)$ on quantum groups are classified due to the property of the generating element $X$ of its differential $d$. There are, on the one hand differential caluli which contain this element $X$ in the basis of one- forms that span $Gamma$, called Inner Differential Calculi. On the other hand, one has the differential caluli which do not contain the generating element $X$ of its differential $d$, thus they are called Outer Differential Calculi. We show that this two classes of differential caluli, for a given quantum group ${\cal A}$, are related by homomorphisms, wh…
The differential calculus on n-dimensional quantum Minkowski space covariant with respect to left action of Kappa-Poincar'e group is constructed and its uniqueness is shown.
In this work, we introduce the ${\mathbb Z}_3$-graded differential algebra, denoted by $Ω(\widetilde{\rm GL}_q(2))$, treated as the ${\mathbb Z}_3$-graded quantum de Rham complex of ${\mathbb Z}_3$-graded quantum group $\widetilde{\rm GL}_q(2)$. In this sense, we construct left-covariant differential calculi on the ${\mathbb Z}_3$-graded quantum group $\widetilde{\rm GL}_q(2)$.
We introduce a large class of bicovariant differential calculi on any quantum group $A$, associated to $Ad$-invariant elements. For example, the deformed trace element on $SL_q(2)$ recovers Woronowicz' $4D_\pm$ calculus. More generally, we obtain a sequence of differential calculi on each quantum group $A(R)$, based on the theory of the corresponding braided groups $B(R)$. Here $R$ is any regular solution of the QYBE.
In this review article the construction of first order coordinate differential calculi on finitely generated and finitely related associative algebras are considered and explicit construction of the bimodule of one form over such algebras is presented. The concept of optimal algebras for such calculi are also discussed. Detailed computations presented will make this note particularly useful for physicists.
We show that the bicovariant first order differential calculi on a factorisable semisimple quantum group are in 1-1 correspondence with irreducible representations $V$ of the quantum group enveloping algebra. The corresponding calculus is constructed and has dimension $dim V^2$. The differential calculi on a finite group algebra $C G$ are also classified and shown to be in correspondence with pairs consisting of an irreducible representation $V$ and a continuous parameter in $C P^{dim V -1}$. They have dimension $dim V$. For a classical Lie group we obtain an infinite family of non-standard ca…
All bicovariant first order differential calculi on the quantum group GLq(3,C) are determined. There are two distinct one-parameter families of calculi. In terms of a suitable basis of 1-forms the commutation relations can be expressed with the help of the R-matrix of GLq(3,C). Some calculi induce bicovariant differential calculi on SLq(3,C) and on real forms of GLq(3,C). For generic deformation parameter q there are six calculi on SLq(3,C), on SUq(3) there are only two. The classical limit q-->1 of bicovariant calculi on SLq(3,C) is not the ordinary calculus on SL(3,C). One obtains a deformat…
A variety of three-dimensional left-covariant differential calculi on the quantum group $SU_q(2)$ is considered using an approach based on global $ U(1) $ -covariance. Explicit representations of possible $q $-Lie algebras are constructed in terms of differential operators. A gauge covariant differential algebra is uniquely determined. The non-standard Leibnitz rule is obtained for a corresponding $q$-Lie algebra.
We introduce a category of noncommutative bundles. To establish geometry in this category we construct suitable noncommutative differential calculi on these bundles and study their basic properties. Furthermore we define the notion of a connection with respect to a differential calculus and consider questions of existence and uniqueness. At the end these constructions are applied to basic examples of noncommutative bundles over a coquasitriangular Hopf algebra.
Let A be a coquasitriangular Hopf algebra and X the subalgebra of A generated by a row of a matrix corepresentation u or by a row of u and a row of the contragredient representation u^c. In the paper left-covariant first order differential calculi on the quantum group A are constructed and the corresponding induced calculi on the left quantum space X are described. The main tool for these constructions are the L-functionals associated with u. The results are applied to the quantum homogeneous space GL_q(N)/GL_q(N-1).
We have introduced q-analogues of bounded symmetric domains in our work q-alg/9703005. Given the simplest ones among those, the works q-alg/9603012 and math.QA/9803110 announce the relations describing the algebras of functions, differential and integral calculi. This paper presents the proofs of these results.
We investigate covariant first order differential calculi on the quantum complex projective spaces CP_q^{N-1} which are quantum homogeneous spaces for the quantum group SU_q(N). Hereby, one more well-studied example of covariant first order differential calculus on a quantum homogeneous space is given. Since the complex projective spaces are subalgebras of the quantum spheres S_q^{2N-1} introduced by Vaksman and Soibelman, we get also an example of the relations between covariant differential calculus on two closely related quantum spaces. Two approaches are combined in obtaining covariant fir…
A differential calculus of the first order over multi-braided quantum groups is developed. In analogy with the standard theory, left/right-covariant and bicovariant differential structures are introduced and investigated. Furthermore, antipodally covariant calculi are studied. The concept of the *-structure on a multi-braided quantum group is formulated, and in particular the structure of left-covariant *-covariant calculi is analyzed. A special attention is given to differential calculi covariant with respect to the action of the associated braid system. In particular it is shown that the lef…
We show that bicovariant bimodules as defined by Woronowicz are in one to one correspondence with the Drinfeld quantum double representations. We then prove that a differential calculus associated to a bicovariant bimodule of dimension n is connected to the existence of a particular (n+1)--dimensional representation of the double. An example of bicovariant differential calculus on the non quasitriangular quantum group E_q(2) is developed. The construction is studied in terms of Hochschild cohomology and a correspondence between differential calculi and 1-cocycles is proved. Some differences of…
We show that a class of braided Hopf algebras, which includes the braided $SU_q(2)$ is obtained by twisting. We show further examples and demonstrate that twisting of bicovariant differential calculi gives braided bicovariant differential calculi.
Differential calculi are obtained for quantum homogeneous spaces by extending Woronowicz' approach to the present context. Representation theoretical properties of the differential calculi are investigated. Connections on quantum homogeneous vector bundles are classified and explicitly constructed by using the theory of projective modules.
We describe Laplacian operators on the quantum group SUq (2) equipped with the four dimensional bicovariant differential calculus of Woronowicz as well as on the quantum homogeneous space S2q with the restricted left covariant three dimensional differential calculus. This is done by giving a family of Hodge dualities on both the exterior algebras of SUq (2) and S2q . We also study gauged Laplacian operators acting on sections of line bundles over the quantum sphere.
Covariant first order differential calculus over quantum complex Grassmann manifolds is considered. It is shown by a Pusz-Woronowicz type argument that under restriction to calculi close to classical Kaehler differentials there exist exactly two such calculi for the homogeneous coordinate ring. Complexification and localization procedures are used to induce covariant first order differential calculi over quantum Grassmann manifolds. It is shown that these differential calculi behave in many respects as their classical counterparts. As an example the q-deformed Chern character of the tautologic…
Let (Γ,d) be the 3D-calculus or the 4D_{\pm}-calculus on the quantum group SU_q(2). We describe all pairs (π, F) of a *-representation πof O(SU_q(2)) and of a symmetric operator F on the representation space satisfying a technical condition concerning its domain such that there exist a homomorphism of first order differential calculi which maps dx into the commutator [iF,π(x)] for x\in O(SU_q(2)). As an application commutator representations of the 2-dimensional left-covariant calculus on Podles' quantum 2-sphere S^2_{qc} with c=0 are given.
A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a one-to-one correspondence, between the module structure of the 1-forms and the metric torsion-free connections on it. In the commutative limit the connection remains as a shadow of the algebraic structure of the 1-forms.
Differential calculi of Poincare-Birkhoff-Witt type on universal enveloping algebras of Lie algebras g are defined. This definition turns out to be independent of the basis chosen in g. The role of automorphisms of g is explained. It is proved that no differential calculus of Poincare-Birkhoff-Witt type exists on semi-simple Lie algebras. Examples are given, namely gl_n, Abelian Lie algebras, the Heisenberg algebra, the Witt and the Virasoro algebra. Completely treated are the 2-dimensional solvable Lie algebra, and the 3-dimensional Heisenberg algebra.
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…
Alma Mater Studiorum - Università di Bologna（博洛尼亚大学）；Istituto Nazionale di Fisica Nucleare Sezione di Bologna（意大利国家核物理研究所博洛尼部分所）；Mathematical Institute of Charles University（查理大学数学研究所）
A method of constructing covariant differential calculi on a quantum homogeneous space is devised. The function algebra X of the quantum homogeneous space is assumed to be a left coideal of a coquasitriangular Hopf algebra H and to contain the coefficients of any matrix over H which is the two-sided inverse of one with entries in X. The method is based on partial derivatives. For the quantum sphere of Podles and the quantizations of symmetric spaces due to Noumi, Dijkhuizen and Sugitani the construction produces the subcalculi of the standard bicovariant calculus on the quantum group.
For transcendental values of $q$ all bicovariant first order differential calculi on the coordinate Hopf algebras of the quantum groups $SL_q(n+1)$ and $Sp_q(2n)$ are classified. It is shown that the irreducible bicovariant first order calculi are determined by an irreducible corepresentation of the quantum group and a complex number $ζ$ such that $ζ^{n+1}=1$ for $SL_q(n+1)$ and $ζ^2=1$ for $Sp_q(2n)$. Any bicovariant calculus is inner and its quantum Lie algebra is generated by a central element. The main technical ingredient is a result of the Hopf algebra $R(G_q)^0$ for arbitrary simple Lie…
Using non canonical braidings, we first introduce a notion of symmetric tensors and corresponding Hodge operators on a class of left covariant 3d differential calculi over the quantum SU(2) group, then we induce Hodge operators on the left covariant 2d exterior algebras over the standard Podles quantum sphere.
Explicit construction of the second order left differential calculi on the quantum group and its subgroups are obtained with the property of the natural reduction: the differential calculus on the quantum group $GL_q(2,C)$ has to contain the 3-dimensional differential calculi on the quantum subgroup $SL_q(2,C)$, the differential calculi on the Borel subgroups $B_{L}^{(2)}(C)$, $B_{U}^{(2)}(C)$ of the lower and of the upper triangular matrices, on the quantum subgroups $U_{q}(2)$, $SU_{q}(2)$, $Sp_{q}(2,C)$, $Sp_{q}(2)$, $T_{q}(2,C)$, $B_{L}(C)$, $B_{U}(C)$, $U_{q}(1)$, $Z_{-}^{(2)}(C)$, $Z_{+}…
We define a noncommutative algebra of four basic objects within a differential calculus on quantum groups: functions, 1-forms, Lie derivatives and inner derivations, as the cross-product algebra associated with Woronowicz's (differential) algebra of functions and forms. This definition properly takes into account the Hopf algebra structure of the Woronowicz calculus. It also provides a direct proof of the Cartan identity.
This paper is devoted to study of differential calculi over quadratic algebras, which arise in the theory of quantum bounded symmetric domains. We prove that in the quantum case dimensions of the homogeneous components of the graded vector spaces of k-forms are the same as in the classical case. This result is well-known for quantum matrices. The quadratic algebras, which we consider in the present paper, are q-analogues of the polynomial algebras on prehomogeneous vector spaces of commutative parabolic type. This enables us to prove that the de Rham complex is isomorphic to the dual of a quan…
A new notion of an optimal algebra for a first order coordinate differential was introduced in \cite{BKO}. Some relevant examples are indicated. Quadratic identities in the optimal algebras and calculi on quadratic algebras are studied. Canonical construction of a quantum de Rham complex for the coordinate differential is proposed. The relations between calculi and various generalizations of the Yang--Baxter equation are established.
Cartan calculi on the extended quantum superplane are given. To this end, the noncommutative differential calculus on the extended quantum superplane is extended by introducing inner derivations and Lie derivatives.
Let A be a cosemisimple Hopf *-algebra with antipode S and let $Γ$ be a left-covariant first order differential *-calculus over A such that $Γ$ is self-dual and invariant under the Hopf algebra automorphism S^2. A quantum Clifford algebra $\Cl(Γ,σ,g)$ is introduced which acts on Woronowicz' external algebra $Γ^\wedge$. A minimal left ideal of $\Cl(Γ,σ,g)$ which is an A-bimodule is called a spinor module. Metrics on spinor modules are investigated. The usual notion of a linear left connection on $Γ$ is extended to quantum Clifford algebras and also to spinor modules. The corresponding Dirac ope…
For differential calculi over certain right coideal subalgebras of quantum groups the notion of quantum tangent space is introduced. In generalization of a result by Woronowicz a one to one correspondence between quantum tangent spaces and covariant first order differential calculi is established. This result is used to classify differential calculi over quantum Grassmann manifolds. It turns out that up to special cases in low dimensions there exists exactly one such calculus of classical dimension 2r(N-r). Keywords: Quantum groups, quantum spaces, quantum Grassmann manifolds, differential cal…
Let (G,d) be a first order differential *-calculus on a *-algebra A. We say that a pair (π,F) of a *-representation πof A on a dense domain D of a Hilbert space and a symmetric operator F on D gives a commutator representation of G if there exists a linear mapping t:G -> L(D) such that t(adb)=π(a)i[F,π(b)], a,b in A. Among others, it is shown that each left-covariant *-calculus G of a compact quantum group Hopf *-algebra A has a faithful commutator representation. For a class of bicovariant *-calculi on A there is a commutator representation such that F is the image of a central element of the…
We construct a noncommutative Cartan calculus on any braided commutative algebra and study its applications in noncommutative geometry. The braided Lie derivative, insertion and de Rham differential are introduced and related via graded braided commutators, also incorporating the braided Schouten-Nijenhuis bracket. The resulting braided Cartan calculus generalizes the Cartan calculus on smooth manifolds and the twisted Cartan calculus. While it is a necessity of derivation based Cartan calculi on noncommutative algebras to employ central bimodules our approach allows to consider bimodules over…
In this report we give an intrinsic treatment of the results we developed in a previous work connecting the differential calculi on Hopf algebras to the Drinfeld double. In the first place we recover that bicovariant bimodules are in one to one correspondence with the Drinfeld double representations; we then introduce a Hochschild cohomology of the algebra of functions and discuss the main result stating that each differential calculus is associated to a 1-cocycle satisfying an additional invariance condition with respect to a natural action. Defining a Hochschild cohomology of the double, the…
We review the construction of the multiparametric quantum group $ISO_{q,r}(N)$ as a projection from $SO_{q,r}(N+2) $ and show that it is a bicovariant bimodule over $SO_{q,r}(N)$. The universal enveloping algebra $U_{q,r}(iso(N))$, characterized as the Hopf algebra of regular functionals on $ISO_{q,r}(N)$, is found as a Hopf subalgebra of $U_{q,r}(so(N+2))$ and is shown to be a bicovariant bimodule over $U_{q,r}(so(N))$. An R-matrix formulation of $U_{q,r}(iso(N))$ is given and we prove the pairing $U_{q,r}(iso(N))\leftrightarrow ISO_{q,r}(N)$. We analyze the subspaces of $U_{q,r}(iso(N))$ tha…
Braided non-commutative differential geometry is studied. In particular we investigate the theory of (bicovariant) differential calculi in braided abelian categories. Previous results on crossed modules and Hopf bimodules in braided categories are used to construct higher order bicovariant differential calculi over braided Hopf algebras out of first order ones. These graded objects are shown to be braided differential Hopf algebras with universal bialgebra properties. The article especially extends Woronowicz's results on (bicovariant) differential calculi to the braided non-commutative case.
The deformed double covering of E(2) group, denoted by $\tilde{E}_κ(2)$, is obtained by contraction from the $SU_μ(2)$. The contraction procedure is then used for producing a new examples of differential calculi: 3D-left covariant calculus on both $\tilde{E}_κ(2)$ and the deformed Euclidean group $E_κ(2)$ and two different 4D-bicovariant calculi on $\tilde{E}_κ(2)$ which correspond to the one 4D-bicovariant calculi on $E_κ(2)$ described in Ref.\cite{b14}.
We construct differential calculi on multiparametric quantum orthogonal planes in any dimension N. These calculi are bicovariant under the action of the full inhomogeneous (multiparametric) quantum group ISO_{q,r}(N), and do contain dilatations. If we require bicovariance only under the quantum orthogonal group SO_{q,r}(N), the calculus on the q-plane can be expressed in terms of its coordinates x^a, differentials dx^a and partial derivatives \partial_a without the need of dilatations, thus generalizing known results to the multiparametric case. Using real forms that lead to the signature (n+1…
We study covariant differential calculus on the quantum spheres S_q^{N-1} which are quantum homogeneous spaces with coactions of the quantum groups O_q(N). The first part of the paper is devoted to first order differential calculus. A classification result is proved which says that for N>=6 there exist exactly two covariant first order differential calculi on S_q^{N-1} which satisfy the classification constraint that the bimodule of one-forms is generated as a free left module by the differentials of the generators of S_q^{N-1}. Both calculi exist also for 3<=N<=5. The same calculi can be cons…
