Bicovariant Differential Calculi Research Articles

Bicovariant differential calculi for finite global quotients

Bicovariant differential calculi for finite global quotients

Braided Hopf algebras from twisting

We show that a class of braided Hopf algebras, which includes the braided [Formula: see text] of [P. Kasprzak, R. Meyer, S. Roy and S. L. Woronowicz, Braided quantum [Formula: see text] groups, J. Noncommut. Geom. 10 (2016) 1161–1125], is obtained by twisting. We show further examples and demonstrate that twisting of bicovariant differential calculi gives braided bicovariant differential calculi.

Some construction of bicovariant differential calculi on weak Hopf algebras

Let H be a weak Hopf algebra with an idempotent element p. Denote by Hp some twisted weak Hopf algebra associated with a generalized Drinfeld twist . Let be a bicovariant differential calculus on H. We give some method to construct a bicovariant differential calculus on Hp by a generalized Drinfeld twist, where is some subspace of Γ determined by . As an application, we present some examples of bicovariant differential calculi on some face algebra and its quantum double.

Toward the classification of differential calculi on κ-Minkowski space and related field theories

Classification of differential forms on $\kappa$-Minkowski space, particularly, the classification of all bicovariant differential calculi of classical dimension is presented. By imposing super-Jacobi identities we derive all possible differential algebras compatible with the $\kappa$-Minkowski algebra for time-like, space-like and light-like deformations. Embedding into the super-Heisenberg algebra is constructed using non-commutative (NC) coordinates and one-forms. Particularly, a class of differential calculi with an undeformed exterior derivative and one-forms is considered. Corresponding NC differential calculi are elaborated. Related class of new Drinfeld twists is proposed. It contains twist leading to $\kappa$-Poincar\'e Hopf algebra for light-like deformation. Corresponding super-algebra and deformed super-Hopf algebras, as well as the symmetries of differential algebras are presented and elaborated. Using the NC differential calculus, we analyze NC field theory, modified dispersion relations, and discuss further physical applications.

Classification of Bicovariant Differential Calculi over free Orthogonal Hopf Algebras

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.

BICOVARIANT DIFFERENTIAL CALCULI ON A WEAK HOPF ALGEBRA

Let $H$ be a weak Hopf algebra with bijective antipode. In this paper we follow Woronowicz's fundamental method to characterize bicovariant differential calculi on $H$. We show that there exists a 1-1 correspondence between bicovariant differential calculi and some right ideals of $H$ contained in $ ker \varepsilon_s $ such that these ideals are right $H$-comodules with coadjoint maps, where $\varepsilon_s$ is the source map of $H$. This is a generalization of well-known Woronowicz's theorem about bicovariant differential calculi on quantum groups.

ON QUANTUM BRST COHOMOLOGY

Within bicovariant differential calculi framework, the BRST operator Ω is constructed. We showed that Ω is nil-potent (Ω2=0).

Q-Deformed quaternions and su(2) instantons

We have recently introduced the notion of a q-quaternion bialgebra and shown its strict link with the SOq(4)-covariant quantum Euclidean space ℝ4q. Adopting the available differential geometric tools on the latter and the quaternion language we have formulated and found solutions of the (anti)selfduality equation [instantons and multi-instantons] of a would-be deformed su(2) Yang-Mills theory on this quantum space. The solutions depend on some noncommuting parameters, indicating that the moduli space of a complete theory should be a noncommutative manifold. We summarize these results and add an explicit comparison between the two SOq(4)-covariant differential calculi on ℝ4q and the two 4-dimensional bicovariant differential calculi on the bi- (resp. Hopf) algebras Mq(2), GLq(2), SUq(2), showing that they essentially coincide.

Braided Lie algebras and bicovariant differential calculi over co-quasitriangular Hopf algebras

We show that if g Γ is the quantum tangent space (or quantum Lie algebra in the sense of Woronowicz) of a bicovariant first-order differential calculus over a co-quasitriangular Hopf algebra (A, r ) , then a certain extension of it is a braided Lie algebra in the category of A -comodules. This is used to show that the Woronowicz quantum universal enveloping algebra U( g Γ ) is a bialgebra in the braided category of A -comodules. We show that this algebra is quadratic when the calculus is inner. Examples with this unexpected property include finite groups and quantum groups with their standard differential calculi. We also find a quantum Lie functor for co-quasitriangular Hopf algebras, which has properties analogous to the classical one. This functor gives trivial results on standard quantum groups O q (G) , but reasonable ones on examples closer to the classical case, such as the cotriangular Jordanian deformations. In addition, we show that split braided Lie algebras define ‘generalized-Lie algebras’ in a different sense of deforming the adjoint representation. We construct these and their enveloping algebras for O q (SL(n)) , recovering the Witten algebra for n =2.

De Rham Cohomology and Hodge Decomposition For Quantum Groups

ISTVAN HECKENBERGER and AXEL SCHULER Abstra t Let Γ = Γτ,z be one of the N -dimensional bicovariant first order differential calculi for the quantum groups GLq(N), SLq(N), SOq(N), or Spq(N), where q is a transcendental complex number and z is a regular parameter. It is shown that the de Rham cohomology of Woronowicz’ external algebra Γ coincides with the de Rham cohomologies of its left-coinvariant, its right-coinvariant and its (twosided) coinvariant subcomplexes. In the cases GLq(N) and SLq(N) the cohomology ring is isomorphic to the coinvariant external algebra Γ Inv and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in these cases. The main technical tool is the spectral decomposition of the quantum Laplace-Beltrami operator.

Two Exterior Algebras for Orthogonal and Symplectic Quantum Groups

Let Γ be one of the N2-dimensional bicovariant first-order differential calculi on the quantum groups Oq(N) or Spq(N), where q is not a root of unity. We show that the second antisymmetrizer exterior algebra sΓ∧ is the quotient of the universal exterior algebra uΓ∧ by the principal ideal generated by θ∧θ. Here ∧ denotes the unique up to scalars bi-invariant 1-form. Moreover, θ∧θ is central in uΓ∧ and uΓ∧ is an inner differential calculus.

Laplace Operator and Hodge Decomposition for Quantum Groups and Quantum Spaces

LetΓ=Γ±,z be one of theN2-dimensional bicovariant first order differential calculi for the quantum groups GLq(N), SLq(N), SOq(N), or Spq(N), whereq is a transcendental complex number andz is a regular parameter. It is shown that the de Rham cohomology of Woronowicz’s external algebraΓ^ coincides with the de Rham cohomologies of its leftinvariant, its right-invariant and its biinvariant subcomplexes. In the cases GLq(N) and SLq(N) the cohomology ring is isomorphic to the biinvariant external algebraΓinv^ and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in these cases. The main technical tool is the spectral decomposition of the quantum Laplace-Beltrami operator. It is also applicable for quantum Euclidean spheres. The eigenvalues of the Laplace-Beltrami operator in cases of the general linear quantum group, the orthogonal quantum group, and the quantum Euclidean spheres are given.

DIFFERENTIAL ALGEBRAS FOR QUANTUM GROUPS OF TYPE B, C, AND D*)

We study higher order bicovariant differential calculi on the quantum groups Oq(N) and Spq(N). We show that the second antisymmetrizer exterior algebra uΓ∧ is the quotient of the universal exterior algebra uΓ∧ by the principal ideal generated by Θ ∧ Θ. Here θ denotes the unique up to scalars biinvariant 1-form. Moreover Θ ∧ Θ is central in uΓ∧ and uΓ∧ is an inner differential calculus. We show that the quadratic dual to the left-invariant algebra sΓL∧ is isomorphic to the reflection equation algebra. Let Γ be an arbitrary left-covariant first order differential calculus. We show that the dimension of the space of left-invariant 2-forms in the universal exterior algebra equals the number of linearly independent quadratic-linear relations in the quantum tangent space.

Quasitriangular and Differential Structures on Bicrossproduct Hopf Algebras

Let X=GM be a finite group factorisation. It is shown that the quantum double D(H) of the associated bicrossproduct Hopf algebra H=kM▷◀k(G) is itself a bicrossproduct kX▷◀k(Y) associated to a group YX, where Y=G×Mop. This provides a class of bicrossproduct Hopf algebras which are quasitriangular. We also construct a subgroup YθXθ associated to every order-reversing automorphism θ of X. The corresponding Hopf algebra kXθ▷◀k(Yθ) has the same coalgebra as H. Using related results, we classify the first order bicovariant differential calculi on H in terms of orbits in a certain quotient space of X.

Classification of differential calculi on Uq(b+), classical limits, and duality

We give a complete classification of bicovariant first order differential calculi on the quantum enveloping algebra Uq(b+) which we view as the quantum function algebra Cq(B+). Here, b+ is the Borel subalgebra of sl2. 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 four-dimensional calculus of κ-Minkowski space and the associated formal integral.

Differential Hopf Algebras on Quantum Groups of Type A

Let A be a Hopf algebra and Γ be a bicovariant first order differential calculus over A. It is known that there are three possibilities to construct a differential Hopf algebra Γ∧=Γ⊗/Jthat contains Γ as its first order part. Corresponding to the three choices of the idealJ, we distinguish the “universal” exterior algebra, the “second antisymmetrizer” exterior algebra, and Woronowicz' external algebra, respectively. Let Γ be one of theN2-dimensional bicovariant first order differential calculi on the quantum groupGLq(N) orSLq(N), and letqbe a transcendental complex number. For Woronowicz' external algebra we determine the dimension of the space of left-invariant and of bi-invariantk-forms, respectively. Bi-invariant forms are closed and represent different de Rham cohomology classes. The algebra of bi-invariant forms is graded anti-commutative. ForN≥3 the three differential Hopf algebras coincide. However, in case of the 4D±-calculi onSLq(2) the universal differential Hopf algebra is strictly larger than Woronowicz' external algebra. The bi-invariant 1-form is not closed.

Hodge and Laplace-Beltrami Operators for Bicovariant Differential Calculi on Quantum Groups

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 (SLq(N)) are computed.

Twisting of Quantum Differentials and¶the Planck Scale Hopf Algebra

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 space quantum mechanics. We further study the noncommutative differential geometry and Fourier theory for this Hopf algebra as a toy model for Planck scale physics. The Fourier theory implements a T-duality like self-duality. The noncommutative geometry turns out to be singular when $\grav\to 0$ and is therefore not visible in flat space quantum mechanics alone.

Classification of bicovariant differential calculi on the Jordanian quantum groups and and quantum Lie algebras

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 classical-like reduction from any one of the three families of 4-dimensional calculi on GL_{h,g}(2). Details of the higher order calculi and also the quantum Lie algebras are presented for all calculi. The quantum Lie algebra obtained from the bicovariant calculus on SL_{h}(2) is shown to be isomorphic to the quantum Lie algebra we obtain as an ad-submodule within the Jordanian universal enveloping algebra U_{h}(sl(2)) and also through a consideration of the decomposition of the tensor product of two copies of the deformed adjoint module. We also obtain the quantum Killing form for this quantum Lie algebra.

Classification of bicovariant differential calculi on the quantum groups SLq(n+1) and Spq(2n)

Article Classification of bicovariant differential calculi on the quantum groups SLq(n+1) and Spq(2n) was published on September 15, 1998 in the journal Journal für die reine und angewandte Mathematik (volume 1998, issue 502).