Salih Celik
We construct a two-parameter covariant differential calculus on the quantum $h$-exterior plane. We also give a deformation of the two-dimensional fermionic phase space.
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We give a two-parameter quantum deformation of the exterior plane and its differential calculus without the use of any R-matrix and relate it to the differential calculus with the R-matrix. We prove that there are two types of solutions of the Yang-Baxter equation whose symmetry group is $GL_{p,q}(2)$. We also give a two-parameter deformation of the fermionic oscillator algebra.
Hom-connections and associated integral forms have been introduced and studied by T.Brzeziński as an adjoint version of the usual notion of a connection in non-commutative geometry. Given a flat hom-connection on a differential calculus $(Ω, d)$ over an algebra $A$ yields the integral complex which for various algebras has been shown to be isomorphic to the noncommutative de Rham complex (in the sense of Brzeziński et al.). In this paper we shed further light on the question when the integral and the de Rham complex are isomorphic for an algebra $A$ with a flat hom-connection. We specialise ou…
For a Hopf algebra A, we define the structures of differential complexes on two dual exterior Hopf algebras: 1) an exterior extension of A and 2) an exterior extension of the dual algebra A^*. The Heisenberg double of these two exterior Hopf algebras defines the differential algebra for the Cartan differential calculus on A. The first differential complex is an analog of the de Rham complex. In the situation when A^* is a universal enveloping of a Lie (super)algebra the second complex coincides with the standard complex. The differential is realized as an (anti)commutator with a BRST- operator…
We investigate a particular realization of generalized q-differential calculus of exterior forms on a smooth manifold based on the assumption that the N-th power (N>2) of exterior differential is equal to zero. It implies the existence of cyclic commutation relations for the differentials of first order and their generalization for the differentials of higher order. Special attention is paid to the cases N=3 and N=4. A covariant basis of the algebra of such q-grade forms is introduced, and the analogues of torsion and curvature of higher order are considered. We also study a graded exterior ca…
A $Z_3$-graded Hopf algebra structure of exterior algebra with two generators is introduced. Two covariant differential calculus on the $Z_3$-graded exterior algebra are presented. Using the generators and their partial derivatives a Grassmann-Heisenberg algebra is constructed. An R-matrix which satisfies graded Yang-Baxter equations is obtained. A $Z_3$-graded universal enveloping algebra $U_q(\widetilde{gl}(2))$ is constructed with the quadratic elements of the Grassmann-Weyl algebra.
Let Γbe one of the N^2-dimensional bicovariant first order differential calculi on the orthogonal or symplectic quantum group O_q(N) or Sp_q(N). The parameter q is not a root of unity. We show that the second antisymmetrizer exterior algebra is the quotient of the universal exterior algebra U by the principal ideal generated by w^2. Here w denotes the unique up to scalars bi-invariant 1-form. Moreover w^2 is central in U and U is an inner differential calculus.
We show that the algebra of the bicovariant differential calculus on a quantum group can be understood as a projection of the cross product between a braided Hopf algebra and the quantum double of the quantum group. The resulting super-Hopf algebra can be reproduced by extending the exterior derivative to tensor products.
The non-commutative differential calculus on the quantum groups $SL_q(N)$ is constructed. The quantum external algebra proposed contains the same number of generators as in the classical case. The exterior derivative defined in the constructive way obeys the modified version of the Leibnitz rules.
We provide the Cartan calculus for bicovariant differential forms on bicrossproduct quantum groups $k(M)\lrbicross kG$ associated to finite group factorizations $X=GM$ and a field $k$. The irreducible calculi are associated to certain conjugacy classes in $X$ and representations of isotropy groups. We find the full exterior algebras and show that they are inner by a bi-invariant 1-form $θ$ which is a generator in the noncommutative de Rham cohomology $H^1$. The special cases where one subgroup is normal are analysed. As an application, we study the noncommutative cohomology on the quantum codo…
A rigid framework for the Cartan calculus of Lie derivatives, inner derivations, functions, and forms is proposed. The construction employs a semi-direct product of two graded Hopf algebras, the respective super-extensions of the deformed universal enveloping algebra and the algebra of functions on a quantum group. Relations in the Cartan calculus follow as consistency conditions. The approach is not a priori based on the Leibniz rule for the exterior derivative and might hence also be of interest in the recent work on its deformations. The Cartan identity for the Lie derivatives is proven. (T…
A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum groups, we combine an exterior derivative, inner derivations, Lie derivatives, forms and functions all into one big algebra. In particular we find a generalized Cartan identity that holds on the whole quantum universal enveloping algebra of the left-invariant vector fields and implicit commutation relations for a left-invariant basis of 1-forms.
We show how one can construct a differential calculus over an algebra where position variables x and momentum variables p have be defined. As the simplest example we consider the one-dimensional q-deformed Heisenberg algebra. This algebra has a subalgebra generated by x and its inverse which we call the coordinate algebra. A physical field is considered to be an element of the completion of this algebra. We can construct a derivative which leaves invariant the coordinate algebra and so takes physical fields into physical fields. A generalized Leibniz rule for this algebra can be found. Based o…
The non-commutative differential calculus on quantum groups can be extended by introducing, in analogy with the classical case, inner product operators and Lie derivatives. For the case of $\GL$ we show how this extended calculus induces by coaction a similar extended calculus, covariant under $\GL$, on the quantum plane. In this way, inner product operators and Lie derivatives can be introduced on the plane as well. The situation with other quantum groups and quantum spaces is briefly discussed. Explicit formulas are given for the two dimensional quantum plane.
A Cartan Calculus of Lie derivatives, differential forms, and inner derivations, based on an undeformed Cartan identity, is constructed. We attempt a classification of various types of quantum Lie algebras and present a fairly general example for their construction, utilizing pure braid methods, proving orthogonality of the adjoint representation and giving a (Killing) metric and the quadratic casimir. A reformulation of the Cartan calculus as a braided algebra and its extension to quantum planes, directly and induced from the group calculus, are provided.
Based on \cite{DH94}, we introduce a bijective correspondence between first order differential calculi and the graph structure of the symmetric lattice that allows one to encode completely the interconnection structure of the graph in the exterior derivative. As a result, we obtain the Grassmannian character of the lattice as well as the mutual commutativity between basic vector-fields on the tangent space. This in turn gives several similarities between the Clifford setting and the algebra of endomorphisms endowed by the graph structure, such as the hermitian structure of the lattice as well …
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