Finite dimensional quantum group covariant differential calculus on a complex matrix algebra

Abstract

Using the fact that the algebra M3(C l ) of 3 × 3 complex matrices can be taken as a reduced quantum plane, we build a differential calculus Ω(S) on the quantum space S defined by the algebra, where M is a space-time manifold. This calculus is covariant under the action and coaction of finite dimensional dual quantum groups. We study the star structures on these quantum groups and the compatible one in M3(C l ). This leads to an invariant scalar product on the later space. We analyse the differential algebra Ω(M3(C l )) in terms of quantum group representations, and consider in particular the space of 1-forms on S since its elements can be considered as generalized gauge fields.

Differential Calculus and Connections on a Quantum Plane at a Cubic Root of Unity

“…Remember we only said the scalar product was such that the stars coincide with the adjointoperators, or such that the actions are given by * -representations. We refer the reader to [14], where it is shown that the * -representation condition on the scalar product,…”

“…In order to find these expressions, it helps to notice that ∂ x and ∂ y are respectively of weight −1/2, 1/2 in Z/3Z (see also the discussion at the end of Section 4.4). Writing the generators as arbitrary differential operators of a fixed weight, one can determine the coefficients by imposing that equations (14), (22) An alternative way of writing these, is to make use of the scaling operators,…”

Hopf stars, twisted Hopf stars and scalar products on quantum spaces

“…We refer the reader to [11] for a more detailed discussion. As the Hopf star doesn't commute with the antipode, since S * = * S −1 , (18) and (19) are, in general, two different conditions.…”

“…For these reasons, and although we decided to write quite a general paper, most explicitly discussed examples will involve the case of the finite dimensional algebra H = U res q (sl(2, C)) at a cubic root of unity. Another motivation for studying the reality structures and the type of scalar products in star representations of quantum groups comes from our previous work [11,12]. Here a new kind of gauge fields was obtained: starting from the observation that the reduced quantum plane (identified with the algebra of N ×N complex matrices) is a module-algebra for the finite dimensional quantum group H, when q N = 1, we built a differential algebra over it by taking an appropriate quotient of the Wess-Zumino differential algebra over the -infinite dimensional-quantum plane; generalized differential forms are then obtained by making the tensor product of the De Rham complex of forms over an arbitrary spacetime manifold times the previous Wess-Zumino reduced differential complex; generalized gauge fields (and curvatures, etc. )…”

The Two-Parameter Higher-Order Differential Calculus and Curvature on a Quantum Plane

“…Many authors [1,2,3,4,5,6] have studied differential calculus with nilpotency d 2 = 0 on quantum spaces with one or two-parameter quantum group symmetry [7,8].…”