Braiding and exponentiating noncommutative vector fields

1995

Czechoslovak journal of physics, 2003

The aim of this paper is to avoid some difficulties, related with the Lie bracket, in the definition of vector fields in a noncommutative setting, as they were defined by Woronowicz, Schmüdgen-Schüler and Aschieri-Schupp. We extend the definition of vector fields to ...

arXiv (Cornell University), 2022

In this paper, we revise the concept of noncommutative vector fields introduced previously in [1, 2], extending the framework, adding new results and clarifying the old ones. Using appropriate algebraic tools certain shortcomings in the previous considerations are filled and made more precise. We focus on the correspondence between so-called Cartan pairs and first-order differentials. The case of free bimodules admitting more friendly "coordinate description" and their braiding is considered in more detail. Bimodules of right/left universal vector fields are explicitly constructed.

2006

1997

A notion of Cartan pairs as an analogy of vector fields in the realm of noncommutative geometry has been proposed in . In this paper we give an outline of the construction of a noncommutative analogy of the algebra of partial differential operators as well as its natural (Fock type) representation. We shall also define co-universal vector fields and covariant derivatives.

Journal of Geometry and Physics, 1993

This is an introduction to the old and new concepts of non-commutative (N.C.) geometry. We review the ideas underlying N.C. measure and topology, N.C. differential calculus, N.C. connections on N.C. vector bundles, and N.C. Riemannian geometry by following A. Connes' point of view.

Journal of Mathematical Physics, 1993

Braided differential operators ∂ i are obtained by differentiating the addition law on the braided covector spaces introduced previously (such as the braided addition law on the quantum plane). These are affiliated to a Yang-Baxter matrix R. The quantum eigenfunctions exp R (x|v) of the ∂ i (braided-plane waves) are introduced in the free case where the position components x i are totally noncommuting. We prove a braided R-binomial theorem and a braided-Taylors theorem exp R (a|∂)f (x) = f (a + x). These various results precisely generalise to a generic R-matrix (and hence to n-dimensions) the well-known properties of the usual 1dimensional q-differential and q-exponential. As a related application, we show that the q-Heisenberg algebra px − qxp = 1 is a braided semidirect product C[x]>⊳C[p] of the braided line acting on itself (a braided Weyl algebra). Similarly for its generalization to an arbitrary R-matrix.

2018

Noncommutative geometry is the idea that when geometry is done in terms of coordinate algebras, one does not really need the algebra to be commutative. We provide an introduction to the relevant mathematics from a constructive ‘differential algebra’ point of view that works over general fields and includes the noncommutative geometry of quantum groups as well as of finite groups. We also mention applications to models of quantum spacetime.

1994

We show that every Lie algebra or superLie algebra has a canonical braiding on it, and that in terms of this its enveloping algebra appears as a flat space with braided-commuting coordinate functions. This also gives a new point of view about q-Minkowski space which arises in a similar way as the enveloping algebra of the braided Lie algebra gl 2,q. Our point of view fixes the signature of the metric on q-Minkowski space and hence also of ordinary Minkowski space at q = 1. We also describe an abstract construction for left-invariant integration on any braided group.