Differential Calculus on Quantum Spaces and Quantum Groups

A review of recent developments in the quantum differential calculus. The quantum group $GL_q(n)$ is treated by considering it as a particular quantum space. Functions on $SL_q(n)$ are defined as a subclass of functions on $GL_q(n)$. The case of $SO_q(n)$ is also briefly considered. These notes cover part of a lecture given at the XIX International Conference on Group Theoretic Methods in Physics, Salamanca, Spain 1992.Comment: 23 page

Non-Abelian Anomalies and Effective Actions for a Homogeneous Space G/HG/H

We consider the problem of constructing the fully gauged effective action in $2n$-dimensional space-time for Nambu-Goldstone bosons valued in a homogeneous space $G/H$, with the requirement that the action be a solution of the anomalous Ward identity and be invariant under the gauge transformations of $H$. We show that this can be done whenever the homotopy group $\pi_{2n}(G/H)$ is trivial, $G/H$ is reductive and $H$ is embedded in $G$ so as to be anomaly free, in particular if $H$ is an anomaly safe group. We construct the necessary generalization of the Bardeen counterterm and give explicit forms for the anomaly and the effective action. When $G/H$ is a symmetric space the counterterm and the anomaly decompose into a parity even and a parity odd part. In this case, for the parity even part of the action, one does not need the anomaly free embedding of $H$.Comment: revise

Reality in the Differential Calculus on q-euclidean Spaces

The nonlinear reality structure of the derivatives and the differentials for the euclidean q-spaces are found. A real Laplacian is constructed and reality properties of the exterior derivative are given.Comment: 10 page

Cartan Calculus on Quantum Lie Algebras

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 spaces we combine an exterior derivative, inner derivations, Lie derivatives, forms and functions all into one big algebra, the ``Cartan Calculus''. (This is an extended version of a talk presented by P. Schupp at the XXII$^{th}$ International Conference on Differential Geometric Methods in Theoretical Physics, Ixtapa, Mexico, September 1993)Comment: 15 pages in LaTeX, LBL-34833 and UCB-PTH-93/3

Holomorphy, Minimal Homotopy and the 4D, N = 1 Supersymmetric Bardeen-Gross-Jackiw Anomaly

By use of a special homotopy operator, we present an explicit, closed-form and simple expression for the left-right Bardeen-Gross-Jackiw anomalies described as the proper superspace integral of a superfunction.Comment: 16 pp, LaTeX, Replacement includes addition comment on WZNW term and one new referenc

Some remarks on unilateral matrix equations

We briefly review the results of our paper hep-th/0009013: we study certain perturbative solutions of left-unilateral matrix equations. These are algebraic equations where the coefficients and the unknown are square matrices of the same order, or, more abstractly, elements of an associative, but possibly noncommutative algebra, and all coefficients are on the left. Recently such equations have appeared in a discussion of generalized Born-Infeld theories. In particular, two equations, their perturbative solutions and the relation between them are studied, applying a unified approach based on the generalized Bezout theorem for matrix polynomials.Comment: latex, 6 pages, 1 figure, talk given at the euroconference "Brane New World and Noncommutative Geometry", Villa Gualino, Torino, Italy, Oct 2-7, 200

Realization of Vector fields for Quantum Groups as Pseudodifferential Operators on Quantum Spaces

The vector fields of the quantum Lie algebra are described for the quantum groups $GL_q(N), SL_q(N)$ and $SO_q(N)$ as pseudodifferential operators on the linear quantum spaces covariant under the corresponding quantum group. Their expressions are simple and compact. It is pointed out that these vector fields satisfy certain characteristic polynomial identities. The real forms $SU_q(N)$ and $SO_q(N,R)$ are discussed in detail.Comment: 16 pages, Latex, no figures, to appear in the Proceedings of the XX International Conference on Group Theory Methods in Physics, Toyonaka, Japan (1994