Non-commutative Euclidean structures in compact spaces

Based on results for real deformation parameter q we introduce a compact non- commutative structure covariant under the quantum group SOq(3) for q being a root of unity. To match the algebra of the q-deformed operators with necesarry conjugation properties it is helpful to define a module over the algebra genera- ted by the powers of q. In a representation where X is diagonal we show how P can be calculated. To manifest some typical properties an example of a one-di- mensional q-deformed Heisenberg algebra is also considered and compared with non-compact case.Comment: Changed conten

Hilbert Space Representation of an Algebra of Observables for q-Deformed Relativistic Quantum Mechanics

Using a representation of the q-deformed Lorentz algebra as differential operators on quantum Minkowski space, we define an algebra of observables for a q-deformed relativistic quantum mechanics with spin zero. We construct a Hilbert space representation of this algebra in which the square of the mass $p^2$ is diagonal.Comment: 13 pages, LMU-TPW 94-

The N=1 superstring as a topological field theory

By "untwisting" the construction of Berkovits and Vafa, one can see that the N=1 superstring contains a topological twisted N=2 algebra, with central charge c^ = 2. We discuss to what extent the superstring is actually a topological theory.Comment: 8 Pages (LaTeX). TAUP-2155-9

Differential Calculus on the Quantum Superspace and Deformation of Phase Space

We investigate non-commutative differential calculus on the supersymmetric version of quantum space where the non-commuting super-coordinates consist of bosonic as well as fermionic (Grassmann) coordinates. Multi-parametric quantum deformation of the general linear supergroup, $GL_q(m|n)$, is studied and the explicit form for the ${\hat R}$-matrix, which is the solution of the Yang-Baxter equation, is presented. We derive the quantum-matrix commutation relation of $GL_q(m|n)$ and the quantum superdeterminant. We apply these results for the $GL_q(m|n)$ to the deformed phase-space of supercoordinates and their momenta, from which we construct the ${\hat R}$-matrix of q-deformed orthosymplectic group $OSp_q(2n|2m)$ and calculate its ${\hat R}$-matrix. Some detailed argument for quantum super-Clifford algebras and the explict expression of the ${\hat R}$-matrix will be presented for the case of $OSp_q(2|2)$.Comment: 17 pages, KUCP-4

On the Decoupling of the Homogeneous and Inhomogeneous Parts in Inhomogeneous Quantum Groups

We show that, if there exists a realization of a Hopf algebra $H$ in a $H$-module algebra $A$, then one can split their cross-product into the tensor product algebra of $A$ itself with a subalgebra isomorphic to $H$ and commuting with $A$. This result applies in particular to the algebra underlying inhomogeneous quantum groups like the Euclidean ones, which are obtained as cross-products of the quantum Euclidean spaces $R_q^N$ with the quantum groups of rotation $U_qso(N)$ of $R_q^N$, for which it has no classical analog.Comment: Latex file, 27 pages. Final version to appear in J. Phys.

Braided Hopf Algebras and 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.Comment: 8 page

More on quantum groups from the the quantization point of view

Star products on the classical double group of a simple Lie group and on corresponding symplectic grupoids are given so that the quantum double and the "quantized tangent bundle" are obtained in the deformation description. "Complex" quantum groups and bicovariant quantum Lie algebras are discused from this point of view. Further we discuss the quantization of the Poisson structure on symmetric algebra $S(g)$ leading to the quantized enveloping algebra $U_{h}(g)$ as an example of biquantization in the sense of Turaev. Description of $U_{h}(g)$ in terms of the generators of the bicovariant differential calculus on $F(G_q)$ is very convenient for this purpose. Finally we interpret in the deformation framework some well known properties of compact quantum groups as simple consequences of corresponding properties of classical compact Lie groups. An analogue of the classical Kirillov's universal character formula is given for the unitary irreducible representation in the compact case.Comment: 18 page

Noncommutative Chiral Anomaly and the Dirac-Ginsparg-Wilson Operator

It is shown that the local axial anomaly in $2-$dimensions emerges naturally if one postulates an underlying noncommutative fuzzy structure of spacetime . In particular the Dirac-Ginsparg-Wilson relation on ${\bf S}^2_F$ is shown to contain an edge effect which corresponds precisely to the ``fuzzy'' $U(1)_A$ axial anomaly on the fuzzy sphere . We also derive a novel gauge-covariant expansion of the quark propagator in the form $\frac{1}{{\cal D}_{AF}}=\frac{a\hat{\Gamma}^L}{2}+\frac{1}{{\cal D}_{Aa}}$ where $a=\frac{2}{2l+1}$ is the lattice spacing on ${\bf S}^2_F$, $\hat{\Gamma}^L$ is the covariant noncommutative chirality and ${\cal D}_{Aa}$ is an effective Dirac operator which has essentially the same IR spectrum as ${\cal D}_{AF}$ but differes from it on the UV modes. Most remarkably is the fact that both operators share the same limit and thus the above covariant expansion is not available in the continuum theory . The first bit in this expansion $\frac{a\hat{\Gamma}^L}{2}$ although it vanishes as it stands in the continuum limit, its contribution to the anomaly is exactly the canonical theta term. The contribution of the propagator $\frac{1}{{\cal D}_{Aa}}$ is on the other hand equal to the toplogical Chern-Simons action which in two dimensions vanishes identically .Comment: 26 pages, latex fil

Metric Properties of the Fuzzy Sphere

The fuzzy sphere, as a quantum metric space, carries a sequence of metrics which we describe in detail. We show that the Bloch coherent states, with these spectral distances, form a sequence of metric spaces that converge to the round sphere in the high-spin limit.Comment: Slightly shortened version, no major changes, two new references, version to appear on Letters in Mathematical Physic

The Fuzzy Ginsparg-Wilson Algebra: A Solution of the Fermion Doubling Problem

The Ginsparg-Wilson algebra is the algebra underlying the Ginsparg-Wilson solution of the fermion doubling problem in lattice gauge theory. The Dirac operator of the fuzzy sphere is not afflicted with this problem. Previously we have indicated that there is a Ginsparg-Wilson operator underlying it as well in the absence of gauge fields and instantons. Here we develop this observation systematically and establish a Dirac operator theory for the fuzzy sphere with or without gauge fields, and always with the Ginsparg-Wilson algebra. There is no fermion doubling in this theory. The association of the Ginsparg-Wilson algebra with the fuzzy sphere is surprising as the latter is not designed with this algebra in mind. The theory reproduces the integrated U(1)_A anomaly and index theory correctly.Comment: references added, typos corrected, section 4.2 simplified. Report.no: SU-4252-769, DFUP-02-1