Two Sorts of Natural Theology

Usually, natural theology is understood as the project of providing arguments for the existence of God. This project is endorsed by Moreland and Craig. McGrath, on the other hand, says that this project fails. In the first part of this article, I show how McGrath’s dismissal of arguments for the existence of God follows from his view of natural theology. In the second part, I argue that McGrath’s natural theology contains an accurate critique of Moreland and Craig’s way of doing natural theology, a critique that exposes two major problems in their treatment of the moral argument for the existence of God. In the third part, I propose a way of providing arguments for the existence of God that avoids the problems pointed out by McGrath, namely a way of arguing that seek to show how theology may improve a certain non-theistic understanding of a natural phenomenon

The center of Uq(nω){\mathcal U}_q({\mathfrak n}_\omega)

We determine the center of a localization of ${\mathcal U}_q({\mathfrak n}_\omega)\subseteq {\mathcal U}^+_q({\mathfrak g})$ by the covariant elements (non-mutable elements) by means of constructions and results from quantum cluster algebras. In our set-up, ${\mathfrak g}$ is any finite-dimensional complex Lie algebra and $\omega$ is any element in the Weyl group $W$. The non-zero complex parameter $q$ is mostly assumed not to be a root of unity, but our method also gives many details in case $q$ is a primitive root of unity. We point to a new and very useful direction of approach to a general set of problems which we exemplify here by obtaining the result that the center is determined by the null space of $1+\omega$. Further, we use this to give a generalization to double Schubert Cell algebras where the center is proved to be given by $\omega^{\mathfrak a}+\omega^{\mathfrak c}$. Another family of quadratic algebras is also considered and the centers determined.Comment: 28 pages LaTeX. Relevant references as well as a new section relating to the root-of-unity case have been added. Now in print with minor change

Special classes of homomorphisms between generalized Verma modules for Uq(su(n,n)){\mathcal U}_q(su(n,n))

We study homomorphisms between quantized generalized Verma modules $M(V_{\Lambda})\stackrel{\phi_{\Lambda,\Lambda_1}}{\rightarrow}M(V_{\Lambda_1})$ for ${\mathcal U}_q(su(n,n))$. There is a natural notion of degree for such maps, and if the map is of degree $k$, we write $\phi^k_{\Lambda,\Lambda_1}$. We examine when one can have a series of such homomorphisms $\phi^1_{\Lambda_{n-1},\Lambda_{n}} \circ \phi^1_{\Lambda_{n-2}, \Lambda_{n-1}} \circ\cdots\circ \phi^1_{\Lambda,\Lambda_1} = \textrm{Det}_q$, where $\textrm{Det}_q$ denotes the map $M(V_{\Lambda})\ni p\rightarrow \textrm{Det}_q\cdot p\in M(V_{\Lambda_n})$. If, classically, $su(n,n)^{\mathbb C}={\mathfrak p}^-\oplus(su(n)\oplus su(n)\oplus {\mathbb C})\oplus {\mathfrak p}^+$, then $\Lambda = (\Lambda_L,\Lambda_R,\lambda)$ and $\Lambda_n =(\Lambda_L,\Lambda_R,\lambda+2)$. The answer is then that $\Lambda$ must be one-sided in the sense that either $\Lambda_L=0$ or $\Lambda_R=0$ (non-exclusively). There are further demands on $\lambda$ if we insist on ${\mathcal U}_q({\mathfrak g}^{\mathbb C})$ homomorphisms. However, it is also interesting to loosen this to considering only ${\mathcal U}^-_q({\mathfrak g}^{\mathbb C})$ homomorphisms, in which case the conditions on $\lambda$ disappear. By duality, there result have implications on covariant quantized differential operators. We finish by giving an explicit, though sketched, determination of the full set of ${\mathcal U}_q({\mathfrak g}^{\mathbb C})$ homomorphisms $\phi^1_{\Lambda,\Lambda_1}$.Comment: 10 pages proceedings of Group 32, Prague 201

Quantized Dirac Operators

We determine what should correspond to the Dirac operator on certain quantized hermitian symmetric spaces and what its properties are. A new insight into the quantized wave operator is obtained.Comment: To appear in the Proceedings of the Quantum Groups And Integrable Systems meeting in Prag, June 22-24 2000. To be published with the Czechoslovak Journal of Physi