This is a readable review of recent work on non-Hermitian bound state problems with complex potentials. A particular example is the generalization of the harmonic oscillator with the potentials: $$V\left(x\right)=\frac{{\omega }^2}{2}{\left(x-\frac{2i\beta }{\omega }\right)}^2-\frac{\omega }{2}.$$ Other examples include complex generalizations of the Morse potential, the spiked radial harmonic potential, the Kratzer-Coulomb potential, the Rosen Morse oscillator and others. Instead of demanding Hermiticity $H=H^{*}$ the condition required is $H=PTHPT$ where $P$ changes the parity and $T$ transforms $i$ to $-i$.

The author constructs the gauged Skyrme model by introducing the skyrmion bundle as follows: instead of considering maps $U:M\to {\text{SU}}_{N_F}$ he thinks of the meson fields as of global sections in a bundle $B(M,{\text{SU}}_{N_F},G)=P(M,G){\times }_G{\text{SU}}_{N_F}$. For calculations within the skyrmion bundle the author introduces by means of the so-called equivariant cohomology an analogue of the topological charge and the Wess-Zumino term. The final result of this paper is the following Theorem. For the skyrmion bundle with $N_F\le 6$, one has $$H^{*}\left(EG{\times }_G{\text{SU}}_{N_F}\right)\cong H^{*}{\left({\text{SU}}_{N_F}\right)}^G\cong \text{S}\left({\underline{G}}^{*}\right)\otimes H^{*}\left({\text{SU}}_{N_F}\right)\cong H^{*}\left(BG\right)\otimes H^{*}\left({\text{SU}}_{N_F}\right),$$ where $EG(BG,G)$ is the universal bundle...

This paper contains the lectures given by the author at the Winter School on “Geometry and Physics” in Srní 2001. These lectures are based on two recent works of the author with A. Korányi and on a forthcoming paper with K. Johnson and A. Korányi. In the paper results are presented concerning equivariant differential operators on homogeneous spaces (section 1), first order equivariant differential operators on boundaries of symmetric spaces (section 2), the Poisson transform (section 3) and complex...

Summary: The author gives the defining relations of a new type of bialgebras that generalize both the quantum groups and braided groups as well as the quantum supergroups. The relations of the algebras are determined by a pair of matrices $(R,Z)$ that solve a system of Yang-Baxter-type equations. The matrix coproduct and counit are of standard matrix form, however, the multiplication in the tensor product of the algebra is defined by virtue of the braiding map given by the matrix $Z$. Besides simple solutions...

Summary: We give an introduction to the Skyrme model from a mathematical point of view. Hereby, we show that it is difficult to solve the field equation even by means of the classical ansatz, the so-called hedgehog ansatz. Our main result is an extended existence proof for solutions of the field equation in the hedgehog ansatz.

A homogeneous Riemannian manifold $M=G/H$ is called a “g.o. space” if every geodesic on $M$ arises as an orbit of a one-parameter subgroup of $G$. Let $M=G/H$ be such a “g.o. space”, and $m$ an $\text{Ad}\left(H\right)$-invariant vector subspace of $\text{Lie}\left(G\right)$ such that $\text{Lie}\left(G\right)=m\oplus \text{Lie}\left(H\right)$. A geodesic graph is a map $\xi :m\to \text{Lie}\left(H\right)$ such that $$t\mapsto exp\left(t\right(X+\xi \left(X\right)\left)\right)\left(eH\right)$$ is a geodesic for every $X\in m\setminus \left\{0\right\}$. The author calculates explicitly such geodesic graphs for certain special 2-step nilpotent Lie groups. More precisely, he deals with “generalized Heisenberg groups” (also known as “H-type groups”) whose center has...