• Journal of the Korean Mathematical Society
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• v.49 no.2
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• pp.343-356
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• 2012

In this paper, we consider the canonical cusps in complex hyperbolic surfaces. We will classify canonical cusps in complex hyper-bolic surfaces and find correspondence between them and 3-dimensiona nilpotent groups. This paper is a sequel of our paper [6].

• Journal of the Korean Mathematical Society
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• v.49 no.4
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• pp.733-743
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• 2012

For the "Hopf bundle" $S^1{\rightarrow}S^{2n,1}{\rightarrow}\mathbb{C}H^n$, horizontal lifts of simple closed curves are studied. Let ${\gamma}$ be a piecewise smooth, simple closed curve on a complete totally geodesic surface $S$ in the base space. Then the holonomy displacement along ${\gamma}$ is given by $$V({\gamma})=e^{{\lambda}A({\gamma})i}$$ where $A({\gamma})$ is the area of the region on the surface $S$ surrounded by ${\gamma}$; ${\lambda}=1/2$ or 0 depending on whether $S$ is a complex submanifold or not. We also carry out a similar investigation for the complex Heisenberg group $\mathbb{R}{\rightarrow}\mathcal{H}^{2n+1}{\rightarrow}\mathbb{C}^n$.

• Kyungpook Mathematical Journal
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• v.59 no.3
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• pp.403-414
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• 2019

Theta functions are the sections of line bundles on a complex torus. Noncommutative versions of theta functions have appeared as theta vectors and quantum theta operators. In this paper we describe a super version of theta vectors and quantum theta operators. This is the natural unification of Manin's result on bosonic operators, and the author's previous result on fermionic operators.