*Paula Tretkoff*

- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691144771
- eISBN:
- 9781400881253
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691144771.003.0007
- Subject:
- Mathematics, Geometry / Topology

This chapter justifies the assumption that ball quotients covering line arrangements exist. It begins with the general case on the existence of finite covers by ball quotients of weighted ...
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This chapter justifies the assumption that ball quotients covering line arrangements exist. It begins with the general case on the existence of finite covers by ball quotients of weighted configurations, focusing on log-canonical divisors and Euler numbers reflecting the weight data on divisors on the blow-up X of P2 at the singular points of a line arrangement. It then uses the Kähler-Einstein property to prove an inequality between Chern forms that, when integrated, gives the appropriate Miyaoka-Yau inequality. It also discusses orbifolds and b-spaces, weighted line arrangements, the problem of the existence of ball quotient finite coverings, log-terminal singularity and log-canonical singularity, and the proof of the main existence theorem for line arrangements. Finally, it considers the isotropy subgroups of the covering group.Less

This chapter justifies the assumption that ball quotients covering line arrangements exist. It begins with the general case on the existence of finite covers by ball quotients of weighted configurations, focusing on log-canonical divisors and Euler numbers reflecting the weight data on divisors on the blow-up *X* of P2 at the singular points of a line arrangement. It then uses the Kähler-Einstein property to prove an inequality between Chern forms that, when integrated, gives the appropriate Miyaoka-Yau inequality. It also discusses orbifolds and *b*-spaces, weighted line arrangements, the problem of the existence of ball quotient finite coverings, log-terminal singularity and log-canonical singularity, and the proof of the main existence theorem for line arrangements. Finally, it considers the isotropy subgroups of the covering group.

*Adam M. Bincer*

- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199662920
- eISBN:
- 9780191745492
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199662920.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

SO(3) — the group of rotations in three dimensions, whose elements are 3x3 orthogonal unimodular real matrices — is introduced and SU(2) is introduced as the universal covering ...
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SO(3) — the group of rotations in three dimensions, whose elements are 3x3 orthogonal unimodular real matrices — is introduced and SU(2) is introduced as the universal covering group of SO(3). This chapter starts with a discussion of rotations in two dimensions, which should be familiar to everybody. Next, it is shown that the determinant of real orthogonal nxn matrices equals +1 or –1 and therefore the O(n) group consists of two disconnected pieces: rotations, for which the determinant equals +1 and reflections, for which the determinant equals –1. Since rotations are generated by angular momentum I identify the infinitesimal generators of SO(n) with the Cartesian components of angular momentum in appropriate units. Explicit evaluation of the structure constants shows the so(3) and su(2) algebras are isomorphic. The corresponding groups are shown to be homomorphic, SO(3) being doubly connected, while SU(2) is the simply connected covering group.Less

*SO*(3) — the group of rotations in three dimensions, whose elements are 3x3 orthogonal unimodular real matrices — is introduced and *SU*(2) is introduced as the universal covering group of *SO*(3). This chapter starts with a discussion of rotations in two dimensions, which should be familiar to everybody. Next, it is shown that the determinant of real orthogonal *n*x*n* matrices equals +1 or –1 and therefore the *O*(*n*) group consists of two disconnected pieces: rotations, for which the determinant equals +1 and reflections, for which the determinant equals –1. Since rotations are generated by angular momentum I identify the infinitesimal generators of *SO*(*n*) with the Cartesian components of angular momentum in appropriate units. Explicit evaluation of the structure constants shows the *so*(3) and *su*(2) algebras are isomorphic. The corresponding groups are shown to be homomorphic, *SO*(3) being doubly connected, while *SU*(2) is the simply connected covering group.