*Loring W. Tu*

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691191751
- eISBN:
- 9780691197487
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691191751.003.0013
- Subject:
- Mathematics, Educational Mathematics

This chapter explores integration on a compact connected Lie group. One of the great advantages of working with a compact Lie group is the possibility of extending the notion of averaging from a ...
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This chapter explores integration on a compact connected Lie group. One of the great advantages of working with a compact Lie group is the possibility of extending the notion of averaging from a finite group to the compact Lie group. If the compact Lie group is connected, then there exists a unique bi-invariant top-degree form with total integral 1, which simplifies the presentation of averaging. The averaging operator is useful for constructing invariant objects. For example, suppose a compact connected Lie group G acts smoothly on the left on a manifold M. Given any C∞ differential k-form ω on M, by averaging all the left translates of ω over G, one can produce a C∞ invariant k-form on M. As another example, on a G-manifold one can average all translates of a Riemannian metric to produce an invariant Riemann metric.Less

This chapter explores integration on a compact connected Lie group. One of the great advantages of working with a compact Lie group is the possibility of extending the notion of averaging from a finite group to the compact Lie group. If the compact Lie group is connected, then there exists a unique bi-invariant top-degree form with total integral 1, which simplifies the presentation of averaging. The averaging operator is useful for constructing invariant objects. For example, suppose a compact connected Lie group *G* acts smoothly on the left on a manifold *M*. Given any *C∞* differential *k*-form ω on *M*, by averaging all the left translates of ω over *G*, one can produce a *C∞* invariant *k*-form on *M*. As another example, on a *G*-manifold one can average all translates of a Riemannian metric to produce an invariant Riemann metric.

*Loring W. Tu*

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691191751
- eISBN:
- 9780691197487
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691191751.003.0008
- Subject:
- Mathematics, Educational Mathematics

This chapter looks at a universal bundle for a compact Lie group. By Milnor's construction, every topological group has a universal bundle. Independently of Milnor's result, the chapter constructs a ...
More

This chapter looks at a universal bundle for a compact Lie group. By Milnor's construction, every topological group has a universal bundle. Independently of Milnor's result, the chapter constructs a universal bundle for any compact Lie group G. This construction is based on the fact that every compact Lie group can be embedded as a subgroup of some orthogonal group O(k). The chapter first constructs a universal O(k)-bundle by finding a weakly contractible space on which O(k) acts freely. The infinite Stiefel variety V (k, ∞) is such a space. As a subgroup of O(k), the compact Lie group G will also act freely on V (k, ∞), thereby producing a universal G-bundle.Less

This chapter looks at a universal bundle for a compact Lie group. By Milnor's construction, every topological group has a universal bundle. Independently of Milnor's result, the chapter constructs a universal bundle for any compact Lie group *G*. This construction is based on the fact that every compact Lie group can be embedded as a subgroup of some orthogonal group O(*k*). The chapter first constructs a universal O(*k*)-bundle by finding a weakly contractible space on which O(*k*) acts freely. The infinite Stiefel variety *V* (*k*, *∞*) is such a space. As a subgroup of O(*k*), the compact Lie group *G* will also act freely on *V* (*k*, *∞*), thereby producing a universal *G*-bundle.

*JEAN ZINN-JUSTIN*

- Published in print:
- 2002
- Published Online:
- January 2010
- ISBN:
- 9780198509233
- eISBN:
- 9780191708732
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509233.003.0013
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter deals with linear continuous symmetries corresponding to compact Lie groups because they imply interesting formal properties; consequences of discrete symmetries can also be studied but ...
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This chapter deals with linear continuous symmetries corresponding to compact Lie groups because they imply interesting formal properties; consequences of discrete symmetries can also be studied but with somewhat different methods. It also deals only with infinitesimal group transformations and, therefore, topological properties of groups will play no role. The chapter proceeds as follows. It first introduces a regularization which preserves the symmetry. It then proves identities — generally called Ward–Takahashi (WT) identities — consequences of the symmetry of the action and satisfied by the generating functional of 1PI correlation functions. These identities imply relations between the divergences of correlation functions and thus between the counter-terms which render the theory finite. From these relations, the generic form of the counter-terms is derived. Such an analysis is based on a loop expansion of perturbation theory. Finally, the chapter reads off the properties of the renormalized action.Less

This chapter deals with linear continuous symmetries corresponding to compact Lie groups because they imply interesting formal properties; consequences of discrete symmetries can also be studied but with somewhat different methods. It also deals only with infinitesimal group transformations and, therefore, topological properties of groups will play no role. The chapter proceeds as follows. It first introduces a regularization which preserves the symmetry. It then proves identities — generally called Ward–Takahashi (WT) identities — consequences of the symmetry of the action and satisfied by the generating functional of 1PI correlation functions. These identities imply relations between the divergences of correlation functions and thus between the counter-terms which render the theory finite. From these relations, the generic form of the counter-terms is derived. Such an analysis is based on a loop expansion of perturbation theory. Finally, the chapter reads off the properties of the renormalized action.