# Multi-valued representation

*of a connected topological group *

An ordinary representation of a connected topological group (cf. Representation of a topological group) such that is isomorphic (as a topological group) to a quotient group of relative to a discrete normal subgroup which is not contained in the kernel of . A multi-valued representation is called -valued if contains exactly elements. By identifying the elements of with the elements of one obtains for the sets , , the relations , , . Multi-valued representations of connected, locally path-connected topological groups exist only for non-simply-connected groups. The most important example of a multi-valued representation is the spinor representation of the complex orthogonal group , ; this representation is a two-valued representation of and is determined by a faithful representation of the universal covering group of .

#### References

[1] | A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) |

[2] | M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) |

**How to Cite This Entry:**

Multi-valued representation. A.I. Shtern (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Multi-valued_representation&oldid=12923