View source for Multi-valued representation

''of a connected topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m0652501.png" />''

An ordinary representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m0652502.png" /> of a connected topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m0652503.png" /> (cf. [[Representation of a topological group|Representation of a topological group]]) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m0652504.png" /> is isomorphic (as a topological group) to a quotient group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m0652505.png" /> relative to a discrete normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m0652506.png" /> which is not contained in the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m0652507.png" />. A multi-valued representation is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m0652509.png" />-valued if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m06525010.png" /> contains exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m06525011.png" /> elements. By identifying the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m06525012.png" /> with the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m06525013.png" /> one obtains for the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m06525014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m06525015.png" />, the relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m06525016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m06525017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m06525018.png" />. Multi-valued representations of connected, locally path-connected topological groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m06525019.png" /> exist only for non-simply-connected groups. The most important example of a multi-valued representation is the [[Spinor representation|spinor representation]] of the complex orthogonal group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m06525020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m06525021.png" />; this representation is a two-valued representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m06525022.png" /> and is determined by a faithful representation of the universal covering group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065250/m06525023.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Kirillov,   "Elements of the theory of representations" , Springer  (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Naimark,   "Theory of group representations" , Springer  (1982)  (Translated from Russian)</TD></TR></table>