Multi-valued representation

of a connected topological group $ G $

An ordinary representation $ \pi $ of a connected topological group $ G _ {1} $( cf. Representation of a topological group) such that $ G $ is isomorphic (as a topological group) to a quotient group of $ G _ {1} $ relative to a discrete normal subgroup $ N $ which is not contained in the kernel of $ \pi $. A multi-valued representation is called $ n $- valued if $ \pi ( N) $ contains exactly $ n $ elements. By identifying the elements of $ G $ with the elements of $ G _ {1} / N $ one obtains for the sets $ \pi ( g) $, $ g \in G = G _ {1} / N $, the relations $ \pi ( e) \ni 1 $, $ \pi ( g _ {1} g _ {2} ) = \pi ( g _ {1} ) \pi ( g _ {2} ) $, $ g _ {1} , g _ {2} \in G $. Multi-valued representations of connected, locally path-connected topological groups $ G $ 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 $ \mathop{\rm SO} ( n , \mathbf C ) $, $ n \geq 2 $; this representation is a two-valued representation of $ \mathop{\rm SO} ( n , \mathbf C ) $ and is determined by a faithful representation of the universal covering group of $ \mathop{\rm SO} ( n , \mathbf C ) $.

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)