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A linear [[Representation of a topological group|representation of a topological group]] in a finite-dimensional vector space. The theory of finite-dimensional representations is one of the most important and most developed parts of the representation theory of groups. An irreducible finite-dimensional representation is completely irreducible (see [[Schur lemma|Schur lemma]]), but an operator-irreducible finite-dimensional representation can be reducible. A measurable finite-dimensional representation of a locally compact group locally coincides almost-everywhere with a continuous finite-dimensional representation. A bounded finite-dimensional representation of a locally compact group is equivalent to a unitary representation. A locally compact group having a faithful finite-dimensional representation is a Lie group [[#References|[7]]].

A unitary finite-dimensional representation is a direct sum of irreducible unitary finite-dimensional representations. The intersection of the kernels of the continuous homomorphisms of a topological group  $  G $
coincides with the intersection of the kernels of the irreducible unitary finite-dimensional representations of  $  G $; 
if this set contains only the identity of  $  G $, 
then there is a continuous monomorphism from  $  G $
into some compact group, and  $  G $
is said to be imbeddable in a compact group, or to be a maximally almost-periodic group (a MAP-group). If  $  G $
is a MAP-group, then the family of matrix entries of irreducible finite-dimensional representations of  $  G $
separates points in  $  G $. 
Commutative groups and compact groups are MAP-groups; a connected locally compact group is a MAP-group if and only if it is the direct product of a connected compact group and  $  \mathbf R  ^ {n} $(
see [[#References|[5]]]). A MAP-group can have infinite-dimensional irreducible unitary representations and need not be a group of type 1. Every continuous irreducible unitary representation of a locally compact group  $  G $
is finite-dimensional if and only if  $  G $
is a projective limit of finite extensions of groups  $  H $
of the form  $  ( K \cdot D) \times V $, 
where  $  K $, 
$  D $
and  $  V $
are closed subgroups of  $  H $
such that  $  V $
is isomorphic to  $  \mathbf R  ^ {n} $, 
$  K $
is compact and  $  D $
is a discrete group that is central in  $  H $[[#References|[8]]]; a sufficient condition is that the quotient group of  $  G $
by its centre be compact. Moreover, for many locally compact groups (in particular, for non-compact simple Lie groups), the only irreducible unitary finite-dimensional representation is the [[trivial representation]].

Non-unitary finite-dimensional representations of topological groups have been classified (up to equivalence) only for special groups; in particular, for  $  \mathbf R $
and  $  \mathbf Z $, 
where the problem of describing finite-dimensional representations is solved by reducing matrices to Jordan form, and — in the case of  $  \mathbf R $— 
is related to the theory of ordinary differential equations with constant coefficients, in that the solution space of such an equation is a finite-dimensional invariant subspace of the regular representation of  $  \mathbf R $
in the space of continuous functions on  $  \mathbf R $. 
In addition, the finite-dimensional representations of connected semi-simple Lie groups are known. More precisely, these are direct sums of irreducible finite-dimensional representations that can be described as follows. If  $  G $
is a semi-simple complex Lie group and  $  U $
is a maximal compact subgroup, then every continuous irreducible unitary representation  $  \pi $
of  $  U $
in a space  $  E $
can be extended: 1) to an irreducible representation  $  \pi  ^ {G} $
of  $  G $
in  $  E $
whose matrix entries are analytic functions on  $  G $; 
and 2) to an irreducible representation  $  \overline \pi \; {}  ^ {G} $
of  $  G $
whose matrix entries are complex conjugates of analytic functions on  $  G $; 
$  \pi  ^ {G} $
and  $  \overline \pi \; {}  ^ {G} $
are determined uniquely by  $  \pi $. 
The tensor product  $  \pi _ {1}  ^ {G} \otimes \overline \pi \; {} _ {2}  ^ {G} $
is an irreducible finite-dimensional representation of  $  G $
for arbitrary irreducible unitary finite-dimensional representations  $  \pi _ {1} $
and  $  \pi _ {2} $
of  $  U $, 
and every irreducible finite-dimensional representation of  $  G $
is equivalent to a representation of the form  $  \pi _ {1}  ^ {G} \otimes \overline \pi \; {} _ {2}  ^ {G} $. 
A description of the finite-dimensional representations of a simply-connected connected complex semi-simple Lie group can also be given in terms of the exponentials of finite-dimensional representations of its Lie algebra, and also by using the [[Gauss decomposition|Gauss decomposition]]  $  G \supset Z _ {-} D Z _ {+} $
of  $  G $: 
Let  $  \alpha $
be a continuous function on  $  G $
such that  $  \alpha ( z _ {-} \delta z _ {+} ) = \alpha ( \delta ) $
for all  $  z _ {-} \in Z _ {-} $, 
$  \delta \in D $, 
$  z _ {+} \in Z _ {+} $, 
and suppose that the linear hull  $  \Phi _  \alpha  $
of the functions  $  g \rightarrow \alpha ( gg _ {0} ) $, 
$  g _ {0} \in G $, 
is finite-dimensional; then the formula  $  [ T _  \alpha  ( g _ {0} ) f] ( g) = f ( gg _ {0} ) $, 
$  g, g _ {0} \in G $, 
$  f \in \Phi _  \alpha  $, 
defines an irreducible finite-dimensional representation of  $  G $, 
and all irreducible finite-dimensional representations of  $  G $
can be obtained in this way. If  $  G $
is a real semi-simple Lie group having complex form  $  G ^ {\mathbf C } $, 
then every irreducible finite-dimensional representation of  $  G $
is the restriction to  $  G $
of some unique irreducible finite-dimensional representation of  $  G ^ {\mathbf C } $
whose matrix entries are analytic on  $  G ^ {\mathbf C } $(
so that the theory of finite-dimensional representations of semi-simple connected Lie groups reduces essentially to that of compact Lie groups). A finite-dimensional representation of an arbitrary Lie group is real analytic; if  $  G $
is a simply-connected Lie group, then there is a one-to-one correspondence between the finite-dimensional representations of  $  G $
and of its Lie algebras (associating to a representation of the Lie group its differential; the inverse mapping associates to a representation of the Lie algebra its exponential). However, the classification of finite-dimensional representations of arbitrary Lie groups is far from being completely solved (1988), even for the special cases of  $  \mathbf R  ^ {n} $, 
$  n \geq  2 $, 
and  $  \mathbf Z  ^ {n} $, 
$  n \geq  2 $(
see [[#References|[6]]]). On the other hand, the irreducible finite-dimensional representations  $  \pi $
of a connected Lie group  $  G $
are known [[#References|[2]]]: They have the form  $  \pi = \chi \otimes \pi _ {0} $, 
where  $  \chi $
is a one-dimensional representation of  $  G $(
that is, essentially of its commutative quotient group by the commutator subgroup), and  $  \pi _ {0} $
is a finite-dimensional representation of the semi-simple quotient group of  $  G $
by the maximal connected solvable normal subgroup of  $  G $(
see [[Levi–Mal'tsev decomposition|Levi–Mal'tsev decomposition]]).

====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">  D.P. Zhelobenko,   "Compact Lie groups and representations" , Amer. Math. Soc.  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.A. Naimark,   "Theory of group representations" , Springer  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J. Dixmier,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040250/f04025087.png" /> algebras" , North-Holland  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Weil,   "l'Intégration dans les groupes topologiques et ses applications" , Hermann  (1940)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  I.M. Gel'fand,   V.A. Ponomarev,   "Remarks on the classification of a pair of commuting linear transformations in a finite-dimensional space"  ''Funct. Anal. Appl.'' , '''3''' :  4  (1969)  pp. 325–326  ''Funktsional. Anal. i Prilozhen.'' , '''3''' :  4  (1969)  pp. 81–82</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.M. Glushkov,   "The structure of locally compact groups and Hilbert's fifth problem"  ''Transl. Amer. Math. Soc.'' , '''15'''  (1960)  pp. 55–93  ''Uspekhi Mat. Nauk'' , '''12''' :  2  (1957)  pp. 3–41</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A.I. Shtern,   "Locally bicompact groups with finite-dimensional irreducible representations"  ''Math. USSR Sb.'' , '''19''' :  1  (1973)  pp. 85–94  ''Mat. Sb.'' , '''90''' :  1  (1973)  pp. 86–95</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.E. Humphreys,   "Introduction to Lie algebras and representation theory" , Springer  (1972)</TD></TR></table>