# Representation of a group

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A homomorphism of the group into the group of all invertible transformations of a set . A representation of a group is called linear if is a vector space over a field and if the transformations , , are linear. Often, linear representations are, for shortness, simply termed representations (cf. Representation theory). In the theory of representations of abstract groups the theory of finite-dimensional representations of finite groups is best developed (cf. Finite group, representation of a; Representation of the symmetric groups).

If is a topological group, then one considers continuous linear representations of on a topological vector space (cf. Continuous representation; Representation of a topological group). If is a Lie group and is a finite-dimensional space over or , then a continuous linear representation is automatically real analytic. Analytic and differentiable representations of a Lie group are defined also in the infinite-dimensional case (cf. Analytic representation; Infinite-dimensional representation). To each differentiable representation of a Lie group corresponds some linear representation of its Lie algebra — the differential representation of (cf. Representation of a Lie algebra). If is moreover connected, then its finite-dimensional representations are completely determined by their differentials. The most developed branch of the representation theory of topological groups is the theory of finite-dimensional linear representations of semi-simple Lie groups, which is often formulated in the language of Lie algebras (cf. Finite-dimensional representation; Representation of the classical groups; Cartan theorem on the highest weight vector), the representation theory of compact groups, and the theory of unitary representations (cf. Representation of a compact group; Unitary representation).

For algebraic groups one has the theory of rational representations (cf. Rational representation), which is in many aspects analogous to the theory of finite-dimensional representations of Lie groups.

#### References

 [1] D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) [2] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) [3] M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) [4] D.P. Zhelobenko, A.I. Shtern, "Representations of Lie groups" , Moscow (1981) (In Russian)