Unitary representation

In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological group and the representations are strongly continuous. In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological group and the representations are strongly continuous. The theory has been widely applied in quantum mechanics since the 1920s, particularly influenced by Hermann Weyl's 1928 book Gruppentheorie und Quantenmechanik. One of the pioneers in constructing a general theory of unitary representations, for any group G rather than just for particular groups useful in applications, was George Mackey. The theory of unitary representations of groups is closely connected with harmonic analysis. In the case of an abelian group G, a fairly complete picture of the representation theory of G is given by Pontryagin duality. In general, the unitary equivalence classes (see below) of irreducible unitary representations of G make up its unitary dual. This set can be identified with the spectrum of the C*-algebra associated to G by the group C*-algebra construction. This is a topological space. The general form of the Plancherel theorem tries to describe the regular representation of G on L2(G) by means of a measure on the unitary dual. For G abelian this is given by the Pontryagin duality theory. For G compact, this is done by the Peter–Weyl theorem; in that case the unitary dual is a discrete space, and the measure attaches an atom to each point of mass equal to its degree. Let G be a topological group. A strongly continuous unitary representation of G on a Hilbert space H is a group homomorphism from G into the unitary group of H, such that g → π(g) ξ is a norm continuous function for every ξ ∈ H. Note that if G is a Lie group, the Hilbert space also admits underlying smooth and analytic structures. A vector ξ in H is said to be smooth or analytic if the map g → π(g) ξ is smooth or analytic (in the norm or weak topologies on H). Smooth vectors are dense in H by a classical argument of Lars Gårding, since convolution by smooth functions of compact support yields smooth vectors. Analytic vectors are dense by a classical argument of Edward Nelson, amplified by Roe Goodman, since vectors in the image of a heat operator e–tD, corresponding to an elliptic differential operator D in the universal enveloping algebra of G, are analytic. Not only do smooth or analytic vectors form dense subspaces; they also form common cores for the unbounded skew-adjoint operators corresponding to the elements of the Lie algebra, in the sense of spectral theory. Two unitary representations π1: G → U(H1), π2: G → U(H2) are said to be unitarily equivalent if there is a unitary transformation A:H1 → H2 such that π1(g) = A* ∘ π2(g) ∘ A for all g in G. When this holds, A is said to be an intertwining operator for the representations (π1,H1), (π2,H2). If π {displaystyle pi } is a representation of a connected Lie group G {displaystyle G} on a finite-dimensional Hilbert space H {displaystyle H} , then π {displaystyle pi } is unitary if and only if the associated Lie algebra representation d π : g → E n d ( H ) {displaystyle dpi :{mathfrak {g}} ightarrow mathrm {End} (H)} maps into the space of skew-self-adjoint operators on H {displaystyle H} .