In mathematics, the unitarian trick (or unitary trick) is a device in the representation theory of Lie groups, introduced by Adolf Hurwitz (1897) for the special linear group and by Hermann Weyl for general semisimple groups. It applies to show that the representation theory of some complex Lie group G is in a qualitative way controlled by that of some compact real Lie group K, and the latter representation theory is easier. An important example is that in which G is the complex general linear group GLn(C), and K the unitary group U(n) acting on vectors of the same size. From the fact that the representations of K are completely reducible, the same is concluded for the complex-analytic representations of G, at least in finite dimensions.
The relationship between G and K that drives this connection is traditionally expressed in the terms that the Lie algebra of K is a real form of that of G. In the theory of algebraic groups, the relationship can also be put that K is a dense subset of G, for the Zariski topology.
The trick works for reductive Lie groups G, of which an important case are semisimple Lie groups.
In mathematics, the unitariantrick (or unitary trick) is a device in the representation theory of Lie groups, introduced by Adolf Hurwitz (1897) for...
irreducible representations. This line of reasoning is called the unitariantrick and was Weyl's original argument for what is now called Weyl's theorem...
subgroups. By contrast, there is a trick, also called Weyl's unitariantrick, but unrelated to the unitariantrick of above showing that all finite-dimensional...
reductive groups can be reduced to the case of compact groups using Weyl's unitariantrick. Given the Reynolds operator, Hilbert's theorem is proved as follows...
algebras depend only on the fact that they are reductive. Notably, the unitariantrick of Hermann Weyl works for reductive Lie algebras. The associated reductive...
compact group theory on non-compact groups was formulated by Weyl in his unitariantrick. Inside a general semisimple Lie group there is a maximal compact subgroup...
Degree 11 (Noun) A form of degree 11 unimodular Having determinant 1 unitariantrick Finite-dimensional representations of a semisimple Lie group are equivalent...
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