Editing Unitarian trick

{{Short description|Device in the representation theory of Lie groups}}
In [[mathematics]], the '''unitarian trick''' (or '''unitary trick''') is a device in the [[representation theory]] of [[Lie group]]s, introduced by {{harvs|txt|authorlink=Adolf Hurwitz|first=Adolf|last= Hurwitz|year=1897}} for the [[special linear group]] and by [[Hermann Weyl]] for general [[Semisimple Lie algebra#Semisimple and reductive groups|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 group|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]] GL<sub>''n''</sub>('''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 group]]s, the relationship can also be put that ''K'' is a [[dense subset]] of ''G'', for the [[Zariski topology]].

The trick works for [[reductive Lie group]]s ''G'', of which an important case are [[semisimple Lie group]]s.

==Formulations==
The "trick" is stated in a number of ways in contemporary mathematics. One such formulation is for ''G'' a reductive group over the complex numbers. Then ''G''<sub>an</sub>, the complex points of ''G'' considered as a Lie group, has a [[compact subgroup]] ''K'' that is Zariski-dense.<ref>{{cite book |last1=Parshin |first1=A. N. |last2=Shafarevich |first2=I. R. |title=Algebraic Geometry IV: Linear Algebraic Groups Invariant Theory |date=6 December 2012 |publisher=Springer Science & Business Media |isbn=978-3-662-03073-8 |page=92 |url=https://books.google.com/books?id=80LtCAAAQBAJ&pg=PA92 |language=en}}</ref> For the case of the special linear group, this result was proved for its special unitary subgroup by [[Issai Schur]] (1924, presaged by earlier work).<ref>{{cite book |last1=Hawkins |first1=Thomas |title=Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics 1869–1926 |date=6 December 2012 |publisher=Springer Science & Business Media |isbn=978-1-4612-1202-7 |page=415 |url=https://books.google.com/books?id=lz3aBwAAQBAJ&pg=PA415 |language=en}}</ref> The special linear group is a complex semisimple Lie group. For any such group ''G'' and maximal compact subgroup ''K'', and ''V'' a complex vector space of finite dimension which is a ''G''-module, its ''G''-submodules and ''K''-submodules are the same.<ref>{{cite book |last1=Santos |first1=Walter Ferrer |last2=Rittatore |first2=Alvaro |title=Actions and Invariants of Algebraic Groups |date=26 April 2005 |publisher=CRC Press |isbn=978-1-4200-3079-2 |page=304 |url=https://books.google.com/books?id=zf_9RdmJFwEC&pg=PA304 |language=en}}</ref>

In the ''[[Encyclopedia of Mathematics]]'', the formulation is

<blockquote>The classical compact Lie groups ... have the same complex linear representations and the same invariant subspaces in tensor spaces as their complex envelopes [...]. Therefore, results of the theory of linear representations obtained for the classical complex Lie groups can be carried over to the corresponding compact groups and vice versa.<ref>{{SpringerEOM |title=Representation of the classical groups  |author-last1= Vinberg |author-first1=E. B. |author-link1=Ernest Vinberg |id=Representation_of_the_classical_groups}}</ref></blockquote>

In terms of [[Tannakian formalism]], [[Claude Chevalley]] interpreted [[Tannaka duality]] starting from a compact Lie group ''K'' as constructing the "complex envelope" ''G'' as the dual reductive algebraic group ''Tn(K)'' over the complex numbers.<ref>{{cite book |last1=Hitchin |first1=Nigel J. |title=The Many Facets of Geometry: A Tribute to Nigel Hitchin |date=July 2010 |publisher=Oxford University Press |isbn=978-0-19-953492-0 |pages=97–98 |url=https://books.google.com/books?id=AwIUDAAAQBAJ&pg=PA97 |language=en}}</ref> [[Veeravalli S. Varadarajan]] wrote of the "unitarian trick" as "the canonical correspondence between compact and complex semisimple complex groups discovered by Weyl", noting the "closely related duality theories of Chevalley and Tannaka", and later developments that followed on [[quantum group]]s.<ref>{{cite book |last1=Doran |first1=Robert S. |title=The Mathematical Legacy of Harish-Chandra: A Celebration of Representation Theory and Harmonic Analysis : an AMS Special Session Honoring the Memory of Harish-Chandra, January 9-10, 1998, Baltimore, Maryland |date=2000 |publisher=American Mathematical Soc. |isbn=978-0-8218-1197-9 |page=3 |url=https://books.google.com/books?id=t8ACCAAAQBAJ&pg=PA3 |language=en}}</ref>

==History==
[[Adolf Hurwitz]] had shown how integration over a [[compact Lie group]] could be used to construct invariants, in the cases of unitary groups and compact [[orthogonal group]]s. Issai Schur in 1924 showed that this technique can be applied to show complete reducibility of representations for such groups via the construction of an invariant inner product. Weyl extended Schur's method to complex semisimple Lie algebras by showing they had a [[compact real form]].<ref>[[Nicolas Bourbaki]], ''Lie groups and Lie algebras'' (1989), p. 426.</ref>

==Weyl's theorem==
{{main|Weyl's theorem on complete reducibility}}

The [[complete reducibility]] of finite-dimensional linear representations of compact groups, or connected [[semisimple Lie group]]s and complex [[semisimple Lie algebra]]s goes sometimes under the name of ''Weyl's theorem''.<ref>{{Springer|id=C/c024000|title=Completely-reducible set}}</ref> A related result, that the [[universal cover]] of a compact semisimple Lie group is also compact, also goes by the same name. It was proved by Weyl a few years before "universal cover" had a formal definition.<ref>{{Springer|id=l/l058610|title=Lie group, compact}}</ref><ref>{{cite book |last1=Bourbaki |first1=Nicolas |title=Lie Groups and Lie Algebras: Chapters 1-3 |date=1989 |publisher=Springer Science & Business Media |isbn=978-3-540-64242-8 |page=426 |url=https://books.google.com/books?id=brSYF_rB2ZcC&pg=PA426 |language=en}}</ref>

==Explicit formulas==

Let <math>\pi: G \rightarrow GL(n,\mathbb{C})</math> be a complex representation of a compact Lie group <math>G</math>. Define <math>P = \int_G \pi(g)\pi(g)^*dg</math>, integrating over <math>G</math> with respect to the Haar measure. Since <math>P</math> is a positive matrix, there exists a square root <math>Q</math> such that <math>P=Q^2</math>. For each <math>g \in G</math>, the matrix <math>\tau(g) = Q^{-1}\pi(g)Q</math> is unitary.

==Notes==
{{reflist}}

==References==
*V. S. Varadarajan, ''An introduction to harmonic analysis on semisimple Lie groups'' (1999), p.&nbsp;49.
*Wulf Rossmann, ''Lie groups: an introduction through linear groups'' (2006), p.&nbsp;225.
*Roe Goodman, Nolan R. Wallach, ''Symmetry, Representations, and Invariants'' (2009), p.&nbsp;171.
*{{Citation | last1=Hurwitz | first1=A. | title=Über die Erzeugung der Invarienten durch Integration | year=1897 | journal=Nachrichten Ges. Wiss. Göttingen | pages=71–90}}

[[Category:Representation theory of Lie groups]]