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Difference between revisions of "Iwasawa decomposition"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> K. Iwasawa,   "On some types of topological groups" ''Ann. of Math.'' , '''50''' (1949) pp. 507–558</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.A. Naimark,   "Theory of group representations" , Springer (1982) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Helgason,   "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> F. Bruhat,   "Sur une classe de sous-groupes compacts maximaux des groupes de Chevalley sur un corps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306044.png" />-adique" ''Publ. Math. IHES'' , '''23''' (1964) pp. 45–74</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> N. Iwahori,   H. Matsumoto,   "On some Bruhat decomposition and the structure of the Hecke rings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306045.png" />-adic Chevalley groups" ''Publ. Math. IHES'' , '''25''' (1965) pp. 5–48</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> K. Iwasawa, "On some types of topological groups" ''Ann. of Math.'' , '''50''' (1949) pp. 507–558 {{MR|0029911}} {{ZBL|0034.01803}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) {{MR|0793377}} {{ZBL|0484.22018}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) {{MR|0514561}} {{ZBL|0451.53038}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> F. Bruhat, "Sur une classe de sous-groupes compacts maximaux des groupes de Chevalley sur un corps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306044.png" />-adique" ''Publ. Math. IHES'' , '''23''' (1964) pp. 45–74 {{MR|179298}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> N. Iwahori, H. Matsumoto, "On some Bruhat decomposition and the structure of the Hecke rings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306045.png" />-adic Chevalley groups" ''Publ. Math. IHES'' , '''25''' (1965) pp. 5–48 {{MR|185016}} {{ZBL|}} </TD></TR></table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Helgason,   "Groups and geometric analysis" , Acad. Press (1984) pp. Chapt. II, Sect. 4</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Helgason, "Groups and geometric analysis" , Acad. Press (1984) pp. Chapt. II, Sect. 4 {{MR|0754767}} {{ZBL|0543.58001}} </TD></TR></table>

Revision as of 14:50, 24 March 2012

The unique representation of an arbitrary element of a non-compact connected semi-simple real Lie group as a product of elements of analytic subgroups , respectively, where , and are defined as follows. Let be a Cartan decomposition of the Lie algebra of ; let be the maximal commutative subspace of the space , and let be a nilpotent Lie subalgebra of such that is the linear hull of the root vectors in some system of positive roots with respect to . The decomposition of the Lie algebra as the direct sum of the subalgebras , and is called the Iwasawa decomposition [1] of the semi-simple real Lie algebra . The groups , and are defined to be the analytic subgroups of corresponding to the subalgebras , and , respectively. The groups , and are closed; and are simply-connected; contains the centre of , and the image of under the adjoint representation of is a maximal compact subgroup of the adjoint group of . The mapping is an analytic diffeomorphism of the manifold onto the Lie group . The Iwasawa decomposition plays a fundamental part in the representation theory of semi-simple Lie groups. The Iwasawa decomposition can be defined also for connected semi-simple algebraic groups over a -adic field (or, more generally, for groups of -adic type) (see [4], [5]).

References

[1] K. Iwasawa, "On some types of topological groups" Ann. of Math. , 50 (1949) pp. 507–558 MR0029911 Zbl 0034.01803
[2] M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) MR0793377 Zbl 0484.22018
[3] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) MR0514561 Zbl 0451.53038
[4] F. Bruhat, "Sur une classe de sous-groupes compacts maximaux des groupes de Chevalley sur un corps -adique" Publ. Math. IHES , 23 (1964) pp. 45–74 MR179298
[5] N. Iwahori, H. Matsumoto, "On some Bruhat decomposition and the structure of the Hecke rings of -adic Chevalley groups" Publ. Math. IHES , 25 (1965) pp. 5–48 MR185016


Comments

An example of an Iwasawa decomposition is with , the subgroup of diagonal matrices of and a lower triangular matrix with 's on the diagonal. So, in particular, every element of gets written as a product of a special orthogonal matrix and a lower triangular matrix.

References

[a1] S. Helgason, "Groups and geometric analysis" , Acad. Press (1984) pp. Chapt. II, Sect. 4 MR0754767 Zbl 0543.58001
How to Cite This Entry:
Iwasawa decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Iwasawa_decomposition&oldid=14077
This article was adapted from an original article by A.S. FedenkoA.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article