View source for Iwasawa decomposition

The unique representation of an arbitrary element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i0530601.png" /> of a non-compact connected semi-simple real [[Lie group|Lie group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i0530602.png" /> as a product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i0530603.png" /> of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i0530604.png" /> of analytic subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i0530605.png" />, respectively, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i0530606.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i0530607.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i0530608.png" /> are defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i0530609.png" /> be a [[Cartan decomposition|Cartan decomposition]] of the [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306011.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306012.png" /> be the maximal commutative subspace of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306013.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306014.png" /> be a nilpotent Lie subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306016.png" /> is the linear hull of the root vectors in some system of positive roots with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306017.png" />. The decomposition of the Lie algebra as the direct sum of the subalgebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306020.png" /> is called the Iwasawa decomposition [[#References|[1]]] of the semi-simple real Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306021.png" />. The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306024.png" /> are defined to be the analytic subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306025.png" /> corresponding to the subalgebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306028.png" />, respectively. The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306031.png" /> are closed; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306033.png" /> are simply-connected; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306034.png" /> contains the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306035.png" />, and the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306036.png" /> under the adjoint representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306037.png" /> is a maximal compact subgroup of the adjoint group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306038.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306039.png" /> is an analytic diffeomorphism of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306040.png" /> onto the Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306041.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306042.png" />-adic field (or, more generally, for groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306043.png" />-adic type) (see [[#References|[4]]], [[#References|[5]]]).

====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 {{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>



====Comments====
An example of an Iwasawa decomposition is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306046.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306048.png" /> the subgroup of diagonal matrices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306050.png" /> a lower triangular matrix with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306051.png" />'s on the diagonal. So, in particular, every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306052.png" /> gets written as a product of a special orthogonal matrix and a lower triangular matrix.

====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 {{MR|0754767}} {{ZBL|0543.58001}} </TD></TR></table>