Iwasawa decomposition
The unique representation of an arbitrary element of a non-compact connected semi-simple real Lie group G as a product g = k an of elements k,\ a,\ n of analytic subgroups K,\ A,\ N , respectively, where K , A and N are defined as follows. Let \mathfrak g = \mathfrak k + \mathfrak P be a Cartan decomposition of the Lie algebra \mathfrak g of G ; let \mathfrak a be the maximal commutative subspace of the space \mathfrak P , and let \mathfrak N be a nilpotent Lie subalgebra of \mathfrak g such that \mathfrak N is the linear hull of the root vectors in some system of positive roots with respect to \mathfrak a . The decomposition of the Lie algebra as the direct sum of the subalgebras \mathfrak k , \mathfrak a and \mathfrak N is called the Iwasawa decomposition [1] of the semi-simple real Lie algebra \mathfrak g . The groups K , A and N are defined to be the analytic subgroups of G corresponding to the subalgebras \mathfrak k , \mathfrak a and \mathfrak N , respectively. The groups K , A and N are closed; A and N are simply-connected; K contains the centre of G , and the image of K under the adjoint representation of G is a maximal compact subgroup of the adjoint group of G . The mapping (k,\ a,\ n) \rightarrow kan is an analytic diffeomorphism of the manifold K \times A \times N onto the Lie group G . 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 p - adic field (or, more generally, for groups of p - adic type) (see [4], [5]).
Comments
An example of an Iwasawa decomposition is \mathop{\rm SL}\nolimits _{n} ( \mathbf R ) = K A N with K = \mathop{\rm SO}\nolimits _{n} ( \mathbf R ) , A the subgroup of diagonal matrices of \mathop{\rm SL}\nolimits _{n} ( \mathbf R ) and N a lower triangular matrix with 1 ' s on the diagonal. So, in particular, every element of \mathop{\rm SL}\nolimits _{n} ( \mathbf R ) gets written as a product of a special orthogonal matrix and a lower triangular matrix.
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 \mathfrak{p}-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 \mathfrak{p}-adic Chevalley groups" Publ. Math. IHES , 25 (1965) pp. 5–48 MR185016 |
[a1] | S. Helgason, "Groups and geometric analysis" , Acad. Press (1984) pp. Chapt. II, Sect. 4 MR0754767 Zbl 0543.58001 |
Iwasawa decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Iwasawa_decomposition&oldid=53634