This paper uses Poisson geometry to study triangular factorization in compact symmetric spaces. The central result is the interpretation of the logarithm of the diagonal as a momentum map. In this summary, we'll describe the results in the context of a concrete example.

Symmetric Spaces

Globally symmetric spaces form a very special class of Riemannian manifolds. These manifolds have very large isometry groups (hence the adjective "symmetric", lots of symmetry!), large enough to act transitively, in fact. Consider Euclidean 3-space for a moment. For each point O we can define an isometry by mapping a point P to the other point P' equidistant to O on the line through O and P. In a general Riemannian manifold, we can define a similar transformation in a neighborhood of each point O by reflecting a point P which is t units along a geodesic through O to the point P' which is t units along the geodesic in the other direction from P. A priori, this transformation of reflection along geodesics through O only makes sense in a neighborhood of O. But, if the geometry of the space is especially nice, this transformation can be defined on the whole space, and moreover, this can be done for each O. Such a space is said to be globally symmetric. Two compact examples are the round 2-sphere and the round 3-sphere. The following films illustrate the geodesic symmetry in Euclidean space and in the 3-sphere.

The conditions that a space must satisify in order to be globally symmetric are very strong. Consequently, a classification is possible and this was worked out by Elie Cartan in the 1920's. Cartan's fundamental idea was to make use of the existence of the transitive action by a group of isometries to transfer the study in the realm of algebra of Lie groups and Lie algebras. The idea was two associate to each point a coset of a subgroup K, in the group of isometries U; the subgroup K being those isometries which fix O. We write X = U/K. Viewing the round 2-sphere as the Riemann sphere in an appropriate metric, the group U is then SU(2), the set of 2 x 2 unitary matrices with determinant one, and K is the diagonal subgroup. It is this connection with Lie groups that we exploit to connect symmetric spaces with triangular factorization.

Triangular Factorization

Every n x n complex matrix of trace zero can be uniquely decomposed as a sum of a strictly lower triangular matrix, a diagonal trace zero matrix, and a strictly upper triangular matrix. The multiplicative analog of this statement for an element g of the group of n x n complex matrices of determinant one, SL(n,C), does not always hold. There is an algebraic condition that must be imposed in order to guarantee that g may be factored uniquely as ldu where l (resp. u) is a lower (resp. upper) triangular unipotent matrix, and d is a diagonal matrix of determinant one. The algorithm that determines this factorization is Gaussian elimination, and the condition is that the principal minors of g cannot vanish. When one or more of the principal minors does vanish, a permutation of the rows or columns must be performed in order to continue Gauss' algorithm. Thus every element of SL(n,C) can be factored as lwdu where w is a permutation matrix. The diagonal factor d, and the permutation w are uniquely determined in every case, but the factors l and u are uniquely determined only in the case w=1. By separating the elements of SL(n,C) into sets corresponding to different permutations w (Weyl group elements) we obtain a decomposition of SL(n,C) into a disjoint union of submanifolds whose codimension increases with the length of the indexing Weyl group element. An important observation is that the non-zero entries of each of the factors l, d, and u are ratios of determinants of submatrices of g.

As with most topics in linear algebra, this decomposition has been generalized in Lie theory where it is known as the Birkhoff decomposition (alternatively, triangular or LDU decomposition). Corresponding to a triangular decomposition of a complex semi-simple Lie algebra, the corresponding group G decomposes as a disjoint union of submanifolds whose codimensions increase with the length of the indexing elements w of the Weyl group W.

Pickrell introduced a generalization of this decomposition for symmetric spaces. Recall that X = U/K. The stability subgroup K is (essentially) the fixed point set of an involution and the complexification of U is a semi-simple complex Lie group G. Through the Cartan embedding, which realizes U/K as a totally geodesic submanifold of U, one obtains a induced decomposition of X from a Birkhoff decomposition of G. Such decomposition depends on the basepoint in X (which determined K and the involution) and the chosen triangular decomposition of G.

Returning to our running example: X is the round 2-sphere; U is SU(2); K is the diagonal subgroup; G is SL(2,C); and $\theta$ is the involution of SU(2) which negates the off-diagonal entries. Equipped with an invariant metric, SU(2) can be identified with the round 3-sphere, and the symmetric space is the base of the Hopf fibration. The Cartan embedding realizes X as an equatorial 2-sphere inside the 3-sphere.

Poisson Geometry

A Poisson structure on a manifold is a bivector field whose components in each local coordinate chart satisfy a certain undetermined system of semi-linear partial differential equations. These equations ensure that the associated bracket {.,.} on the algebra of smooth functions satisfies the Jacobi identity and thus defines a Poisson bracket. Geometrically, this means that the manifold is foliated by even dimensional submanifolds, each of which is symplectic with symplectic structure equal to the inverse of the Poisson bivector along the submanifold. From a Poisson bivector one obtains a natural map from the cotangent bundle to the tangent bundle whose image is an integrable general distribution. This distribution determines the symplectic foliation. The additional ingredient in homogeneous Poisson geometry is a Poisson Lie group structure on U. This is a Poisson structure on U with the property that the multiplication map U x U --> U is a Poisson map. With respect to such a structure, a Poisson structure on X is homogeneous if the action map U x X --> X is a Poisson map. The movie below helps you to visualized a standard Poisson Lie group structure on SU(2). The zero dimensional symplectic leaves are the points of the diagonal subgroup of SU(2). The other symplectic leaves are disks whose boundary meet this circle.

In this paper we first observe that each standard Poisson Lie group structure on U is determined by a triangular decomposition of the Lie algebra of G. This structure vanishes on the Cartan torus T which is the intersection of H with U. Using a general construction due to Evens and Lu we introduce a Poisson structure on X which is homogeneous relative to a standard Poisson Lie group structure on U. In our running example, the foliation of the of the 2-sphere has two open symplectic leaves, the upper and lower hemispheres, and a circle worth of zero dimensional symplectic leaves making up the equator.

In this paper we accomplish the following:

• We give a complete description of the symplectic foliation of this Poisson structure by exhibiting a connection with triangular factorization. Specifically, we choose the basepoint in X so that the triangular decomposition of the Lie algebra of G is stable with respect to the involution. The symplectic foliation then aligns with the associated Birkhoff decomposition of X in the following sense. If we consider the layer of the Birkhoff decomposition of X corresponding to a Weyl group element w, then the Poisson structure is regular there. Morover, the symplectic leaves are contractible and we construct a geometric model for these spaces. The followin film animates this alignment in our running example.

• The symplectic leaves in the layer corresponding to w are indexed by elements w in the intersection of the image of the Cartan embedding with w inside of U. The Cartan embedded images of the elements in a leaf indexed by w have a triangular factorization of the form lwdu. Each such leaf is acted on in a Hamiltonian fashion by a subgroup T_w of T and, essentially, the log of the diagonal factor gives rise to a momentum map for the action of T_w on the leaf.
• Finally we display the first explicit formulas for this structure in several families of classical examples.