automorphisms of a Siegel domain. In particular two Siegel domains are isomorphic if and only if they are isomorphic by an affine transformation. j-algebras Suppose that G is the Lie algebra of a transitive connected group of analytic automorphisms of a bounded homogeneous domain X, and let K be the subalgebra fixing a point x. Then the almost complex structure j on X induces a vector space endomorphism j of G such that j2=–1 on G/K [x,y] + j[jx,y] + j[x,jy] – [jx,jy] = 0 in G/K; this follows from the fact that the almost complex structure of X is integrable There is a linear form ω on G such that ω[jx,jy]=ω[x,y] and ω[jx,x]>0 if x∉K if L is a compact subalgebra of G with jL⊆K+L then L⊆K A j-algebra is a Lie algebra G with a subalgebra K and a linear map j satisfying the properties above. The Lie algebra of a connected Lie group acting transitively on a homogeneous bounded domain is a j-algebra, which is not surprising as j-algebras are defined to have the obvious properties of such a Lie algebra. The converse is also true: any j-algebra is the Lie algebra of some transitive group of automorphisms of a homogeneous bounded domain. This does not give a 1:1 correspondence between homogeneous bounded domains and j-algebras, because a homogeneous bounded domain can have several different Lie groups acting transitively on it. References There is an English translation in the appendix of . Complex manifolds