Hessian Geometry of the Homogeneous Graph Domain

Left-symmetric algebras and homogeneous improper affine spheres

“…showing that x ∈ ker λ. This shows (5). Suppose J is a nontrivial left ideal in (A, ⋆) and 0 = x ∈ J ∩ ker λ.…”

“…As this holds if and only if x ∈ ker D, n( D ) = ker D ⊕ D . This shows(5).If D is invertible, then for x ∈ B there exists z ∈ B such that x = Dz = D ⋆ z − z ⋆ D. This shows that B ⊂ [A, A]. Since the opposite containment is clear from (6.2), this shows B = [A, A].…”

“…(5) ⇐⇒ (6). That (6) =⇒ (5) is immediate, so suppose there holds (5). Since P (0) = 1 and P is a polynomial, log P is real analytic in a neighborhood U of 0.…”

“…Claim ( 7) is straightforward. Claims (2) and (3) are stated explicitly as Proposition 2.6 and Corollary 2.7 of [5]. When h is positive definite the content of Lemma 4.1 is due to Vinberg in [23] (see pages 369 − 370), and is also in [22]; the proofs for general h are essentially the same, although the conclusions are not as strong as for positive definite h. If h is positive definite then that R(u) − R(u) 2 be nilpotent and self-adjoint means it is zero, so R(u) is a projection operator.…”

“…They ask if these properties characterize these hypersurfaces, and with the additional assumption that the domain above the hypersurface is homogeneous this was proved by Y. Choi and H. Kim in [6] using the theory of LSAs. Relations between homogeneous improper affine spheres, LSAs, and Monge-Ampère equations like that in Theorem 1.1 and Theorems 4.3 and 6.10 in section 3 have been studied by Choi and Kim and K. Chang in the papers [4], [5], and [6], that address a characterization of the generalized Cayley hypersurfaces conjectured in [7]. Their work, as well as that of H. Shima [22] and A. Mizuhara [19,20], provided motivation for the material described here.…”