リー群上の左不変平坦な射影構造と概均質ベクトル空間

We show the correspondence between left invariant flat projective structures on Lie groups and certain prehomogeneous vector spaces. Moreover by using the classification theory of prehomogeneous vector spaces, we classify complex Lie groups admitting irreducible left invariant flat complex projective structures. As a result, direct sums of special linear Lie algebras sl(2) ⊕ sl(m_1) ⊕ ⋅ ⋅ ⋅ ⊕ sl(m_k) admit left invariant flat complex projective structures if the equality 4 + m^2_1 + ⋅ ⋅ ⋅ + m^2_k - k - 4m_1m_2 ⋅ ⋅ ⋅ m_k = 0 holds. These contain sl(2), sl(2) ⊕ sl(3), sl(2) ⊕ sl(3) ⊕ sl(11) for example.広島大学(Hiroshima University)博士(理学)Physical Sciencedoctora

Low dimensional Lie groups admitting left invariant flat projective or affine structures

AbstractWe prove that any real Lie group of dimension ⩽5 admits a left invariant flat projective structure. We also prove that a real Lie group L of dimension ⩽5 admits a left invariant flat affine structure if and only if the Lie algebra of L is not perfect

Hypergonadotropic hypogonadism and hypersegmented neutrophils in a patient with ataxia-telangiectasia-like disorder: Potential diagnostic clues?

Ataxia-telangiectasia-like disorder (ATLD) is a rare autosomal recessive disorder, and has symptoms similar to ataxia-telangiectasia (AT). ATLD is caused by mutations in the MRE11 gene, involved in DNA double-strand break repair (DSBR). In contrast to AT, ATLD patients lack key clinical features, such as telangiectasia or immunodeficiency, and are therefore difficult to be diagnosed. We report a female ATLD patient presenting with hypergonadotropic hypogonadism and hypersegmented neutrophils, previously undescribed features in this disorder, and potential diagnostic clues to differentiate ATLD from other conditions. The patient showed slowly progressive cerebellar ataxia from 2 years of age, and MRI revealed atrophy of the cerebellum, oculomotor apraxia, mild cognitive impairment, writing dystonia, hypergonadotropic hypogonadism with primary amenorrhea, and hypersegmented neutrophils. Western blot assay demonstrated total loss of MRE11 and reduction of ATM-dependent phosphorylation; thus, we diagnosed ATLD. Genetically, a novel missense mutation (c.140C>T) was detected in the MRE11 gene, but no other mutation was found in the patient. Our presenting patient suggests that impaired DSBR may be associated with hypergonadotropic hypogonadism and neutrophil hypersegmentation. In conclusion, when assessing patients with ataxia of unknown cause, ATLD should be considered, and the gonadal state and peripheral blood smear samples evaluated