GELATI · Geometry of exceptional Lie algebras à la Tits
7РП — „Хора“ (Действия „Мария Кюри“)
- Период
- 2013-04-01 → 2015-03-31
- Финансиране от ЕС
- 221 606 €
- Участници
- 1
- Схема
- MC-IEF
Линиите свързват координатора с партньорите.
Накратко на български
Геометрията на специалните алгебри на Ли изследва връзката между алгебричните структури и геометричните обекти, като например проективните равнини. Това помага за по-доброто разбиране на свойствата на многомерните пространства и техните математически характеристики.
Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.
Резултати накратко
Geometry of exceptional Lie algebras à la Tits
The magic square was constructed by Freudenthal and Tits starting from a pair (A, B) of composition algebras and forming a corresponding Lie algebra. There exist interpretations (both split and non-split) of this table over arbitrary fields, as well as interpretations emphasizing the geometries related to these algebras and groups, i.e. the Tits buildings. In recent years the geometries appearing in the magic square have been studied quite exten- sively, mostly over the complex numbers, using algebraic geometry and representation theory. In 1984 Mazzocca and Melone formulated axioms involving conics which characterize finite quadratic Veronesean varieties over fields of odd order (the A2 case). In joint work with Van Maldeghem the applicant established that a more abstract version of these axioms turns out to be characteristic for the second row of the magic square over arbitrary fields. The highlight of this research so far is the characterization of the split version of the second row over arbitrary fields as projective planes over split composition algebras [SVM4] 1. The varieties obtained are the analogues over arbitrary fields of the ones in the following theorem by Zak: Over an algebraically closed field of characteristic zero each Severi (secant defective) variety is projectively equivalent to either a quadratic Veronese variety in 5 dimensions, or a Segre variety in 8 dimensions, or a line Grassmannian variety in 14 dimensions, or the Cartan variety in 26 dimensions. In 1901 Severi already proved Zak’s theorem for surfaces. In 2014, in joint work with Krauss and Van Maldeghem the applicant obtained also a characterization of the non-split version of the second row as the Veronesean representations of Moufang planes over quadratic alternative division algebras. In particular this characterized the Veronesean representation of the Moufang projective plane P(O) related to any Cayley- Dickson division algebra O, which is the geometry of the real form E28 of the simple Lie group 6,2 of exceptional type E6.
Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз
Цел на проекта
The proposal concerns algebraic groups and their associated geometries, in particular those of exceptional type. The main goal of the proposal is to give a uniform axiomatic description of the embeddings in projective space of the varieties occurring in the Freudenthal-Tits magic square.For instance, the second row comprises Severi-Brauer varieties, which have applications in Galois cohomology. Of special interest are the geometries of exceptional Lie type over arbitrary fields, where we would obtain a purely geometric characterization of F4, E6, E7 and E8. In particular this involves a direct construction of the 248-dimensional E8-module.In the spirit of the work of Tits (Abel prize 2008) and Aschbacher (Wolf Prize 2012), there is a nice interaction between geometry and groups. The embeddings (geometry) will provide fruitful information about the subgroup structure of finite simple groups and groups of Lie type over arbitrary fields, and conversely, the expert knowledge of Prof. Liebeck on algebraic groups will help describe the embeddings.""
Оригинален текст от CORDIS (на английски).
Участници
- IMPERIAL COLLEGE OF SCIENCE TECHNOLOGY AND MEDICINE · LondonКоординаторОбединеното кралство
Връзки
Данни: CORDIS, © Европейски съюз