TY  - THES
AB  - In this thesis, we use an algebro-geometric approach to study solutions to a generalized version of the classical Yang-Baxter equation (CYBE) for central simple Lie algebras over arbitrary fields of characteristic 0. We assign to these solutions certain geometric data including a cohomology-free sheaf of Lie algebras on a projective curve. The application of geometric methods leads to a new proof of the Belavin-Drinfeld trichotomy, which states that non-degenerate solutions of the CYBE for complex simple Lie algebras are either elliptic, trigonometric, or rational. We give more explicit descriptions of the geometric data as well as the structure theory for solutions from each of these three classes. We also derive a purely geometric version of the Belavin-Drinfeld trichotomy which works over any field of characteristic 0. Moreover, we prove that every non-skew-symmetric solution of the generalized CYBE for central simple Lie algebras over an arbitrary field of characteristic 0 corresponds to a projective curve normalized by $\mathbb{P}^1$ and extends to a rational function by passing to an \'etale $\mathbb{P}^1$-scheme.
AU  - Abedin, Raschid
DO  - 10.17619/UNIPB/1-1580
DP  - Universität Paderborn
LA  - eng
PB  - Veröffentlichungen der Universität
T2  - Institut für Mathematik
TI  - Algebraic geometry of the classical Yang-Baxter equation and its generalizations
UR  - https://nbn-resolving.org/urn:nbn:de:hbz:466:2-43413
Y2  - 2026-02-05T05:47:48
ER  -