CEU Electronic Theses and Dissertations, 2018
| Author | Mészáros, Szabolcs |
|---|---|
| Title | On Constructions of Matrix Bialgebras |
| Summary | The thesis consists of two parts. In the first part consisting of Chapter 2 and 3, matrix bialgebras, generalizations of the quantized coordinate ring of n by n matrices are considered. The defining parameter of the construction is an endomorphism of the tensor-square of a vector space. In the investigations this endomorphism is assumed to be either an idempotent or nilpotent of order two. In Theorem 2.2.2, 2.3.2 and 2.5.3 it is proved that the Yang-Baxter equation gives not only a sufficient condition – as it was known before – for certain regularity properties of matrix bialgebras, such as the Poincaré-Birkhoff-Witt basis property or the Koszul property, but it is also necessary, under some technical assumptions. The proofs are based on the methods of the representation theory of finite-dimensional algebras. In the second part consisting of Chapter 4 and 5, the quantized coordinate rings of matrices, the general linear group and the special linear group are considered, together with the corresponding Poisson algebras called semiclassical limit Poisson algebras. In Theorem 4.1.1 and 5.1.1 it is proved that the subalgebra of cocommutative elements in the above mentioned algebras and Poisson algebras are maximal commutative, and maximal Poisson-commutative subalgebras respectively. The proofs are based on graded-filtered arguments. |
| Supervisor | Domokos, Mátyás |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2018/meszaros_szabolcs.pdf |
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