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The Brauer Monoid is a generalization of the Brauer Group using the Galois cohomological definition by allowing 0 to be in the image of the cocycles. These generalized cocycles are called weak 2-cocycles. This thesis introduces two definitions that help with the computation of the Brauer Monoid for certain idempotents, uses weak Galois cohomology to solve a problem in the theory of finite dimensional algebras, and includes a MATLAB program that produce non-tree idempotents for cyclic Groups.
The Brauer Monoid of a finite Galois field extension K/F with G = Gal(K, F) is denoted by M^{2}( G,K). Each weak 2-cocycle ƒ has an associated weak crossed product algebra A_{ƒ} constructed in an analogous method to that of crossed product algebras. To each idempotent e in the Brauer Monoid there is a group [special characters omitted] with identity e; the Brauer Monoid is the disjoint union of these groups. For a specific ring R_{e} associated with an idempotent e, there is a chain complex of Re-modules, M_{e}, that gives us that [special characters omitted]) [7]. The idempotents e with inertial subgroup H_{e} are in one-to-one correspondence with lower subtractive partial orders on G/H_{e} and hence with a partial order graph.
This thesis builds on the above. We introduce the definition of a lower subtractive partially ordered subset S of (G, ≤_{ e}) and its associated lower subtractive subgraph. We associate a complex M^{S} to S that is a subcomplex of M^{e}. For S and T lower subtractive partially ordered subsets of G, such that [special characters omitted](G, ≤_{e}) and S, T≠ G, then there exists a split Mayer-Vietoris-like short exact sequence on the complexes:[special characters omitted]
This gives us a long exact sequence on cohomology, which can aid with the computation of H^{2}(Hom^{Re }(M^{e},K^{ ×}))≇ [special characters omitted]
We also define a connected lower subtractive partially ordered subset of G and its associated connected lower subtractive subgraph. We provide two constructions and a theorem to determine when a nontrivial decomposition into connected lower subtractive subgraphs exists to be able to apply the short exact sequence to these subsets.
Given a finite Galois extension K/F and a central simple K –algebra B, we classify the F–algebras A that contain B and are weakly Azumaya with respect to B. We use weak Galois cohomology to do this classification.
We also include a MATLAB program that outputs matrices associated with idempotents on a cyclic Galois group such that each idempotent corresponds to a non-tree lower subtractive partial order graph on G. These idempotents contribute nontrivially to M^{2}( G,K), whereas Br(K/F) is trivial when K/F is a finite Galois extension of finite fields.
Advisor: | Haile, Darrell |
Commitee: | Larsen, Michael, Lindenstrauss, Ayelet, Strauch, Matthias |
School: | Indiana University |
Department: | Mathematics |
School Location: | United States -- Indiana |
Source: | DAI-B 75/03(E), Dissertation Abstracts International |
Source Type: | DISSERTATION |
Subjects: | Mathematics |
Keywords: | Azumaya algebras, Brauer Monoid, Galois extension |
Publication Number: | 3603929 |
ISBN: | 9781303588761 |