On characteristic p Verma modules and subalgebras of the hyperalgebra
<p>Let <em>G</em> be a finite dimensional semisimple Lie algebra; we study the class of infinite dimensional representations of <em>G</em>called characteristic <em>p</em> Verma modules.</p> <p>To obtain information about the structure of the Ver...
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| Format: | Thesis |
| Language: | English |
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1994
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| _version_ | 1797094015673303040 |
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| author | Carstensen, V Erdmann, K Carstensen, Vivi |
| author2 | Erdmann, K |
| author_facet | Erdmann, K Carstensen, V Erdmann, K Carstensen, Vivi |
| author_sort | Carstensen, V |
| collection | OXFORD |
| description | <p>Let <em>G</em> be a finite dimensional semisimple Lie algebra; we study the class of infinite dimensional representations of <em>G</em>called characteristic <em>p</em> Verma modules.</p> <p>To obtain information about the structure of the Verma module <em>Z</em>(λ) we find primitive weights μ such that a non-zero homomorphism from <em>Z</em>(μ) to <em>Z</em>(λ) exists. For λ + ρ dominant, where ρ is the sum of the fundamental roots, there exist only finitely many primitive weights, and they all appear in a convex, bounded area. In the case of λ + ρ not dominant, and the characteristic <em>p</em> a good prime, there exist infinitely many primitive weights for the Lie algebra. For <em>G</em> = <em>sl<sub>3</sub></em> we explicitly present a large, but not necessarily complete, set of primitive weights. A method to obtain the Verma module as the tensor product of Steinberg modules and Frobenius twisted <em>Z</em>(λ<sub>1</sub>) is given for certain weights, λ = p<sup>n</sup> λ<sub>1</sub> + (p<sup>n</sup> — 1)ρ. Furthermore, a result about exact sequences of Weyl modules is carried over to Verma modules for <em>sl<sub>2</sub></em>.</p> <p>Finally, the connection between the subalgebra u¯<sub>1</sub> of the hyperalgebra <em>U</em> for a finite dimensional semisimple Lie algebra, and a group algebra <em>KG</em> for some suitable <em>p</em>-group G is studied. No isomorphism exists, when the characteristic of the field is larger than the Coxeter number. However, in the case of <em>p</em> — 2 we find u¯<sub>1</sub><em>sl<sub>3</sub></em>≈ KG. Furthermore, we determine the centre ofu¯<sub>n</sub><em>sl<sub>3</sub></em>, and we obtain an alternative <em>K</em>-basis of <em>U-</em>.</p> |
| first_indexed | 2024-03-07T04:08:17Z |
| format | Thesis |
| id | oxford-uuid:c6f5db9a-db94-4d58-9eb1-dd402c8846c5 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T04:08:17Z |
| publishDate | 1994 |
| record_format | dspace |
| spelling | oxford-uuid:c6f5db9a-db94-4d58-9eb1-dd402c8846c52022-03-27T06:41:35ZOn characteristic p Verma modules and subalgebras of the hyperalgebraThesishttp://purl.org/coar/resource_type/c_db06uuid:c6f5db9a-db94-4d58-9eb1-dd402c8846c5Modules (Algebra)EnglishPolonsky Theses Digitisation Project1994Carstensen, VErdmann, KCarstensen, ViviErdmann, K<p>Let <em>G</em> be a finite dimensional semisimple Lie algebra; we study the class of infinite dimensional representations of <em>G</em>called characteristic <em>p</em> Verma modules.</p> <p>To obtain information about the structure of the Verma module <em>Z</em>(λ) we find primitive weights μ such that a non-zero homomorphism from <em>Z</em>(μ) to <em>Z</em>(λ) exists. For λ + ρ dominant, where ρ is the sum of the fundamental roots, there exist only finitely many primitive weights, and they all appear in a convex, bounded area. In the case of λ + ρ not dominant, and the characteristic <em>p</em> a good prime, there exist infinitely many primitive weights for the Lie algebra. For <em>G</em> = <em>sl<sub>3</sub></em> we explicitly present a large, but not necessarily complete, set of primitive weights. A method to obtain the Verma module as the tensor product of Steinberg modules and Frobenius twisted <em>Z</em>(λ<sub>1</sub>) is given for certain weights, λ = p<sup>n</sup> λ<sub>1</sub> + (p<sup>n</sup> — 1)ρ. Furthermore, a result about exact sequences of Weyl modules is carried over to Verma modules for <em>sl<sub>2</sub></em>.</p> <p>Finally, the connection between the subalgebra u¯<sub>1</sub> of the hyperalgebra <em>U</em> for a finite dimensional semisimple Lie algebra, and a group algebra <em>KG</em> for some suitable <em>p</em>-group G is studied. No isomorphism exists, when the characteristic of the field is larger than the Coxeter number. However, in the case of <em>p</em> — 2 we find u¯<sub>1</sub><em>sl<sub>3</sub></em>≈ KG. Furthermore, we determine the centre ofu¯<sub>n</sub><em>sl<sub>3</sub></em>, and we obtain an alternative <em>K</em>-basis of <em>U-</em>.</p> |
| spellingShingle | Modules (Algebra) Carstensen, V Erdmann, K Carstensen, Vivi On characteristic p Verma modules and subalgebras of the hyperalgebra |
| title | On characteristic p Verma modules and subalgebras of the hyperalgebra |
| title_full | On characteristic p Verma modules and subalgebras of the hyperalgebra |
| title_fullStr | On characteristic p Verma modules and subalgebras of the hyperalgebra |
| title_full_unstemmed | On characteristic p Verma modules and subalgebras of the hyperalgebra |
| title_short | On characteristic p Verma modules and subalgebras of the hyperalgebra |
| title_sort | on characteristic p verma modules and subalgebras of the hyperalgebra |
| topic | Modules (Algebra) |
| work_keys_str_mv | AT carstensenv oncharacteristicpvermamodulesandsubalgebrasofthehyperalgebra AT erdmannk oncharacteristicpvermamodulesandsubalgebrasofthehyperalgebra AT carstensenvivi oncharacteristicpvermamodulesandsubalgebrasofthehyperalgebra |