A computational approach to 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type

A computational approach to 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type

Let g be a simple complex Lie algebra and let e be a nilpotent element of g. It was conjectured by Premet in [P07i] that the finite W-algebra U(g; e) admits a 1-dimensional representation, and further work [L10, P08] has reduced this conjecture to the case where g is of exceptional type and e lies in a rigid nilpotent orbit in g. Using the PBW-theorem for U(g; e) we give an algorithm for determining a presentation for U(g; e) which allows us to determine the 1-dimensional representations for U(g; e). Implementing this algorithm in GAP4 we verify the conjecture in the case that g is of type G2, F4 or E6. Using a result of Premet in [P08], we can use these results to deduce that reduced enveloping algebras of those types admit representations of minimal dimension, and using the explicit presentations we can determine for which characteristics this will hold. Further, we show that we can determine the 1-dimensional representations of U(g; e) from a smaller set of relations than is required for a presentation. From calculating these sets of relations, we show that in the case that g is of type E7 and e lies in any rigid nilpotent orbit, or in the case that g is of type E8 and e lies in one of 14 (out of 17) rigid nilpotent orbits, that U(g; e) admits a 1-dimensional representation.

Ubly, Glenn

b3dfde0f-50fe-4fde-aeb5-4c8b643d868b

January 2010

Ubly, Glenn

b3dfde0f-50fe-4fde-aeb5-4c8b643d868b

Bernhard, Koeck

84d11519-7828-43a6-852b-0c1b80edeef9

Ubly, Glenn
(2010)
A computational approach to 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type.
*University of Southampton, School of Mathematics, Doctoral Thesis*, 177pp.

Record type:
Thesis
(Doctoral)

## Abstract

Let g be a simple complex Lie algebra and let e be a nilpotent element of g. It was conjectured by Premet in [P07i] that the finite W-algebra U(g; e) admits a 1-dimensional representation, and further work [L10, P08] has reduced this conjecture to the case where g is of exceptional type and e lies in a rigid nilpotent orbit in g. Using the PBW-theorem for U(g; e) we give an algorithm for determining a presentation for U(g; e) which allows us to determine the 1-dimensional representations for U(g; e). Implementing this algorithm in GAP4 we verify the conjecture in the case that g is of type G2, F4 or E6. Using a result of Premet in [P08], we can use these results to deduce that reduced enveloping algebras of those types admit representations of minimal dimension, and using the explicit presentations we can determine for which characteristics this will hold. Further, we show that we can determine the 1-dimensional representations of U(g; e) from a smaller set of relations than is required for a presentation. From calculating these sets of relations, we show that in the case that g is of type E7 and e lies in any rigid nilpotent orbit, or in the case that g is of type E8 and e lies in one of 14 (out of 17) rigid nilpotent orbits, that U(g; e) admits a 1-dimensional representation.

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## More information

Published date: January 2010

Organisations:
University of Southampton

## Identifiers

Local EPrints ID: 160239

URI: http://eprints.soton.ac.uk/id/eprint/160239

PURE UUID: 0f378982-cef6-49c4-9009-37a3fdb1fb68

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Date deposited: 15 Jul 2010 15:26

Last modified: 30 Jan 2020 01:31

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## Contributors

Author:
Glenn Ubly

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