Principal spin-bundles and triality

In this paper we construct a family of spin Lie groups G with an outer automorphism of order three (triality automorphism) and we describe the subgroups of fixed points for this kind of automorphisms. We will take advantage of this work to study the action of the group of outer automorphisms of G on the moduli space of principal G-bundles and describe the subvariety of fixed points in M(G) for the action of the outer automorphism of order three of G. Finally, we further study the case of Spin(8, C).

DOI: https://doi.org/10.15446/recolma.v49n2.60442

Alvaro Antón Sancho¹

¹ University of Valladolid, Valladolid, Spain
e-mail: alvaro.anton@eumfrayluis.com

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Abstract

In this paper we construct a family of spin Lie groups G with an outer automorphism of order three (triality automorphism) and we describe the subgroups of fixed points for this kind of automorphisms. We will take advantage of this work to study the action of the group of outer automorphisms of G on the moduli space of principal G-bundles and describe the subvariety of fixed points in M(G) for the action of the outer automorphism of order three of G. Finally, we further study the case of Spin(8, C).

Key words and phrases. triality, Spin-principal bundles, moduli space, fixed points, stability.

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2010 Mathematics Subject Classification. 14D20.

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Resumen

En este artículo construimos una familia de grupos de Lie espinoriales G dotados de un automorfismo externo de orden tres (trialidad) y describimos los subgrupos de puntos fijos para esta clase de automorfismos. Usaremos esto para estudiar la acción del grupo de automorfismos externos de G en el espacio de moduli de G-fibrados principales y describir la subvariedad de puntos fijos en M(G) para la acción del automorfismo externo de orden tres de G. Finalmente, profundizaremos en el estudio del caso Spin(8, C).

Palabras y frases clave. Trialidad, Spin-fibrado principal, espacio de moduli, puntos fijos, estabilidad.

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(Recibido en mayo de 2015. Aceptado en diciembre de 2015)