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Revista Colombiana de Matemáticas
versão impressa ISSN 0034-7426
Rev.colomb.mat. vol.53 supl.1 Bogotá dez. 2019 Epub 24-Mar-2020
https://doi.org/10.15446/recolma.v53nsupl.84013
Artículos originales
A short survey on observability
1 Universidad de la República, Uruguay
The exploration of the notion of observability exhibits transparently the rich interplay between algebraic and geometric ideas in geometric invariant theory. The concept of observable subgroup was introduced in the early 1960s with the purpose of studying extensions of representations from an afine algebraic subgroup to the whole group. The extent of its importance in representation and invariant theory in particular for Hilbert's 14th problem was noticed almost immediately. An important strenghtening appeared in the mid 1970s when the concept of strong observability was introduced and it was shown that the notion of observability can be understood as an intermediate step in the notion of reductivity (or semisimplicity), when adequately generalized. More recently starting in 2010, the concept of observable subgroup was expanded to include the concept of observable action of an afine algebraic group on an afine variety, launching a series of new applications and opening a surge of very interesting activity. In another direction around 2006, the related concept of observable adjunction was introduced, and its application to module categories over tensor categories was noticed. In the current survey, we follow (approximately) the historical development of the subject introducing along the way, the definitions and some of the main results including some of the proofs. For the unproven parts, precise references are mentioned.
Keywords: observability; invariants; actions
El estudio de la noción de observabilidad muestra de modo transparente la rica interacción entre las ideas algebraicas y geométricas en la teoría geométrica de invariantes. El concepto de subgrupo observable fue introducido al inicio de la década de 1960 con el propósito de estudiar las extensiones de representaciones desde un subgrupo algebraico afín a todo el grupo (también algebraico afín). La importancia de la noción de subgrupo observable en la teoría de representaciones y la teoría de invariantes, en particular para el estudio del 14to problema de Hilbert, fue observada de inmediato. En la mitad de la década de 1970 apareció un refinamiento importante de esta noción: el concepto de observabilidad fuerte fue introducido y se mostró que la noción de observabilidad puede entenderse como un paso intermedio hacia la noción de reductividad (o semi-simplicidad), haciendo las generalizaciones adecuadas. Recientemente, al inicio de la década de 2010, el concepto de subgrupo observable fue expandido de modo de incluir la noción de acción observable de un grupo algebraico afín en una varidad algebraica afín. Esta generalización inició una serie de trabajos interesantes, con varias aplicaciones novedosas. En otra dirección, cerca de 2006 fue introducido el concepto de adjunción observable, que tuvo aplicación inmediata en el estudio de las módulo categorías sobre categorías tensoriales. En la revisión que sique, seguimos de modo aproximado el desarrollo histórico de esta temática, introduciendo a lo largo del camino las definiciones y los resultados centrales, junto con algunas de las pruebas. Para los resultados sin demostración, se mencionan referencias precisas.
Palabras clave: observabilidad; invariantes; acciones
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Received: September 13, 2018; Accepted: February 11, 2019
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