Abstract
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.
Highlights
The concept of observable subgroup of an affine algebraic group G was introduced by A
It can be proved that G/H is isomorphic to the G–orbit of m0 in M and as such it is a quasi–affine variety
Assume that H ⊆ G is a closed inclusion of affine algebraic groups
Summary
The concept of observable subgroup of an affine algebraic group G was introduced by A. Assume that H ⊆ G is a closed inclusion of affine algebraic groups, if 0 = I ⊆ k[G] is an H stable ideal, there is a non zero element f ∈ I and an extendable character χ of H such that x · f = χ(x)f for all x ∈ H and that f (1) = 0. It can be proved that G/H is isomorphic to the G–orbit of m0 in M (result that is obvious in the case of zero characteristic, but that in general a proof of the separabililty of the action in this situation is needed) and as such it is a quasi–affine variety (for more details see [46, Section 8.3])
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