In mathematics, the formalism of B-admissible representations provides constructions of full Tannakian subcategories of the category of representations of a group G on finite-dimensional vector spaces over a given field E. In this theory, B is chosen to be a so-called (E, G)-regular ring, i.e. an E-algebra with an E-linear action of G satisfying certain conditions given below. This theory is most prominently used in p-adic Hodge theory to define important subcategories of p-adic Galois representations of the absolute Galois group of local and global fields.

(E, G)-rings and the functor D

Let G be a group and E a field. Let Rep(G) denote a non-trivial strictly full subcategory of the Tannakian category of E-linear representations of G on finite-dimensional vector spaces over E stable under subobjects, quotient objects, direct sums, tensor products, and duals. An (E, G)-ring is a commutative ring B that is an E-algebra with an E-linear action of G. Let F = BG be the G-invariants of B. The covariant functor DB : Rep(G) → ModF defined by is E-linear (ModF denotes the category of F-modules). The inclusion of DB(V) in B ⊗EV induces a homomorphism called the comparison morphism.

Regular (E, G)-rings and B-admissible representations

An (E, G)-ring B is called regular if The third condition implies F is a field. If B is a field, it is automatically regular. When B is regular, with equality if, and only if, αB,V is an isomorphism. A representation V ∈ Rep(G) is called B-admissible if αB,V is an isomorphism. The full subcategory of B-admissible representations, denoted RepB(G), is Tannakian. If B has extra structure, such as a filtration or an E-linear endomorphism, then DB(V) inherits this structure and the functor DB can be viewed as taking values in the corresponding category.

Examples

Potentially B-admissible representations

A potentially B-admissible representation captures the idea of a representation that becomes B-admissible when restricted to some subgroup of G.