Quantum Galois Correspondence for Subfactors

Galois correspondence for counting quantifiers

We introduce a new type of closure operator on the set of relations, max-implementation, and its weaker analog max-quantification. Then we show that approximation preserving reductions between counting constraint satisfaction problems (#CSPs) are preserved by these two types of closure operators. Together with some previous results this means that the approximation complexity of counting CSPs i...

The Galois Correspondence

Example 1.1. Two R-automorphisms of C are the identity z 7→ z and complex conjugation z 7→ z. We will show they are the only ones. If σ : C→ C is an R-automorphism, then for any real a and b we have σ(a+ bi) = σ(a) +σ(b)σ(i) = a+ bσ(i), so σ is determined by σ(i) and i = −1 =⇒ σ(i) = σ(−1) =⇒ σ(i) = −1 =⇒ σ(i) = ±i. If σ(i) = i, then σ(z) = z for all z ∈ C and if σ(i) = −i, then σ(z) = z for al...

On the Galois correspondence for Hopf Galois structures

We study the question of the surjectivity of the Galois correspondence from subHopf algebras to subfields given by the Fundamental Theorem of Galois Theory for abelian Hopf Galois structures on a Galois extension of fields with Galois group Γ, a finite abelian p-group. Applying the connection between regular subgroups of the holomorph of a finite abelian p-group (G,+) and associative, commutati...

The Galois Correspondence at Work

Any σ ∈ Gal(L1L2/K) restricted to L1 or L2 is an automorphism since L1 and L2 are both Galois over K. So we get a function R : Gal(L1L2/K)→ Gal(L1/K)×Gal(L2/K) by R(σ) = (σ|L1 , σ|L2). We will show R is an injective homomorphism. To show R is a homomorphism, it suffices to check the separate restriction maps σ 7→ σ|L1 and σ 7→ σ|L2 are each homomorphisms from Gal(L1L2/K) to Gal(L1/K) and Gal(L2...

Introduction to the Galois correspondence