Galois Algebras Research Articles

On Galois Algebras Satisfying the Fundamental Theorem

Let B be a Galois algebra over a commutative ring R with Galois group G such that B H is a separable subalgebra of B for each subgroup H of G. Then it is shown that B satisfies the fundamental theorem if and only if B is one of the following three types: (1) B is an indecomposable commutative Galois algebra, (2) B = Re ⊕ R(1 − e) where e and 1 − e are minimal central idempotents in B, and (3) B is an indecomposable Galois algebra such that for each separable subalgebra A, V B (A) = ⨁∑ g∈G(A) J g , and the centers of A and B G(A) are the same where V B (A) is the commutator subring of A in B, J g = {b ∈ B | bx = g(x)b for each x ∈ B} for a g ∈ G, and G(A) = {g ∈ G | g(a) = a for all a ∈ A}.

Squeeze transformation and optics after Einstein

Squeeze transformation and optics after Einstein

Galois algebras of squeezed quantum phase states

Coding, transmission and recovery of quantum states with high security and efficiency, and with as low fluctuations as possible, is the main goal of quantum information protocols and their proper technical implementations. The paper deals with this topic, focusing on the quantum states related to Galois algebras. We first review the constructions of complete sets of mutually unbiased bases in a Hilbert space of dimension q = pm, with p being a prime and m a positive integer, employing the properties of Galois fields Fq (for p>2) and/or Galois rings of characteristic four R4m (for p = 2). We then discuss the Gauss sums and their role in describing quantum phase fluctuations. Finally, we examine an intricate connection between the concepts of mutual unbiasedness and maximal entanglement.

The Galois algebra with Galois group which is the automorphism group

Let B be a Galois algebra over a commutative ring R with Galois group G. Then it is shown that G = Aut R ( B ) if and only if B is commutative with no idempotents but 0 and 1, or B ≅ R ⊕ R where R contains no idempotents but 0 and 1.

The DeMeyer-Kanzaki Galois extension and its skew group ring

Several characterizations are given for a ring B being a DeMeyer-Kanzaki Galois extension with Galois group G in terms of the skew group ring B G. Consequently, the results of S. Ikehata on commu- tative Galois algebras are generalized.

The Boolean algebra of Galois algebras

Let B be a Galois algebra with Galois group G, Jg = {b ∈ B | bx = g(x)b for all x ∈ B} for each g ∈ G, and BJg = Beg for a central idempotent eg, Ba the Boolean algebra generated by {0, eg | g ∈ G}, e a nonzero element in Ba, and He = {g ∈ G | eeg = e}. Then, a monomial e is characterized, and the Galois extension Be, generated by e with Galois group He, is investigated.

Quartic exercises

A correspondence between quartic étale algebras over a field and quadratic étale extensions of cubic étale algebras is set up and investigated. The basic constructions are laid out in general for sets with a profinite group action and for torsors, and translated in terms of étale algebras and Galois algebras. A parametrization of cyclic quartic algebras is given.

On Hopf DeMeyer‐Kanzaki Galois extensions

Let H be a finite‐dimensional Hopf algebra over a field k, B a left H‐module algebra, and H∗ the dual Hopf algebra of H. For an H∗‐Azumaya Galois extension B with center C, it is shown that B is an H∗‐DeMeyer‐Kanzaki Galois extension if and only if C is a maximal commutative separable subalgebra of the smash product B#H. Moreover, the characterization of a commutative Galois algebra as given by S. Ikehata (1981) is generalized.

The temporal logic of coalgebras via Galois algebras

This paper introduces a temporal logic for coalgebras. Nexttime and lasttime operators are defined for a coalgebra, acting on predicates on the state space. They give rise to what is called a Galois algebra. Galois algebras form models of temporal logics like Linear Temporal Logic (LTL) and Computation Tree Logic (CTL). The mapping from coalgebras to Galois algebras turns out to be functorial, yielding indexed categorical structures. This construction gives many examples, for coalgebras of polynomial functors on sets. More generally, it will be shown how ‘fuzzy’ predicates on metric spaces, and predicates on presheaves, yield indexed Galois algebras, in basically the same coalgebraic manner.

The Galois extensions induced by idempotents in a Galois algebra

Let B be a Galois algebra with Galois group G, Jg = {b ∈ B|bx = g(x)b for all x ∈ B} for each g ∈ G, eg the central idempotent such that BJg = Beg, and for a subgroup K of G. Then BeK is a Galois extension with the Galois group G(eK)( = {g ∈ G | g(eK) = eK}) containing K and the normalizer N(K) of K in G. An equivalence condition is also given for G(eK) = N(K), and BeG is shown to be a direct sum of all Bei generated by a minimal idempotent ei. Moreover, a characterization for a Galois extension B is shown in terms of the Galois extension BeG and B(1 − eG).

The Galois algebras and the Azumay Galois extensions

Let B be a Galois algebra over a commutative ring R with Galois group G, C the center of B, K = {g ∈ G | g(c) = c for all c ∈ C}, Jg{b ∈ B | bx = g(x)b for all x ∈ B} for each g ∈ K, and BK = (⊕∑g∈K Jg). Then BK is a central weakly Galois algebra with Galois group induced by K. Moreover, an Azumaya Galois extension B with Galois group K is characterized by using BK.

Galois Algebras and Monoidal Functors between Categories of Representations of Finite Groups

We use relations between Galois algebras and monoidal functors to describe monoidal functors between categories of representations of finite groups. We pay special attention to two kinds of these monoidal functors: monoidal functors to vector spaces and monoidal equivalences between categories of representations. The functors of the second kind induce isomorphisms of character tables. We show that pairs of groups with the same character table obtained in this way are a generalization of the construction proposed by B. Fischer (1988, Rend. Circ. Mat. Palermo (2) Suppl.19, 71–77).

The Generalized Center Galois Extensions of Rings

Let B be a ring with 1, C the center of B, G a finite automorphism group of B, and Ii = {c - gi(c) | c ∈ C} for each gi ∈ G. Then, B is called a center Galois extension with Galois group G if BIi = B for each gi ≠ 1 in G, and a weak center Galois extension with group G if BIi = Bei for some nonzero idempotent ei in C for each gi ≠ 1 in G. When ei is a minimal element in the Boolean algebra generated by {ei | gi ∈ G} Bei is a center Galois extension with Galois group Hi for some subgroup Hi of G. Moreover, the central Galois algebra B(1 – ei) is characterized when B is a Galois algebra with Galois group G.

The Structure of Galois Algebras

Let B be a ring with 1 and G an automorphism group of B of order n for some integer n. It is shown that if B is a Galois algebra with Galois group G, then B is either a direct sum of central Galois algebras or a direct sum of central Galois algebras and a commutative Galois algebra. Moreover, when G is inner, B is either a direct sum of Azumaya projective group algebras or a direct sum of Azumaya projective group algebras and a commutative Galois algebra. Examples are given for these structures.

The Boolean algebra and central Galois algebras

Let B be a Galois algebra with Galois group G, Jg = {b ∈ B | bx = g(x)b for all x ∈ B} for g ∈ G, and BJg = Beg for a central idempotent eg. Then a relation is given between the set of elements in the Boolean algebra (Ba, ≤) generated by {0, eg | g ∈ G} and a set of subgroups of G, and a central Galois algebra Be with a Galois subgroup of G is characterized for an e ∈ Ba.

On weak center Galois extensions of rings

Let B be a ring with 1, C the center of B, G a finite automorphism group of B, and BG the set of elements in B fixed under each element in G. Then, the notion of a center Galois extension of BG with Galois group G (i.e., C is a Galois algebra over CG with Galois group G|C≅G) is generalized to a weak center Galois extension with group G, where B is called a weak center Galois extension with group G if BIi = Bei for some idempotent in C and Ii = {c − gi(c) | c ∈ C} for each gi ≠ 1 in G. It is shown that B is a weak center Galois extension with group G if and only if for each gi ≠ 1 in G there exists an idempotent ei in C and {bkei ∈ Bei; ckei ∈ Cei, k = 1, 2, …, m} such that and gi restricted to C(1 − ei) is an identity, and a structure of a weak center Galois extension with group G is also given.

On Azumaya algebras with a finite automorphism group

Let B be a ring with 1, C the center of B, and G a finite automorphism group of B. It is shown that if B is an Azumaya algebra such that B = ⊕∑g∈GJg where Jg = {b ∈ B|bx = g(x)b for all x ∈ B}, then there exist orthogonal central idempotents {fi ∈ C|i = 1, 2, …, m for some integer m} and subgroups Hi of G such that where Bfi is a central Galois algebra with Galois group for each i = 1, 2, …, m and D is contained in C.

The general Ikehata theorem for H‐separable crossed products

Let B be a ring with 1, C the center of B, G an automorphism group of B of order n for some integer n, CG the set of elements in C fixed under G, Δ = Δ(B, G, f) a crossed product over B where f is a factor set from G × G to U(CG). It is shown that Δ is an H‐separable extension of B and VΔ(B) is a commutative subring of Δ if and only if C is a Galois algebra over CG with Galois group G|C≅G.

On characterizations of a center Galois extension

Let B be a ring with 1, C the center of B, G a finite automorphism group of B, and BG the set of elements in B fixed under each element in G. Then, it is shown that B is a center Galois extension of BG (that is, C is a Galois algebra over CG with Galois group G|C≅G) if and only if the ideal of B generated by {c − g(c) | c ∈ C} is B for each g ≠ 1 in G. This generalizes the well known characterization of a commutative Galois extension C that C is a Galois extension of CG with Galois group G if and only if the ideal generated by {c − g(c) | c ∈ C} is C for each g ≠ 1 in G. Some more characterizations of a center Galois extension B are also given.

On central commutator Galois extensions of rings

LetBbe a ring with1,Ga finite automorphism group ofBof ordernfor some integern,BGthe set of elements inBfixed under each element inG, andΔ=VB(BG)the commutator subring ofBGinB. Then the type of central commutator Galois extensions is studied. This type includes the types of Azumaya Galois extensions and GaloisH-separable extensions. Several characterizations of a central commutator Galois extension are given. Moreover, it is shown that whenGis inner,Bis a central commutator Galois extension ofBGif and only ifBis anH-separable projective group ringBGGf. This generalizes the structure theorem for central Galois algebras with an inner Galois group proved by DeMeyer.