Lattices and Parameter Reduction in Division Algebras

Transactions of the American Mathematical Society, 1994

Given a family of separable finite dimensional extensions {L,} of a field A:, we construct a division algebra n2 over its center which is freely generated over k by the fields {L,} .

Труды математического института им. Стеклова, 2016

Let D be a central division algebra of degree n over a field K. One defines the genus gen(D) as the set of classes [D ′ ] ∈ Br(K) in the Brauer group of K represented by central division algebras D ′ of degree n over K having the same maximal subfields as D. We prove that if the field K is finitely generated and n is prime to its characteristic then gen(D) is finite, and give explicit estimations of its size in certain situations.

manuscripta mathematica, 2021

We study the partial ordering on isomorphism classes of central simple algebras over a given field F, defined by setting A 1 ≤ A 2 if deg A 1 = deg A 2 and every étale subalgebra of A 1 is isomorphic to a subalgebra of A 2 , and generalizations of this notion to algebras with involution. In particular, we show that this partial ordering is invariant under passing to the completion of the base field with respect to a discrete valuation, and we explore how this partial ordering relates to the exponents of algebras.

manuscripta mathematica, 2010

We show that if a field K of characteristic = 2 satisfies the following property (*) for any two central quaternion division algebras D1 and D2 over K, the fact that D1 and D2 have the same maximal subfields implies that D1 ≃ D2 over K, then the field of rational functions K(x) also satisfies (*). This, in particular, provides an alternative proof for the result of S. Garibaldi and D. Saltman that the fields of rational functions k(x1,. .. , xr), where k is a number field, satisfy (*). We also show that K = k(x1,. .. , xr), where k is either a totally complex number field with a single diadic place (e.g. k = Q(√ −1)) or a finite field of characteristic = 2, satisfies the analog of (*) for all central division algebras having exponent two in the Brauer group Br(K).

Given a central division algebra $D$ of degree $d$ over a field $F$, we associate to any standard polynomial $\phi(z)=z^n+c_{n-1} z^{n-1}+\dots+c_0$ over $D$ a ``companion polynomial" $\Phi(z)$ of degree $n d$ with coefficients in $F$ whose roots are exactly the conjugacy classes of the roots of $\phi(z)$. We explain how in case $D$ is a quaternion algebra, all the roots of $\phi(z)$ can be recovered from the roots of $\Phi(z)$. On the way, we also generalize certain theorems that were known for $\mathbb{H}$ to any division algebra, such as the connection between the right eigenvalues of a matrix and the roots of its characteristic polynomial, and the connection between the roots of a standard polynomial and left eigenvalues of the companion matrix.

Journal of Pure and Applied Algebra, 2013

Let D be a division ring. We say that D is left algebraic over a (not necessarily central) subfield K of D if every x ∈ D satisfies a polynomial equation x n + α n−1 x n−1 + · · · + α 0 = 0 with α 0 , . . . , α n−1 ∈ K. We show that if D is a division ring that is left algebraic over a subfield K of bounded degree d then D is at most d 2 -dimensional over its center. This generalizes a result of Kaplansky. For the proof we give a new version of the combinatorial theorem of Shirshov that sufficiently long words over a finite alphabet contain either a q-decomposable subword or a high power of a non-trivial subword. We show that if the word does not contain high powers then the factors in the q-decomposition may be chosen to be of almost the same length. We conclude by giving a list of problems for algebras that are left algebraic over a commutative subring.

Communications in Algebra, 2005

Let D be a cyclic division algebra over its centre F of index n. Consider the group CK1(D) = D * /F * D where D * is the group of invertible elements of D and D is its commutator subgroup. In this note we shall show that the group CK1(D) is trivial if and only if D is an ordinary quaternion division algebra over a real Pythagorean field F. This in particular shows that if the index of D is an odd prime p, then the exponent of CK1 is p. We show that the converse does not hold by exhibiting a division algebra D and a division subalgebra A ⊂ D such that CK1(A) ∼ = CK1(D). Using valuation theory, the group CK1(D) is computed for some valued division algebras.

Transactions of the American Mathematical Society, 2004

Let G be a finite group, let M be a ZG-lattice, and let F be a field of characteristic zero containing primitive p th roots of 1. Let F (M ) be the quotient field of the group algebra of the abelian group M . It is well known that if M is quasi-permutation and G-faithful, then F (M ) G is stably equivalent to F (ZG) G . Let Cn be the center of the division ring of n × n generic matrices over F . Let Sn be the symmetric group on n symbols. Let p be a prime. We show that there exist a split group extension G of Sp by a p-elementary group, a G -faithful quasi-permutation ZG -lattice M , and a one-cocycle α in Ext 1 G (M, F * ) such that Cp is stably isomorphic to Fα(M ) G . This represents a reduction of the problem since we have a quasi-permutation action; however, the twist introduces a new level of complexity. The second result, which is a consequence of the first, is that, if F is algebraically closed, there is a group extension E of Sp by an abelian p-group such that Cp is stably equivalent to the invariants of the Noether setting F (E).

We establish the absence of zero divisors in the reduction algebra of a Lie algebra g with respect to its reductive Lie sub-algebra k. The class of reduction algebras include the Lie algebras (they arise when k is trivial) and the Gelfand–Kirillov conjecture extends naturally to the reduction algebras. We formulate the conjecture for the diagonal reduction algebras of sl type and verify it on a simplest example. 1 Preliminaries Let k be a reductive Lie subalgebra of a Lie algebra g; that is, the adjoint action of k on g is completely reducible (in particular, k is reductive). Fix a triangular decomposition of the Lie algebra k, k = n − + h + n +. (1.1) Denote by ∆ + and ∆ − the sets of positive and negative roots in the root system ∆ = ∆ + ∪ ∆ − of k. For each root α ∈ ∆ let h α = α ∨ ∈ h be the corresponding coroot vector. Denote by U(h) the ring of fractions of the commutative algebra U(h) relative to the set of denominators { h α + l | α ∈ ∆, l ∈ Z }. (1.2) The elements of this ring can also be regarded as rational functions on the vector space h *. The elements of U(h) ⊂ U(h) are then regarded as polynomial functions on h *. Let U(k) ⊂ ¯ A = U(g) be the rings of fractions of the algebras U(k) and A = U(g) relative to the set of denominators (1.2). These rings are well defined, because both U(k) and U(g) satisfy the Ore condition relative to (1.2); we give a short proof in the second part of Appendix. Define Z(g, k) to be the double coset space of ¯ A by its left ideal I + := ¯ An + , generated by elements of n + , and the right ideal I − := n − ¯ A, generated by elements of n − , Z(g, k) := 1 On leave of absence from P.N. Lebedev Physical Institute, Theoretical Department, Leninsky prospekt 53, 119991 Moscow, Russia 2 Unité Mixte de Recherche (UMR 6207) du CNRS et des Universités Aix–Marseille I, Aix–Marseille II et du Sud Toulon – Var; laboratoire affiliéà la FRUMAM (FR 2291)

Journal of Algebra, 2021

Let G be a finite group and D a division algebra faithfully Ggraded, finite dimensional over its center K, where char(K) = 0. Let e ∈ G denote the identity element and suppose K 0 = K ∩ De, the e-center of D, contains ζn G , a primitive n G-th root of unity, where n G is the exponent of G. To such a G-grading on D we associate a normal abelian subgroup H of G, a positive integer d and an element of Hom(M (H), µn H) G/H. Here µn H denotes the group of n H-th roots of unity, n H = exp(H), and M (H) is the Schur multiplier of H. The action of G/H on µn H is trivial and the action on M (H) is induced by the action of G on H. Our main theorem is the converse: Given an extension 1 → H → G → Q → 1, where H is abelian, a positive integer d, and an element of Hom(M (H), µn H) Q , there is a division algebra as above that realizes these data. We apply this result to classify the G-graded simple algebras whose e-center is an algebraically closed field of characteristic zero that admit a division algebra form whose e-center contains µn G .

Bulletin of the Australian Mathematical Society, 1980

Mathematical Proceedings of the Royal Irish Academy, 2014

Exactly 170 years ago, the construction of the real quaternion algebra by William Hamilton was announced in the Proceedings of the Royal Irish Academy. It became the first example of non-commutative division rings and a major turning point of algebra. To this day, the multiplicative group structure of quaternion algebras have not completely been understood. This article is a long survey of the recent developments on the multiplicative group structure of division rings. (2) Since D is totally ramified, clearly we have D * /F * D ′ = 1. So by Proposition 6.4 we obtain CK 1 (D) ∼ = Γ D /Γ F . Also since in this case D ′ = µ e (F ) from Corollary 6.5 we conclude that SK 1 (D) ∼ = µ n (F )/µ e (F ). (3) From Proposition 6.4 it is enough to prove To show this, consider the norm function N D/F : D * → F * . Now, if x ∈ U D then by Ershov's formula it follows that Nrd D (x) = N D/F (x). This shows that D ′ ⊆ ker N D/F . Conversely, if x ∈ ker N D/F then by Hilbert Theorem 90, there is a b such that x = bσ(b) -1 where σ is the generator of Gal(D/F ). Now, since the fundamental homomorphism D * → Gal(Z(D/F )) is surjective, it follows that σ is of the form σ(a) = cac -1 , for some c ∈ D * . This implies that x ∈ D ′ and so ker N D/F = D ′ . Therefore Here, we are going to use the above results in computing of CK 1 and SK 1 for some division algebras. Example 6.9. Recall that a field F is called real Pythagorean if -1 ∈ F * 2 and the sum of any two square elements of F is a square in F . Let F be a real Pythagorean field and Q be the ordinary quaternion algebra over F , i.e.,

International Journal of Algebra

New simpler statements of the known theorems of Frobenius and Zorn via identities of the form (x 2 , y 2 , z 2) = 0 and (x 2 , y 2 , y 2) = (y 2 , y 2 , x 2) = 0 are given. Also, the 124 B. Aharmim, O. Fayz, E. Idnarour and A. Rochdi identity (x 2 , x 2 , x 2) = 0 in a real division algebra with a non-zero central element forces the power-commutativity.

Israel Journal of Mathematics, 2020

We prove that the genus of a finite-dimensional division algebra is finite whenever the center is a finitely generated field of any characteristic. We also discuss potential applications of our method to other problems, including the finiteness of the genus of simple algebraic groups of type G2. These applications involve the double cosets of adele groups of algebraic groups over arbitrary finitely generated fields: while over number fields these double cosets are associated with the class numbers of algebraic groups and hence have been actively analyzed, similar questions over more general fields seem to come up for the first time. In the Appendix, we link the double cosets withČech cohomology and indicate connections between certain finiteness properties involving double cosets (Condition (T)) and Bass's finiteness conjecture in K-theory. To Louis Rowen on the occasion of his retirement