Finite-dimensional Central Division Algebra Research Articles

A non-commutative Nullstellensatz

Let [Formula: see text] be a field and [Formula: see text] be a finite-dimensional central division algebra over [Formula: see text]. We prove a variant of the Nullstellensatz for [Formula: see text]-sided ideals in the ring of polynomial maps [Formula: see text]. In the case where [Formula: see text] is commutative, our main result reduces to the [Formula: see text]-Nullstellensatz of Laksov and Adkins–Gianni–Tognoli. In the case, where [Formula: see text] is the field of real numbers and [Formula: see text] is the algebra of Hamilton quaternions, it reduces to the quaternionic Nullstellensatz recently proved by Alon and Paran.

On genus of division algebras

The genus $gen(D)$ of a finite-dimensional central division algebra $D$ over a field $F$ is defined as the collection of classes $[D']\in Br(F)$, where $D'$ is a central division $F$-algebra having the same maximal subfields as $D$. We show that the fact that quaternion division algebras $D$ and $D'$ have the same maximal subfields does not imply that the matrix algebras $M_l(D)$ and $M_l(D')$ have the same maximal subfields for $l>1$. Moreover, for any odd $n>1$, we construct a field $L$ such that there are two quaternion division $L$-algebras $D$ and $D'$ and a central division $L$-algebra $C$ of degree and exponent $n$ such that $gen(D) = gen(D')$ but $gen(D \otimes C) \ne gen(D' \otimes C)$.

A note on Lie ideals of prime rings

In this paper, we characterize Lie ideals, which are either finitely generated [Formula: see text]-modules or maximal, in (centrally closed) prime rings. As consequences, we extend the results proved in [1] for finite dimensional central division algebras of characteristic not [Formula: see text] to simple rings of arbitrary characteristic.

Nonassociative Differential Extensions of Characteristic varvec{p}

Let F be a field of characteristic p. We define and investigate nonassociative differential extensions of F and of a finite-dimensional central division algebra over F and give a criterium for these algebras to be division. As special cases, we obtain classical results for associative algebras by Amitsur and Jacobson. We construct families of nonassociative division algebras which can be viewed as generalizations of associative cyclic extensions of a purely inseparable field extension of exponent one or a central division algebra. Division algebras which are nonassociative cyclic extensions of a purely inseparable field extension of exponent one are particularly easy to obtain.

Sliceable groups and towers of fields

Abstract Let l be a prime number, K a finite extension of ℚ l , and D a finite-dimensional central division algebra over K. We prove that the profinite group G = D × / K × ${G=D^\times /K^\times }$ is finitely sliceable, i.e. G has finitely many closed subgroups H 1,...,Hn of infinite index such that G = ⋃ i = 1 n H i G ${G=\bigcup _{i=1}^nH_i^G}$ . Here, H i G = { h g ∣ h ∈ H i , g ∈ G } ${H_i^G=\lbrace h^g\mid h\in H_i, \, g\in G\rbrace }$ . On the other hand, we prove for l ≠ 2 that no open subgroup of GL2(ℤ l ) is finitely sliceable and we give an arithmetic interpretation to this result, based on the possibility of realizing GL2(ℤ l ) as a Galois group over ℚ. Nevertheless, we prove that G = GL2(ℤ l ) has an infinite slicing, that is G = ⋃ i = 1 ∞ H i G ${G=\bigcup _{i=1}^\infty H_i^G}$ , where each Hi is a closed subgroup of G of infinite index and Hi ∩ Hj has infinite index in both Hi and Hj if i ≠ j.

Division algebras of prime degree with infinite genus

The genus gen(D) of a finite-dimensional central division algebra D over a field F is defined as the collection of classes [D′] ∈ Br(F), where D′ is a central division F-algebra having the same maximal subfields as D. For any prime p, we construct a division algebra of degree p with infinite genus. Moreover, we show that there exists a field K such that there are infinitely many nonisomorphic central division K-algebras of degree p and any two such algebras have the same genus.

Noncrossed product bounds over Henselian fields

The existence of finite dimensional central division algebras with no maximal subfield that is Galois over the center (called noncrossed products), was for a time the biggest open problem in the theory of division algebras, before it was settled by Amitsur. Motivated by Brussel's discovery of noncrossed products over Q((t)), we describe the "location" of noncrossed products in the Brauer group of general Henselian valued fields with arbitrary value group and global residue field. We show that within the fibers defined canonically by Witt's decomposition of the Brauer group of such fields, crossed products and noncrossed products are, roughly speaking, separated by an index bound. This generalizes a result of the first and third author for rank 1 valued Henselian fields. Furthermore, we prove that all fibers which are not covered by the rank 1 case, and where the characteristic of the residue field does not interfere, contain noncrossed products. We show by example that, unlike in the rank 1 case, the value of the index bound does not depend on the number of roots of unity that are present. Thus, the index bounds are in general of a different nature than in the rank 1 case.

Irreducibility criterion for representations induced by essentially unitary ones (case of non-archimedean GL(n,𝒜))

The fundamental representations for the description of the unitary dual of general linear groups over a local field F are the Speh representations, which are denoted by u(d, m). Let A be a finite dimensional central division algebra over F. One can define in a similar way representations u(d, m) for general linear groups over A, and they play a similar fundamental role for the unitary duals of these groups. In the non-archimedean case, for two such representations u(d, m) and u(d', m'), and for real numbers a and b, we obtain a necessary and sufficient condition that the representation parabolically induced by the tensor product of |det|^a u(d, m) and |\det|^b u(d', m') is irreducible. Our motivation for working on this problem comes from our work on unitary duals of classical (symplectic, orthogonal and unitary) groups, related to constructions of complementary series for these groups.

On abstract representations of the groups of rational points of algebraic groups and their deformations

In this paper, we continue our study, begun in an earlier paper, of abstract representations of elementary subgroups of Chevalley groups of rank ≥2. First, we extend the methods to analyze representations of elementary groups over arbitrary associative rings and, as a consequence, prove the conjecture of Borel and Tits on abstract homomorphisms of the groups of rational points of algebraic groups for groups of the form SLn,D, where D is a finite-dimensional central division algebra over a field of characteristic 0. Second, we apply the previous results to study deformations of representations of elementary subgroups of universal Chevalley groups of rank ≥2 over finitely generated commutative rings.

Reduced Whitehead Groups and the Conjugacy Problem for Special Unitary Groups of Anisotropic Hermitian Forms

Let K/k be a separable field extension of degree 2, D be a finite-dimensional central division algebra over K with a K/k-involution τ, h be an Hermitian anisotropic form on a right D-vector space with respect to τ, and let U(h) be the unitary group of h. Then the reduced Whitehead group of its special linear subgroup is defined as follows: $$ \mathrm{SUK}_1^{\mathrm{an}}(h)={{{\mathrm{SU}(h)}} \left/ {{\left[ {U(h),U(h)} \right]}} \right.} $$ , where [U(h), U(h)] is the commutator subgroup of U(h). The first main result establishes a link between the above group and its analog SUK1(h) for the case of isotropic h (with respect to the same τ). Theorem. There exists a surjective homomorphism from $$ \mathrm{SUK}_1^{\mathrm{an}}(h) $$ to SUK1(h). Furthermore, we also give a solution of the conjugacy problem for special unitary subgroups of anisotropic Hermitian forms over quaternion division algebras as subgroups of their multiplicative groups. Bibliography: 32 titles.

The genus of a division algebra and the unramified Brauer group

Let \(D\) be a finite-dimensional central division algebra over a field \(K\). We define the genus \(\mathbf{gen}(D)\) of \(D\) to be the collection of classes \([D^{\prime }] \in \mathrm{Br}(K)\), where \(D^{\prime }\) is a central division \(K\)-algebra having the same maximal subfields as \(D\). In this paper, we describe a general approach to proving the finiteness of \(\mathbf{gen}(D)\) and estimating its size that involves the unramified Brauer group with respect to an appropriate set of discrete valuations of \(K\). This approach is then implemented in some concrete situations, yielding in particular an extension of the Stability Theorem of A. Rapinchuk and I. Rapinchuk (Manuscr. Math. 132:273–293, 2010) from quaternion algebras to arbitrary algebras of exponent two. We also consider an example where the size of the genus can be estimated explicitly. Finally, we offer two generalizations of the genus problem for division algebras: one deals with absolutely almost simple algebraic \(K\)-groups having the same isomorphism/isogeny classes of maximal \(K\)-tori, and the other with the analysis of weakly commensurable Zariski-dense subgroups.

On the genus of a division algebra

We define the genus gen(D) of a finite-dimensional central division algebra D over a field K as the set of all classes [D′] in the Brauer group Br(K) that are represented by central division K-algebras D′ having the same maximal subfields as D. We give examples where gen(D) is reduced to a single element, and other examples where it is finite.

The relative Brauer group of a cyclic cover of affine space

We study the finite-dimensional central division algebras over the rational function field in several variables over an algebraically closed field. We describe the division algebras that are split by the cyclic covering obtained by adjoining the n th root of a polynomial. The relative Brauer group is described in terms of the Picard group of the cyclic covering and its Galois group. Many examples are given and in most cases division algebras are presented that represent generators of the relative Brauer group.

Proof of the Tadić conjecture (U0) on the unitary dual of GL m (D)

Let F be a non-Archimedean local field of characteristic 0, and let D be a finite-dimensional central division algebra over F. We prove that any unitary irreducible smooth representation of a Levi subgroup of GLm(D), with m ≧ 1, induces irreducibly to GLm(D). This ends the classification of the unitary dual of GLm(D) initiated by Tadić.

Division algebras with an anti-automorphism but with no involution

In this note we give examples of division rings which posses an anti-automorphism but no involution. The motivation for such examples comes from geometry. If D is a division ring and V a finite-dimensional right D-vector space of dimension ≥ 3, then the projective geometry P(V ) has a duality (resp. polarity) if and only if D has an anti-automorphism (resp. involution) [2, p. 97, p. 111]. Thus, the existence of a division ring with an anti-automorphism but no involution gives examples of projective geometries with dualities but no polarities. All the division rings considered in this paper are finite-dimensional algebras over their center. The anti-automorphisms we construct are not linear over the center (i.e. they do not restrict to the identity on the center) since a theorem of Albert [1, Theorem 10.19] shows that every finite-dimensional central division algebra with a linear anti-automorphism has an involution. Several proofs of this result can be found in the literature, see for instance [5] or [10, Chapter 8, §8]. The paper is organized as follows. Section 2 collects some background information on central division algebras. Section 3 establishes the existence of division algebras over algebraic number fields which have anti-automorphisms but no involution, and gives two explicit constructions of such algebras. The proofs rely on deep classical results on the Brauer group of number fields. Section 4 gives examples whose center has a more complicated structure (they are Laurent series fields over local fields), but the proof that these algebras have no involution is more elementary. Sections 3 and 4 are independent of each other.

Nicely semiramified division algebras over Henselian fields

This paper deals with the structure of nicely semiramified valued division algebras. We prove that any defectless finite‐dimensional central division algebra over a Henselian field E with an inertial maximal subfield and a totally ramified maximal subfield (not necessarily of radical type) (resp., split by inertial and totally ramified field extensions of E) is nicely semiramified.

Derivations with algebraic values of bounded degree

Let Rbe a prime algebra over a field .F, d a nonzero derivation of Rand ρ a nonzero right ideal of R. Suppose that for every x∈ ρ,d(x) is algebraic over Fof bounded degree. Then Ris a primitive ring with a minimal right ideal eR, where e=e2 ∈Rand eReis a finite-dimensional central division algebra, except when dis an inner derivation induced by an element a in the two-sided Martindale quotient ring of Rsuch that aρp = 0. An analogous result is also proved for the Lie ideal case.

The minimal polynomial of a matrix*

Let Rbe a finite dimensional central simple algebra over a field F A be any n× n matrix over R. By using the method of matrix representation, this paper obtains the structure formula of the minimal polynomial qA (λ) of A over F. By using qA (λ), this paper discusses the structure of right (left) eigenvalues set of A, and obtains the necessary and sufficient condition that a matrix over a finite dimensional central division algebra is similar to a diagonal matrix.

Levels of division algebras

In [7] the level, sublevel, and product level of finite dimensional central division algebras D over a field F were calculated when F is a local or global field. In Theorem 1.4 of this paper we calculate the same quantities if all finite extensions K of F satisfy ū(K) ≤2, where ū is the Hasse number of a field as defined in [2]. This occurs, for example, if F is an algebraic extension of the function field R(x) where R is a real closed field or hereditarily Euclidean field (see [4]).

Q-admissibility of certain nonsolvable groups

A finite groupG is calledQ-admissible if there exists a finite dimensional central division algebra overQ, containing a maximal subfield which is a Galois extension ofQ with Galois group isomorphic toG. It is proved thatS5−, one of the two nontrivial central extensions ofS5 byZ/2Z, isQ-admissible. As a consequence of that result and previous results of Sonn and Stern, every finite Sylow-metacyclic group, havingA5 as a composition factor, isQ-admissible.