Let G be a finite group and D a division algebra faithfully G-graded, finite dimensional over its center K, where char(K)=0. Let e∈G denote the identity element and suppose K0=K∩De, the e-center of ...D, contains ζnG, a primitive nG-th root of unity, where nG 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),μnH)G/H. Here μnH denotes the group of nH-th roots of unity, nH=exp(H), and M(H) is the Schur multiplier of H. The action of G/H on μnH 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),μnH)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 μnG.
Let K be a complete discretely valued field of rank one, with residue field Qp. It is well known that period equals index in Br(K). We prove that when p=2 there exist noncyclic K-division algebras of ...every 2-power degree divisible by four. Otherwise, every K-division algebra is cyclic. This gives the first published example of a field whose Brauer dimension and cyclic length are not equal.
We describe the construction of a specific class of disconnected locally compact near-fields. They are so-called Dickson near-fields and derived from
p
-adic division algebras by means of a special ...kind of homomorphisms or antihomomorphisms from the multiplicative group into the group of inner automorphisms of the division algebra. So let
F
be a local field and
D
be a finite-dimensional central division algebra over
F
. We presuppose that
D
/
F
is tamely ramified. In the first part of this paper we determine all finite subgroups of
D
∗
/
F
∗
. Based on that, we then determine all homomorphic and antihomomorphic couplings
D
∗
→
Inn
(
D
)
=
D
∗
/
F
∗
with finite image. With each of these couplings a locally compact near-field can be constructed from
D
. Apart from isomorphism, there is only a finite number of them. Compared to a previous publication, we omit the assumption that the image of the couplings is an Abelian group.
The theory of Moufang sets essentially deals with groups having a split BN-pair of rank one. Every quadratic Jordan division algebra gives rise to a Moufang set such that its root groups are abelian ...and a certain condition called special is satisfied. It is a major open question if also the converse is true, i.e. if every special Moufang set with abelian root groups comes from a quadratic Jordan division algebra. De Medts and Segev Amer. Math. Soc. 360 (2008), pp. 5831–5852 proved in Theorem 5.11 that this is the case for special Moufang set satisfying two conditions. In this paper we prove that these conditions are in fact equivalent and hence either of them suffices. Even more, we can replace them by weaker conditions.
Real Nullstellensatz is a classical result from Real Algebraic Geometry. It has recently been extended to quaternionic polynomials by Alon and Paran 1. The aim of this paper is to extend their ...Quaternionic Nullstellensatz to matrix polynomials. We also obtain an improvement of the Real Nullstellensatz for matrix polynomials from 4 in the sense that we simplify the definition of a real left ideal. We use the methods from the proof of the matrix version of Hilbert's Nullstellensatz 5 and we obtain their extensions to a mildly non-commutative case and to the real case.
Functional identity on division algebras Ferreira, Bruno Leonardo Macedo; Dantas, Alex Carrazedo; Moraes, Gabriela C.
Bollettino della Unione matematica italiana (2008),
12/2023
Journal Article
Division algebras of slice functions Ghiloni, Riccardo; Perotti, Alessandro; Stoppato, Caterina
Proceedings of the Royal Society of Edinburgh. Section A. Mathematics,
08/2020, Volume:
150, Issue:
4
Journal Article
Peer reviewed
Open access
This work studies slice functions over finite-dimensional division algebras. Their zero sets are studied in detail along with their multiplicative inverses, for which some unexpected phenomena are ...discovered. The results are applied to prove some useful properties of the subclass of slice regular functions, previously known only over quaternions. Firstly, they are applied to derive from the maximum modulus principle a version of the minimum modulus principle, which is in turn applied to prove the open mapping theorem. Secondly, they are applied to prove, in the context of the classification of singularities, the counterpart of the Casorati-Weierstrass theorem.
On generic G-graded Azumaya algebras Aljadeff, Eli; Karasik, Yakov
Advances in mathematics (New York. 1965),
04/2022, Volume:
399
Journal Article
Peer reviewed
Open access
Let F be an algebraically closed field of characteristic zero and let G be a finite group. Consider G-graded simple algebras A which are finite dimensional and e-central over F, i.e. ...Z(A)e:=Z(A)∩Ae=F. For any such algebra we construct a generic G-graded algebra U which is Azumaya in the following sense. (1) (Correspondence of ideals): There is one to one correspondence between the G-graded ideals of U and the ideals of the ring R, the e-center of U. (2) Artin-Procesi condition: U satisfies the G-graded identities of A and no nonzero G-graded homomorphic image of U satisfies properly more identities. (3) Generic: If B is a G-graded algebra over a field then it is a specialization of U along an ideal a∈spec(Z(U)e) if and only if it is a G-graded form of A over its e-center.
We apply this to characterize finite dimensional G-graded simple algebras over F that admit a G-graded division algebra form over their e-center.
Let
A
A
be a central division algebra of prime degree
p
p
over
Q
\mathbb {Q}
. We obtain subconvex hybrid bounds, uniform in both the eigenvalue and the discriminant, for the sup-norm of Hecke-Maaß ...forms on the compact quotients of
SL
p
(
R
)
/
SO
(
p
)
\operatorname {SL}_p(\mathbb {R})/\operatorname {SO}(p)
by unit groups of orders in
A
A
. The exponents in the bounds are explicit and polynomial in
p
p
. We also prove subconvex hybrid bounds in the case of certain Eichler-type orders in division algebras of arbitrary odd degree.
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