Non-cyclic algebras with 𝑛-central elements

Journal of Algebra, 2017

For a maximal separable subfield K of a central simple algebra A, we provide a semiring isomorphism between K-K-bimodules A and H-H bisets of G = Gal(L/F ), where F = Z(A), L is the Galois closure of K/F , and H = Gal(L/K). This leads to a combinatorial interpretation of the growth of dim K ((KaK) i ), for fixed a ∈ A, especially in terms of Kummer sets.

Algebras and Representation Theory

We develop a comprehensive theory of algebras over a field which are locally both finite dimensional and central simple. We generalize fundamental concepts of the theory of finite dimensional central simple algebras, and introduce supernatural matrix algebras, the supernatural degree and matrix degree, and so on. We define a Brauer monoid, whose unique maximal subgroup is the classical Brauer group, and show that once infinite dimensional division algebras exist over the field, they are abundant.

Journal of Number Theory, 2019

In a quaternion order of class number one, an element can be factored in multiple ways depending on the order of the factorization of its reduced norm. The fact that multiplication is not commutative causes an element to induce a permutation on the set of primes of a given reduced norm. We discuss this permutation and previously known results about the cycle structure, sign, and number of fixed points for quaternion orders. We generalize these results to other orders in central simple algebras over global fields.

Analyzing square-central elements in central simple algebras of degree 4, we show that every two elementary abelian Galois maximal subfields are connected by a chain of nontrivially-intersecting pairs. Similar results are proved for non-central quaternion subalgebras, and for central quaternion subalgebras when they exist. Along these lines we classify the maximal square-central subspaces. We also show that every two standard quadruples of generators of a biquaternion algebra are connected by a chain of basic steps, in each of which at most two generators are being changed.

arXiv: Rings and Algebras, 2020

This paper shows that $p$ primary components of certain generic crossed products are not crossed products. This applies in particular to primary components of prime degree, thus producing examples of division algebras of prime degree that are not cyclic.

2020

This paper is primarily concerned with studying finite-dimensional anti-commutative nonassociative algebras in which every centralizer is an ideal. These are shown to be anti-associative and are classified over a general field $F$; in particular, they are nilpotent of class at most $3$ and metabelian. These results are then applied to show that a Leibniz algebra over a field of charactersitic zero in which all centralizers are ideals is solvable.

Manuscripta Mathematica, 2013

Given a central simple algebra A of degree 3 over a field of characteristic 3, we prove that there is a unique symmetric composition algebra extending the commutator operation on the trace-zero part modulo scalars. This is analogous to Okubo's construction of symmetric composition algebras in the case of characteristic not 3. We apply the composition algebra tools to obtain a classification of maximal 3-central spaces and maximal Galois hyperplanes of A, and prove a new common slot lemma for such algebras.

Transactions of the American Mathematical Society, 1958

RICHARD BLOCK 1. Introduction. For a number of years, the only known simple Lie algepras of characteristic p>0 not analogues of the classical algebras of characteristic zero were those of dimension pn of H. Zassenhaus , and of dimension np" of N. Jacobson , both generalizations of the ^-dimensional algebras of E. Witt. During the last few years, a number of new classes of simple Lie algebras have been determined. These are due to Kaplansky, who in noted the existence of a class of algebras of dimension rpn, 1 gr g w, generalizing those of Zassenhaus and Jacobson; to M. S. Frank, who in [2]obtained algebras ©" of dimension (n -l)(pn -l); and to A. A. Albert and M. S. Frank, who in [l ] determined new classes of simple algebras Xn, %$m, So, and Sj, of dimensions (n -l)pn, p2m -2, pn -l and pn -2 respectively, and a generalization of the Zassenhaus algebras, of dimension pn. In this paper we obtain a new class of simple Lie algebras, which we shall designate by the symbol £(®, 5, /). Here ® is an additive group, of order pn> 1, which is a direct sum of a finite number of finite elementary ^-groups ®oi • • • , ®m-These groups are then finite dimensional vector spaces over the prime field %p. We let m(@) denote the index m, allow ®0 to be zero, and assume that either p>2 or ® ?^®i. For i = l, • • • , m, we take 5< to be a nonzero element of ®<, and write 5=5i+ • • • +Sm. We index a basis of 8(®, 5,/) over a field $ of characteristic p by the elements of ® other than 0 and -S, denoting by v(a) the basis element corresponding to a. Furthermore we assume given, for each i, a nondegenerate skew-symmetric biadditive function /,-on (®,-, ®j) to §, such that, for i = l, • ■ • , m, there are additive functions gu hi, on ®i to %, with f,-(5.) =0 and/<(a, 8) = gi(°t)hi(0) -gi(0)h{(a) for every a and 8 in ®<-The definition of the algebra 8(®, S,f) may then be completed by defining multiplication in the algebra by v(a)v(8) = JZ?-ofi(ai, 0d ■v(a+8 -8i), where a< and 8i are the components of a and 0 in ®<, and where 50 and v(0) denote zero. Then we shall prove that 8(®, 5,/) is a central simple Lie algebra. Its dimension is pn -2 except when ® = ®0, in which case the

Pacific Journal of Mathematics, 1975

2004

Let p be a prime and suppose that K/F is a cyclic extension of degree p with group G. Let J be the FpG-module K/K of pth-power classes. In our previous paper we established precise conditions for J to contain an indecomposable direct summand of dimension not a power of p. At most one such summand exists, and its dimension must be p + 1 for some 0 ≤ i &lt; n. We show that for all primes p and all 0 ≤ i &lt; n, there exists a field extension K/F with a summand of dimension p + 1. Let p be a prime and K/F a cyclic extension of fields of degree p with Galois group G. Let K be the multiplicative group of nonzero elements of K and J = J(K/F ) := K/K be the group of pthpower classes of K. We see that J is naturally an FpG-module. In our previous paper [MSS] we established the decomposition of J into indecomposables, as follows. For i ∈ N let ξpi denote a primitive pth root of unity, and for 0 ≤ i ≤ n let Ki/F be the subextension of degree p, with Gi = Gal(Ki/F ). We adopt the convention th...