Darrell Haile

Graded identities of matrix algebras and the universal graded algebra

In the last decade, group gradings and graded identities of finite dimensional central simple algebras have been an active area of research. We refer the reader to Bahturin, et al [6] and [7]. There are two basic kinds of group grading, elementary and fine. It was proved by Bahturin and Zaicev [7] that any group grading of Mn(C) is given by a certain composition of an elementary grading and a fine grading. In this paper we are concerned with fine gradings on Mn(C) and their corresponding graded identities. Let R be a simple algebra, finite dimensional over its center k and G a finite group. We say that R is fine graded by G if R ∼= ⊕g∈GRg is a grading and dimk(Rg) ≤ 1. Thus any component is either 0 or isomorphic to k as a k–vector space. It is easy to show that Supp(R), the subset of elements of G for which Rg is not 0, is a subgroup of G. Moreover

ALGEBRAS OF ODD DEGREE WITH INVOLUTION, TRACE FORMS AND DIHEDRAL EXTENSIONS

A 3-fold Pfister form is associated to every involution of the second kind on a central simple algebra of degree 3. This quadratic form is associated to the restriction of the reduced trace quadratic form to the space of symmetric elements; it is shown to classify involutions up to conjugation. Subfields with dihedral Galois group in central simple algebras of arbitrary odd degree with involution of the second kind are investigated. A complete set of cohomological invariants for algebras of degree 3 with involution of the second kind is given.

On Azumaya algebras arising from Clifford algebras

In this paper we continue the investigation begun in Haile [4] and Tesser [12] into the structure of the Clifford algebra of a form. If f is a form of degree d in n variables over a field F, then the Clifford algebra off is the F-algebra C,= R/T where R = Fix,, . . . . x,} is the free associative F-algebra in n variables and T is the ideal generated by {(a,x,+ ..’ + %X,Y-f(~1, ---, cx,,)jcxl ,..., cl,~F}. If d=2 this is the classical Clifford algebra of a quadratic form. If d > 2 then this is also called the “generalized” Clifford algebra and has been studied by various authors (see Roby [ 111, Revoy [lo], Childs [2] ). In Haile [4] it was shown that if d= 3 and n = 2 (i.e., iffis a binary cubic form) and the characteristic of the field F is not 2 or 3, then the Clifford algebra C,of the form is Azymaya with center the affine coordinate ring of an explicitly given elliptic curve over F (the center was also determined by Heerema in [6]). Moreover the algebra C, can be extended to an Azumaya algebra over the projective closure of the elliptic curve. Applying the pairing investigated by Lichtenbaum [4] and Manin [9], one obtains then for each field extension K/F a homomorphism from the group of K-rational points on the elliptic curve to the Brauer group of K, given by specialization. In [lo] Revoy, using results of Roby [ 111, exhibited an explicit F-basis for the Clifford algebra of an arbitrary form. Using this basis it is easy to

Simple

Let G be any group and F an algebraically closed field of characteristic zero. We show that any G-graded finite dimensional associative G-simple algebra over F is determined up to a G-graded isomorphism by its G-graded polynomial identities. This result was proved by Koshlukov and Zaicev in case G is abelian.

G

This paper continues part of the investigation begun in [3], henceforth referred to as HLS, into the structure of the algebras arising from the relaxation of the 2-cocycle condition of Galois cohomology. Specifically let K/F be a finite Galois extension of fields and let G = Gal(K/F). A weak 2-cocycle is a map f: G x G --+ K such that for all u, t, y E G we have f”(r, y) f(o, ry) = f(a, r) f(ar, r) and f( 1, a) = f(a, 1) = 1. The difference between these and the usual 2cocycles is that the weak 2-cocycle is allowed to take on the value zero. Two weak cocycles f and g are called cohomologous, written f N g, if as in the classical theory there is a function a: G -+ KX such that f(o, r> = (a(o) a(~>“/a(~~)>s( u r , ) f or all u, t E G. Associated to a weak cocycle f there is an F-algebra A, defined as follows: For each u E G introduce a one-dimensional left K-space Kx, generated by the indeterminate x,. Let A,= OoeG Kx, as a left K-vector space and multiply subject to the rules x,k = u(k) X, for all u E G, k E K and x,x,=f(u, r) x,,for all u, r E G. The cocycle condition guarantees that A, is an associative F-algebra with unit element x, and center Fx, (which we identify with F). Two cocycles f and g are cohomologous if and only if there is an F-algebra isomorphism

-graded algebras and their polynomial identities

:Af+A, such that &=identity. Let H= {uEGI f(u,u-‘)#O}. It was shown in Section 10 of HLS that H is a subgroup of G, called the inertial subgroup for f; and that A = BoeH Kx, is a central simple KH-algebra. Moreover J = Oo.+* Kx, is the Jacobson radical of A, and so A, = A 0 J (as F-vector spaces) is the Wedderburn splitting of A,. Each “crossed product” algebra A, is isomorphic to K OF K as a left K OF K-module with the canonical action and in fact the algebras arising this way can be characterized as those F-algebras B such that B 1 K and B g K OF K as a K OF K-

On crossed product algebras arising from weak cocycles

Let F be a field and let V be a valuation ring in F. If A is a central simple F-algebra then V can be extended to a Dubrovin valuation ring in A. In this paper we consider the structure of Dubrovin valuation rings with center V in crossed product algebras (K/F, G, f) where K/F is a finite Galois extension with Galois group G unramified over V and f is a normalized two-cocycle. In the case where V is indecomposed in K we introduce a family of orders naturally associated to f, examine their basic properties, and determine which of these orders is Dubrovin. In the case where V is decomposed we determine the structure in the case of certain special discrete, finite rank valuations

On Dubrovin valuation rings in crossed product algebras

Let ƒ(u, v)) be a nondegenerate binary form over a field F of characteristic not two or three. We prove that the Clifford algebra of ƒ(u, v) over F is split if and only if the ternary form w3 − ƒ(u, v) has a nontrivial F-rational point.

When is the Clifford algebra of a binary cubic form split

In this article we introduce an invariant of a field F, the Brauer monoid. The elements of this monoid are based on certain F-central finite dimensional algebras, the “strongly primary” algebras over F, which are more general than central simple F-algebras. Under a suitable equivalence relation and product the classes of these algebras form a monoid, denoted M(F). The group of invertible elements of this monoid is the Brauer group of F. This article is based on ideas first introduced by Sweedler in [6] and later developed and extended by Sweedler, Larsen and the present author in [ 2. 3 1. In these articles a cohomology theory was introduced which generalizes the usual Galois and Amitsur cohomology theories. In Section 2 of this article it is shown that these new cohomology monoids play the same role for the Brauer monoid as the usual cohomology groups play for the Brauer group; that is, a given cohomology monoid M’(Gal(K/F), K). for K a finite Galois extension of F, is isomorphic to the submonoid of elements of M(F) that are split by K in a suitable sense. In the first section we define strongly primary algebras, examine their properties, and construct the monoid. In the second section we show the connection with cohomology. In the third section we begin a dissection of the Brauer monoid of F and show in particular that it is the union of the groups M,(F), where for each idempotent E in M(F) we let M,(F)= (XEM(F)IXE=E and XY=E for some YEA-f(F)}. As in [2] we classify the possible idempotents and show that associated to each one is a finite graph. Moreover we state a result (proved in Section 4) that determines M,(F) in the case where the associated graph is a tree. At the end of Section 3 we present some examples. We work throughout over a fixed base field F. Unlabelled tensors are over

The Brauer monoid of a field

In 1921 in a paper entitled “On Division Algebras,” J. H. M. Wedderburn proved that every division algebra of degree 3 over a field of characteristic not 3 is cyclic. In the process he proved two other remarkable results. First he showed that if D is a division algebra over a field F and Q E D F then the minimal polynomial g(L) of 0 over F factors over D into linear factors; that is, in the polynomial ring D[lr] (where 1 commutes with the elements of D) we have g(1) = (2 0,)(L l3,._ 1) ... (2 0,) for some tii E D, where B, = 8. Each ei is a root of g(A) and hence a conjugate in D of 8. Wedderburn then proceeded to prove that if D has degree 3 then one can do better. In that case there is an element 4 in D” such that g(A) = (A t -28t2)(A [P’et)(l Q) in D[A]. It follows that t3 E F”, and using this he quickly derived his main result, the cyclicity of division algebras of degree 3. In this paper we introduce the notion of a conjugate splitting for a polynomial with coefficients in some field F over an F-algebra A-it is precisely the kind of decomposition Wedderburn proved for division algebras of degree 3. We then prove that the Clifford algebra of the homogenization of the polynomial is universal for these conjugate splittings (see Theorem 1.2). The rest of the paper consists of consequences and interpretations of this result along with a number of examples. In the first section we present examples of conjugate splittings including examples for polynomials of degree greater than 3. In addition we show among other things that Wedderburn’s theorem extends to M,(F), the split central simple F-algebra of degree 3. In the second section we restrict attention to cubic polynomials and use the known structure of the Clifford