Showing papers on "Algebra representation published in 1982"

Some varieties containing relation algebras

Abstract: Three varieties of algebras are introduced which extend the variety RA of relation algebras. They are obtained from RA by weakening the associative law for relative product, and are consequently called nonassociative, weakly-associative and semiassociative relation algebras, or NA, WA, and SA, respectively. Each of these varieties arises naturally in solving various problems concerning relation algebras. We show, for example, that WA is the only one of these varieties which is closed under the formation of complex algebras of atom structures of algebras, and that WA is the closure of the variety of representable RA's under relativization. The paper also contains a study of the elementary theories of these varieties, various representation theorems, and numerous examples. 0. Introduction. Relation algebras (RA's) have a binary operation; which serves as an abstract algebraic analogue of the relative product of binary relations. (The relative product of R, S C U X U, is R I S = {(x, z):(x, y) E R and (y, z) E S for somey E U).) The relative product is associative, and one of the postulates for RA's is that ; is associative. The associativity of relative product can be expressed by a sentence in a first-order language with binary relation symbols, namely ( 1 ) ~VV [3z(3y(Rxy A Syz) A Tzy) -3z(Rxz A 3X(Szx A Txy))]. Although this sentence has three variables, it cannot be proved from the ordinary axioms of first-order logic without using four variables. In contrast, all the other postulates for relation algebras can not only be expressed but proved using only three variables. (These facts were first proved by Tarski. For a proof that (1) requires four variables to prove, see [3].) Tarski asked whether there are any other equations whose translations into first-order sentences can be expressed and proved using only three variables, but which are not derivable from the postulates for RA's without using the associativity of ;. There are such equations. One of them is a special case of the associative law for; called the "semiassociative law", (2) x; (1; 1) = (x; 1); 1 . The class SA of semiassociative relation algebras is defined by the postulates for RA's with the associative law for; replaced by (2). This class properly includes RA, as will be shown in this paper. Received by the editors November 12, 1980. 1980 Mathematics Subject Classification. Primary 03G25, 06E99.

Moving frames and prolongation algebras

Abstract: Differential ideals generated by sets of 2-forms which can be written with constant coefficients in a canonical basis of 1-forms are considered. By setting up a Cartan-Ehresmann connection, in a fiber bundle over a base space in which the 2-forms live, one finds an incomplete Lie algebra of vector fields in the fields in the fibers. Conversely, given this algebra (a prolongation algebra), one can derive the differential ideal. The two constructs are thus dual, and analysis of either derives properties of both. Such systems arise in the classical differential geometry of moving frames. Examples of this are discussed, together with examples arising more recently: the Korteweg-de Vries and Harrison-Ernst systems.

A study of the local components of the Hecke algebra mod $l$

Abstract: We use information about modular forms mod / to study the local structure of the Hecke ring. In particular, we find nontrivial lower bounds for the dimensions of the Zariski tangent spaces of the local components of the Hecke ring mod /. These results suggest that the local components of the Hecke ring mod / are more complex than originally expected. We also investigate the inverse limits of the Hecke rings of weight k mod / as k varies within a fixed congruence class mod / 1. As an immediate corollary to some of the above results, we show that when k is sufficiently large, an arbitrary prime / must divide the index of the classical Hecke ring Tk in the ring of integers of 1k ® Q.

Kac-Moody algebra in the self-dual Yang-Mills equation

Abstract: In the $J$ formulation of self-dual Yang-Mills equations, we propose a parametric infinitesimal transformation, which generates new solutions from any old ones and satisfies the equations of the Bianchi-B\"acklund transformation with parameter. Expanding in the parameter, we obtain an infinite number of transformations, all of which leave the self-dual Yang-Mills equation invariant. We discuss the group properties for these transformations, and find that they form a Lie group, to which the Lie algebra is an infinite-dimensional Kac-Moody algebra, a mathematical structure encountered in the recent development of principal chiral theories.

Representations of the symmetric group and varieties of linear algebras

Abstract: The representation theory of the symmetric group is used to study varieties of linear algebras over a field of characteristic 0. The relatively free algebras and the lattice of subvarieties of the variety of Lie algebras are described. An example of an almost finitely based variety of linear algebras if constructed. A continuous set of locally finite varieties forming a chain with respect to inclusion is indicated. Information is obtained on the variety of Lie algebras (resp., associative algebras with 1) generated by the second-order matrix algebra. In particular, distributivity of the lattice of subvarieties is proved, and in the Lie case a relatively free algebra is described.Bibliography: 16 titles.

The Virasoro algebra

Abstract: Three theroems are proved. With suitable hypotheses in each case, characterizations are found for the Virasoro algebra, for some of its representations, and for the Ramond-Neveu-Schwarz superalgebra built around the Virasoro algebra.

Real Flexible Division Algebras

Abstract: In this paper we classify finite-dimensional flexible division algebras over the real numbers. We show that every such algebra is either (i) commutative and of dimension one or two, (ii) a slight variant of a noncommutative Jordan algebra of degree two, or (iii) an algebra defined by putting a certain product on the 3 × 3 complex skew-Hermitian matrices of trace zero. A precise statement of this result is given at the end of this section after we have developed the necessary background and terminology. In Section 3 we show that, if one also assumes that the algebra is Lie-admissible, then the structure follows rapidly from results in [2] and [3]. All algebras in this paper will be assumed to be finite-dimensional. A nonassociative algebra A is called flexible if (xy)x = x(yx) for all x, y ∈ A.

The radical in a finitely generated P. I. algebra

Abstract: Kemer, in [K], announced a proof of Theorem B with the additional assumption that char(F) = 0. His proof relies on a result of Razmyslov [Ra, Theorem 3] and on certain arguments related to the connection between P.I. ring theory and the theory of representation of the symmetric group Sn over F, char F = 0. Both results rely heavily on the assumption that char F = 0, so they do not seem to generalize directly to arbitrary F. The previously best known results concerning Theorem A are in [Ra, Theorems 1, 3, Sc, Theorem 2] . The proof of Theorem C is a straightforward application of Theorem B and a theorem of J. Lewin [Le, Theorem 10].

Dihedral algebras are cyclic

Abstract: In his book [1], Albert has a proof that every division algebra of degree 3 is cyclic. In this paper we will generalize this result, and derive the theorem below. Our argument is very close to that of Albert, and arose as part of a close examination of his proof. Fix n to be an odd positive integer, and F a field of characteristic prime to n. Denote by Dn the dihedral group of order 2n. We assume the reader is familiar with the basics of the theory of finite dimensional simple algebras as presented, for example, in Albert's book.

Narrowness implies uniformity

Abstract: This paper is about varietiesV of universal algebras which satisfy the following numerical condition on the spectrum: there are only finitely many prime integersp such thatp is a divisor of the cardinality of some finite algebra inV. Such varieties are callednarrow. The variety (or equational class) generated by a classK of similar algebras is denoted by V(K)=HSPK. We define Pr (K) as the set of prime integers which divide the cardinality of a (some) finite member ofK. We callK narrow if Pr (K) is finite. The key result proved here states that for any finite setK of finite algebras of the same type, the following are equivalent: (1) SPK is a narrow class. (2) V(K) has uniform congruence relations. (3) SK has uniform congruences and (3) SK has permuting congruences. (4) Pr (V(K))= Pr(SK). A varietyV is calleddirectly representable if there is a finite setK of finite algebras such thatV= V(K) and such that all finite algebras inV belong to PK. An equivalent definition states thatV is finitely generated and, up to isomorphism,V has only finitely many finite directly indecomposable algebras. Directly representable varieties are narrow and hence congruence modular. The machinery of modular commutators is applied in this paper to derive the following results for any directly representable varietyV. Each finite, directly indecomposable algebra inV is either simple or abelian.V satisfies the commutator identity [x,y]=x·y·[1,1] holding for congruencesx andy over any member ofV. The problem of characterizing finite algebras which generate directly representable varieties is reduced to a problem of ring theory on which there exists an extensive literature: to characterize those finite ringsR with identity element for which the variety of all unitary leftR-modules is directly representable. (In the terminology of [7], the condition is thatR has finite representation type.) We show that the directly representable varieties of groups are precisely the finitely generated abelian varieties, and that a finite, subdirectly irreducible, ring generates a directly representable variety iff the ring is a field or a zero ring.

Noncommutative Jordan C*-algebras

Abstract: We introduce noncommutative JB*-algebras which generalize both B*-algebras and JB*-algebras and set up the bases for a representation theory of noncommutative JB*-algebras. To this end we define noncommutative JB*-factors and study the factor representations of a noncommutative JB*-algebra. The particular case of alternative B*-factors is discussed in detail and a Gelfand-Naimark theorem for alternative B*-algebras is given.

Extensions for AF * algebras and dimension groups

Abstract: Let A, C be approximately finite dimensional (AF) C* algebras, with A nonunital and C unital; suppose that either (i) A is the algebra of compact operators, or (ii) both A, C are simple. The classification of extensions of A by C is studied here, by means of Elliott's dimension groups. In case (i), the weak Ext group of C is shown to be Extz(KO(C), Z), and the strong Ext group is an extension of a cyclic group by the weak Ext group; conditions under which either Ext group is trivial are determined. In case (ii), there is an unnatural and complicated group structure on the classes of extensions when A has only finitely many pure finite traces (and somewhat more generally). Our motivating theme is to consider extensions of C* algebras by other than the algebra of compact operators. Because AF algebras are describable in terms of partially ordered groups, they seem particularly suitable for this extension theory. As the ordered groups arising from simple AF algebras are fairly well understood, it turns out that one can solve completely the problem of classifying simple by simple AF algebras, when the finite trace space of the bottom AF algebra has finite dimensional dual. In the course of doing this, we establish formulae for the usual strong and weak Ext groups of AF algebras; our homological approach to this differs from the computational viewpoint of Pimsner and Popa [14, 15]. We consider short exact sequences ("extensions") of C* algebras, A B C, with B an AF algebra. There is a translation to extensions within a class of partially ordered abelian groups and distinguished subset, known as dimension groups with interval, via the functor Ko0 This translation is reversible (owing to a recent result of L. Brown that an extension of an AF algebra by an AF algebra is AF), so all C* extensions are represented as dimension group extensions. With the appropriate notion of equivalence (extending strong equivalence as defined in [3, p. 268], when A is the algebra of compact operators on a separable Hilbert space), the equivalence classes admit a limited additive operation, often forming a disjoint union of groups. A single group results if, for example, both A and C are simple (with A unitless, but not necessarily stable). ?1 deals with the appropriate definitions of extensions, dimension groups, equivalence, and the translation between AF algebras and dimension groups. Much of this is well known. In the second section, it is shown that if A is simple stable, and C is Received by the editors December 11, 1979 and, in revised form, March 25, 1981. 1980 Mathematics Subject Classification. Primary 46L35; Secondary 54A22, 16A54, 16A56, 06F20. 'Supported in part by an operating grant from NSERC of Canada. (D1982 American Mathematical Society 0002-9947/82/0000-1 059/$09.75

On the derivations of gametic algebras for polyploidy with multiple alleles

Abstract: 1.1. In this paper we obtain some results concerning derivations of a class of genetic algebras. Let us set down our terminology and notation. First of all, our base field is the field R of real numbers. All genetic algebras are algebras over this field and "algebra" means R-algebra, "linear mapping" means "R-linear mapping". We adopt Gonshor's definition: A genetic algebra is a commutative algebra for which there exists a basis Co, C1, ..., C, with a multiplication table satisfying the following conditions:

Classification of flexible composition algebras, II

Abstract: The following theorem has been proven. Let A be a finite-dimensional flexible composition algebra over a field F of characteristic not equal to 2, and not equal to 3. Then, A must be either a Hurwitz, or a para-Hurwitz, or a pseudo-octonion algebra. If the field F is of characteristic 2, then the dimension of a composition algebra is arbitrary in contrast to the standard dimensionality of 1, 2, 4, or 8. Finally, it is shown that any finite-dimensional flexible composition algebra over a field F of any characteristic is automatically a Malcev-admissible algebra.

Residually small varieties ofK-algebras

Abstract: We focus on those universal algebras which ring theorists call associative algebras overK, whereK is a commutative ring having an identity element. They are calledK-algebras in this paper. For fixedK there are two full varieties to consider namelyK-algebras with, or without, an identity element as a formal constant.

On the symmetric algebra of an ideal

Abstract: The symmetric algebra of an ideal I may be compared to the Rees algebra via the canonical epimorphism α:Sym(I)→R(I). A necessary and sufficient criterion is given for a to be an isomorphism, and sequential conditions on the symmetric algebra are studied. Some applications are given to Proj α:ProjR(I)→Pro'j Sym(I) and to the theory of approximation complexes.

The Associative Algebra

Abstract: Our objective in this chapter is to show off a few examples of algebras that occur naturally. After a brief orientation toward concepts and notation, the reader is introduced to group algebras, endomorphism algebras, matrix algebras, and quaternion algebras. Along the way, there is a brief digression, which contains a hint of the connection between algebraic geometry and the theory of finite dimensional algebras over a field.