Type E7 Research Articles

Strictly regular elements in Freudenthal triple systems

Strictly regular elements play a role in the structure theory of Freudenthal triple systems analogous to that played by idempotents in nonassociative algebras with identity. In this paper we study the coordinatization of reduced triple systems relative to a connected pair of strictly regular elements and use the explicit form of strictly regular elements in terms of the coordinatization to prove uniqueness of the coordinatizing Jordan algebra, as well as several generalizations of known results regarding groups of transformations related to triple systems. Finally, we classify forms of a particularly important triple system (the representation module for the Lie algebra E7) over finite, p-adic or real fields. Following Freudenthal [71, Meyberg [91 and Brown [11 have introduced an axiomatic approach to the study of the ternary algebraic structure of the minimal dimensional module for the lie algebra E7. In particular, this module is one of a class of ternary algebras we refer to as Freudenthal Triple Systems (FTS). In this paper we analyze further the internal structure of such algebras using as a basic tool the Peirce decomposition relative to a pair of supplementary strictly regular elements. Using this decomposition we obtain, in a manner similar to [1 ], a coordinatization theorem-every simple, reduced FTS is isomorphic to VQ), ?S a member of a small class of Jordan algebras specified in ?1. In ?6 we analyze the forms of strictly regular elements in K2?) and use the results to show I) lN(S) if and only if ?and A are norm equivalent. In ?7 the action of the group of q-similarities on the strictly regular elements is studied, yielding conjugacy theorems generalizing a result of Brown [1], as well as information regarding ratios of similitudes generalizing results of Faulkner [41 and Seligman (unpublished). Results of this section are useful also in the classification problem for Lie algebras of type E7. In the final section we classify completely all forms of lQ(?), 3 exceptional central simple, for finite, p-adic, or real fields. In [51, symplectic algebras (an asymmetric version of FTS's) are studied, yielding different proofs of several results obtained in this paper. Received by the editors December 22, 1971. AMS (MOS) subject classifications (1970). Primary 17A30; Secondary 17B60.

On the Schur Indices of Characters of the Exceptional Weyl Groups

We will be concerned in this paper with the irreducible characters of the Weyl groups of type E6, E7, and E8. Character tables for these groups have been given by J. S. Frame [6, 7]. In all cases, the characters are rationalvalued. It is fairly easy to show that each character has a real splitting field. The object of this paper is to show that each character has a rational representation. Other authors have shown that the same results hold for other irreducible Weyl groups. A. Young [13] studied the families of groups of type A, and B, as permutation groups. He devised a method for constructing all representations and showed that they could be constructed over the rational field Q. W. Specht [11] did the same for the family of groups of type D. This result is easily shown for the group of type G2, which is dihedral of order 12. T. Kondo [9] presented a character table for the group of type F4 and showed that each character has a rational representation. Our present results complete the proof that any representation of a Weyl group of a finite dimensional semi-simple complex Lie algebra is equivalent to a rational representation. Our methods are character-theory methods and we restate our results in terms of the Schur index. Let X be an irreducible character of a group G and let F be a field. The Schur index m,(X) of X over F is the smallest positive integer m such that mX is a character afforded by an F(X)-representation of G.

A construction of Lie algebras from a class of ternary algebras

A class of algebras with a ternary composition and alternating bilinear form is defined. The construction of a Lie algebra from a member of this class is given, and the Lie algebra is shown to be simple if the form is nondegenerate. A characterization of the Lie algebras so constructed in terms of their structure as modules for the three-dimensional simple Lie algebra is obtained in the case the base ring contains 1/2. Finally, some of the Lie algebras are identified; in particular, Lie algebras of type E8 are obtained. A construction of Lie algebras from Jordan algebras discovered independently by J. Tits [7] and M. Koecher [4] has been useful in the study of both kinds of algebras. In this paper, we give a similar construction of Lie algebras from a ternary algebra with a skew bilinear form satisfying certain axioms. These ternary algebras are a variation on the Freudenthal triple systems considered in [1]. Most of the results we obtain for our construction are parallel to those for the TitsKoecher construction (see [3, Chapter VIII]). In ?1, we define the ternary algebras, derive some basic results about them, and give two examples of such algebras. In ?2, the Lie algebras are constructed and shown to be simple if and only if the skew bilinear form is nondegenerate. In ?3, we give a characterization, in the case the base ring contains 1/2, of the Lie algebras obtained by our construction in terms of their structure as modules for the threedimensional simple Lie algebra. Finally, in ?4, we identify some of the simple Lie algebras obtained by our construction from the examples of ?1. In particular, we show that we can construct a Lie algebra of type E8 from a 56-dimensional space which is a module for a Lie algebra of type E7. A similar construction was given by H. Freudenthal in [2]. 1. A class of ternary algebras. We shall be interested in a module TM over an arbitrary commutative associative ring D with 1 which possesses an alternating bilinear form and a ternary product which satisfy (TI) = + z for x,y, ze M; (T2) = + x for x, y, z e 9; (T3) , w> = , z> + for x, y, z, w e 9; Received by the editors March 12, 1970 and, in revised form, June 19, 1970. AMS 1969 subject classifications. Primary 1730.

Zur konstruktion von Lie-Algebren aus Jordan-Tripelsystemen

In this paper we study certain Lie algebras which are constructed from the (-1)-eigenspaees of an involution of a Jordan algebra. The construction is a generalisation of the Koecher-Tits-construction. We give necessary conditions in terms of the Jordan algebras for the Lie algebras being simple. If the (-1)-spaces are Peirce-1/2-components then we obtain a close relation between the Lie algebras under consideration and the structure algebras of Jordan algebras. We finally give a list of those types of simple Lie algebras which can be formed by this construction; among them are Lie algebras of type E6 and E7.

Groups of type E7.

Automorphisms of order 2 are studied in order to understand generalized symmetric spaces. The groups of type E6 we consider here can be realized as both the group of linear maps that leave a certain determinant invariant, and also as the identity component of the automorphism group of a class of structurable algebras known as Brown algebras. We will classify the k-involutions of these groups of type E6 using aspects of both descriptions, and give detailed descriptions of representatives over certain fields including algebraically closed fields, R, Fp, and Qp.

GEOMETRIES OVER THE ALGEBRA OF ANTIOCTAVES

We give a rigorous construction for a projective and a noneuclidean geometry over the alternative algebra of antioctaves (split octaves). This construction generalizes Freudenthal's definition of the projective plane over the algebra of octaves (Cayley numbers). It is proved that the groups of automorphisms of the projective and the noneuclidean plane are simple noncompact Lie groups of types E6 and F4, respectively.

On Lie algebras of type 𝐸₆

On Lie algebras of type 𝐸₆

Conformal connections and conformal transformations

Introduction. In the previous paper [5], we have studied the groups of projective transformations of affinely connected manifolds by the application of the theory of normal projective connections. The main purpose of the present paper is to study the conformal properties of complete Riemannian spaces of some special type by the application of the theory of normal conformal connections. We shall introduce a family of Riemannian spaces, the elements of which are characterized by conditions on the Ricci tensor fields and will be called of type e (Definition 4). Einstein spaces are of type e and, in general, a Riemannian space of type e is locally isomorphic with the direct product of two Einstein spaces of different signs. Our main result (Theorem 1) states that two conformally equivalent Riemannian metrics on a manifold which are complete and of type e necessarily coincide, except the case where either of the two metrics is an Einstein metric with the vanishing or negative Ricci tensor field. Some results are also obtained in the exceptional case. We see from this theorem that the group of all conformal transformations of a complete Riemannian space of type e5 coincides with the group of all isometries, except the case of an Einstein space with the vanishing or negative Ricci tensor field (Theorem 2).

On character sheaves and characters of reductive groups at unipotent classes

With a view to determining character values of finite reductive groups at unipotent elements, we prove a number of results concerning inner products of generalised Gelfand-Graev characters with characteristic functions of character sheaves, here called Lusztig functions. These are used to determine projections of generalised Gelfand-Graev characters to the space of unipo- tent characters, and to the space of characters with a given wave front set. Such projections are expressed largely in terms of Weyl group data. We show how the values of characters at their unipotent support or wave front set are determined by such data. In some exceptional groups we show that the projection of a generalised Gelfand-Graev character to a family with the same wave front set is (up to sign) the dual of a Mellin transform. Using these results, in certain cases we are able to determine roots of unity which relate almost characters to the characteristic functions. In particular we show how to compute the values of all unipotent characters at all unipotent classes for the exceptional adjoint groups of type G2, F4, E6, E7 and E8. We also pro- vide an appendix which gives a complete list of the cuspidal character sheaves on all quasi-simple groups.