Exceptional Algebras Research Articles

Ghost triality and superstring partition functions

In three or more space-time dimensions a heterotic string can have just four kinds of space-time supersymmetry, corresponding to the exceptional algebras E 6, E 7, E 8 and F 4. We give a simple argument why this correspondence, first observed in the context of the covariant lattice construction, holds for any heterotic superstring, no matter how it is constructed. Using the fact that SO(8) triality is an inner automorphism of the exceptional algebras, we derive a set of identities for character-valued partition functions, valid for any superstring in any space-time dimension and at any genus. These identities have exactly the structure of the Riemann identities, and are identical to the Riemann identities in the case of ten-dimensional strings. Their relavance for proving vanishing of the cosmological constant and non-renormalization theorems is discussed.

Decompositions of representations of exceptional affine algebras with respect to conformal subalgebras.

We use modular-invariance and conformal-invariance constraints on the highest-weight representations of affine algebras as well as asymptotic behavior of their characters to deduce an algorithm for the decomposition of these representations with respect to conformal subalgebras. We use the algorithm to find explicit decompositions of level-1 representations of all affine exceptional algebras with respect to all their conformal subalgebras. The crucial point of this work is a connection of the decomposition problem with the Frobenius theory of positive matrices, which hints to a possible connection with Markov chains. The problem of computing branching coefficients for conformal subalgebras arises in the string compactifications. We show that a solution to this problem allows one to construct modular-invariant partition functions on a group manifold. An important new general theoretical result of the paper is the uniqueness of the vacuum state in the basic representation of an affine algebra for its conformal subalgebra.

String theories and the Jordan algebras

We investigate the sequence of algebras m 3 n , the Jordan algebras consisting of 3x3 hermitean matrices over R , C , Q and O . We show that the variables of the superstring can be interpreted as elements of the algebra m 3 8. The other algebras in the m 3 n sequence correspond to classical superstring theories. In our approach, transverse Lorentz rotations are contained in the autormorphism group of the algebra m 3 n . Our approach, although apparently equivalent to the usual formulations, is suggestive of an underlying connection between the exceptional algebra m 3 8 and string theory.

Exceptional Jordan pairs

A Jordan algebra is called special if it can be embedded in an associative algebra, and is called strongly special if its splittings by arbitrary bimodule extensions are special. The question of strong speciality of simple finite-dimensional Jordan algebras has been studied by N. Jacobson and K. McCrimmon (cf. [i, Chap. 7]). In this class, the only algebras which are not strongly special are the exceptional algebra H(C~ and the algebra of 6 × 6-matrices fixed with respect to a symmetric involution, as well as their forms.

Adjoint representations of exceptional Lie algebras

The recent development of superstring theory (1) has drawn attention to the potential importance of the exceptional Lie algebras (particularly E/sub 6/ and E/sub 8/) in a future unified theory. The paper consists of three parts. In the first, the authors consider normed algebras and the properties of their transformations. In the second they give the construction of the magic square and the corresponding commutation relations. The construction is anticipated by two symmetric constructions, the symbiosis of which it is. In the final part, the exceptional algebras of the series E are decomposed. The proofs of the Jacobi identities and the identification of the algebras in the magic square are omitted since the calculations are lengthy.

Three graded exceptional algebras and symmetric spaces

The exceptional algebras of typeE7 are studied from the point of view of their three graded structure. The connection between three-grading and the Jordan Pair structure of such Lie algebras is analyzed. The Jordan Pair content is in turn related to the symmetric spaces and . Coset spaces of this type have been recently suggested as possible scalar manifolds in supergravity. We develop a way of representingE7 in a matrix form, which makes the Jordan Pair content of theE7 real and complex forms quite transparent and shows whether such forms admit a three graded structure.

Springer forms and the first Tits construction of exceptional Jordan division algebras

In this paper, a certain quadratic form, originally due to Springer [15], which may be associated with any separable cubic subfield living inside an exceptional simple Jordan algebra is related to the coordinate algebra of an appropriate scalar extension. We use this relation to show that, in the presence of the third roots of unity, exceptional Jordan division algebras arising from the first Tits construction are precisely those where reducing fields and splitting fields agree, and that all isotopes of a first construction exceptional division algebra are isomorphic.

The semisimple subalgebras of Lie algebras

Techniques for obtaining generators of nonregular maximal subalgebras of Lie algebras, alternative to those of Dynkin, are developed. They exploit the use of orthogonal bases in weight space, which are related to quark weights. The projection from an algebra G to its nonregular subalgebras g is related to an orthogonal matrix. The roots of G which project onto roots of g can be simply specified in the orthogonal bases. The phases of the expansion coefficients of generators of g in terms of generators of G are specified in such a manner that, for a given G, the entire set { g} of maximal subalgebras have consistent phases. The condition e−β =e°β is satisfied for all generators of g. The generators of all maximal nonregular subalgebras of all exceptional algebras are exhibited.

Cubic Subfields of Exceptional Simple Jordan Algebras

Let $E/k$ be a cubic field extension and $J$ a simple exceptional Jordan algebra of degree 3 over $k$. Then $E$ is a reducing field of $J$ if and only if $E$ is isomorphic to a (maximal) subfield of some isotope of $J$. If $k$ has characteristic not 2 or 3 and contains the third roots of unity then every simple exceptional Jordan division algebra of degree 3 over $k$ contains a cyclic cubic subfield.

Generic reducing fields of Jordan pairs

Generic reducing fields of Jordan pairs, generalizing at the same time generic splitting fields of associative algebras and generic zero fields of quadratic forms, are intrinsically defined and constructed. The most elementary properties are derived, and the relationship with other generic constructions, particularly those linked to Brauer-Severi varieties, are investigated. As an application it is shown that there exist nonisomorphic exceptional Jordan division algebras having the same splitting fields.

Cubic subfields of exceptional simple Jordan algebras

Let E / k E/k be a cubic field extension and J J a simple exceptional Jordan algebra of degree 3 over k k . Then E E is a reducing field of J J if and only if E E is isomorphic to a (maximal) subfield of some isotope of J J . If k k has characteristic not 2 or 3 and contains the third roots of unity then every simple exceptional Jordan division algebra of degree 3 over k k contains a cyclic cubic subfield.

Zelmanov's prime theorem for quadratic Jordan algebras

E. I. Zelmanov has recently shown that any prime linear Jordan algebra without nil ideals (but no finiteness conditions imposed) is either a homomorphic image of a special algebra or is a form of the 27-dimensional exceptional algebra. In particular, all division algebras and all simple unital algebras are either derived from associative algebras or are 27.dimensional over their centers. This effectively ends the search for infinite-dimensional exceptional algebras. In the present paper, Zelmanov’s result is extended to quadratic Jordan algebras. Jordan algebras were invented in the 1930s in the search for an exceptional algebraic setting for quantum mechnics: an algebraic system which behaved like the usual algebra of quantum mechanical observables (hermitian operators on Hilbert space), but was exceptional in the sense that its structure was not determined behind the scenes by some unobservable associative algebra. In their pioneering paper of 1933, [ 91 Jordan, von Neumann, and Wigner classified the finite-dimensional formally real linear Jordan algebras, and A. A. Albert showed [ 11 that the only simple algebra in the list which was exceptional was a certain 27.dimensional algebra of 3 X 3 hermitian matrices with entries from an gdimensional Cayley algebra. All subsequent investigations of exceptional Jordan algebras led back to this same Albert algebra. The next breakthrough in the structure of Jordan algebras was N. Jacobson’s 1965 introduction [6 J of inner ideals and the resultant quadratilication of the theory (in terms of the quadratic product xyx instead of the linear product xy + .vx). The final breakthrough was the astounding 1979 result of a young Russian mathematician, E. I. Zelmanov, that the only simple exceptional linear Jordan are Albert algebras. Thus the search for an infinite-dimensional setting for exceptional quantum mechanics is doomed to failure: the only simple exceptional structure allotted to mortals is the 27-dimensional Albert algebra. This came as a complete surprise to Western researchers (who hoped the free Jordan algebra might lead to an 291 002 I-8693/82/060297-30$02.00/0

The role of noncompact Lie algebras in relativistic wave equations. I

The role of real Lie algebras in the study of relativistic wave equations of the form (αμ∂μ+iκ)ψ(x) = 0 is considered. To a finite-dimensional equation there corresponds a Lie algebra S containing so(4,C) and a vector operator {αμ}. The importance of finding all possible real forms of S containing the Lorentz Lie algebra so(3,1) is discussed. This problem is solved in detail for certain ’’generic’’ cases, namely S = sp(n,C), so(n,C), and sl(n,C). The exceptional algebras G2, F4, and E6 are also considered.

Natural labelling schemes for simple roots and irreducible representations of exceptional Lie algebras

The simple root system of each exceptional, simple Lie algebra is explicitly constructed in a variety of forms. Each construction is based on the natural embedding in the exceptional algebra of a classical, semi-simple algebra of the same rank. The procedure adopted leads to the discovery of a number of new chains of subalgebras illustrated by means of supplemented Dynkin diagrams. The various sets of simple roots are then used to determine natural labels for the irreducible representations of each of the exceptional algebras. For each such labelling scheme the modification rules for dealing with non-standard representation labels are tabulated. The connection between the natural labels and Dynkin labels is given in detail and a comparison is made with the labels of B.G. Wybourne and M.J. Bowick (1977).

Classification of all simple graded Lie algebras whose Lie algebra is reductive. II. Construction of the exceptional algebras

The exceptional simple graded Lie algebras whose existence is suggested by the results of the preceding paper are explicitly constructed. In this way the classification of all simple graded Lie algebras whose Lie algebra is reductive is completed.

Exceptional Jordan division algebras over a field with a discrete valuation.

Article Exceptional Jordan division algebras over a field with a discrete valuation. was published on January 1, 1975 in the journal Journal für die reine und angewandte Mathematik (volume 1975, issue 274-275).

Study of the Renormalization Group for Small Coupling Constants

A renormalizable quantum field theory is said to be stagnant if it is asymptotically free. We study the renormalization group for small coupling constants. In particular the pseudoscalar-fermion theory with an internal-symmetry group $G$ and in which the coupling matrix furnishes a representation of $G$ is considered. A representation is said to be stagnant if the associated theory is stagnant. We show that Cartan's four families $A$, $B$, $C$, $D$ and the exceptional algebra ${\mathrm{G}}_{2}$ possess no stagnant representation. On the basis of this result we conjecture that there are no asymptotically free quantum field theories in four dimensions. Some possible asymptotic behaviors of field theories are also described.

The Freudenthal-Springer-Tits constructions revisited

In [3] some constructions of exceptional Jordan algebras due to H. Freudenthal, T. A. Springer, and J. Tits were carried over to quadratic Jordan algebras (as in [4]) of arbitrary characteristic. The question was left open whether the Tits Constructions yielded all exceptional finite-dimensional central simple algebras in characteristic 2 (it was known that they do for characteristics #2). In this paper we settle this question in the affirmative. This result completes the structure theory for finite-dimensional quadratic Jordan algebras. The Tits Constructions as given in [3] involve the construction of a norm form. To prove that all the exceptional algebras arise from these constructions we need to show that all such algebras have a suitable norm form. This necessitates a slight detour in ??1 and 2 to verify that Jordan algebras in characteristic 2 have generic norms with the same properties as in the other characteristics. In ?3 we define the centroid for quadratic Jordan algebras and the corresponding notion of central simple algebras. In the next section we establish certain conditions under which a central simple Jordan algebra remains simple upon extension of the base field. In ?5 we apply these results to show that every exceptional finite-dimensional central simple algebra is a form of the 27-dimensional exceptional algebra ?((i3) (i.e. becomes ( upon suitable extension of the base field). In the final section we show that the Tits Constructions yield precisely all the exceptional finite-dimensional central simple Jordan algebras. Our proofs will be valid for all characteristics.

The Freudenthal-Springer-Tits constructions of exceptional Jordan algebras

Constructions of 27-dimensional exceptional simple Jordan algebras have been given by H. Freudenthal [2], [3], T. A. Springer [8], and J. Tits [9]. In the first two approaches the cubic generic norm form plays a central role, with applications to projective geometry and algebraic groups; the third approach gives a simple method for constructing all exceptional simple algebras. The constructions are limited to fields of characteristic #2 as usual for Jordan algebras defined in terms of a bilinear multiplication, and in order to polarize the cubic norm form the characteristic must be # 3. Recently a definition of Jordan algebras has been proposed [5] which is based on a cubic composition involving U-operators. A unital Jordan algebra over a commutative associative ring &P is a triple : = (X, U, 1) where X is a &P-module, U a quadratic mapping x -> Ux of X into Homo (X, X), and 1 an element of X satisfying the axioms

On Lie algebras of classical type

G. B. Seligman has proved in [1] the following result: If V is a simple restricted Lie algebra over an algebraically closed field of characteristic p> 7, and if 2 has a restricted representation with nondegenerate trace form, then V is of classical type. By an algebra of classical type is meant an analogue over a field of characteristic p of one of the simple Lie algebras (including the five exceptional algebras) of characteristic 0; for the precise statement, see [1]. We shall show here that the above result of Seligman may be proved without the assumption of restrictedness of the algebra and its representation. We begin with a lemma on matrices. Let Sn(hW) denote the space of all nXn matrices over a field !, and consider this as a Lie algebra (under commutation) over .