Alternating triple systems with simple Lie algebras of derivations

We prove a formula for the multiplicity of the irreducible representation V (n) of sl(2, C) as a direct summand of its own exterior cube ΛV (n). From this we determine that V (n) occurs exactly once as a summand of ΛV (n) if and only if n = 3, 5, 6, 7, 8, 10. These representations admit a unique sl(2)-invariant alternating ternary structure obtained from the projection ΛV (n) → V (n). We calculate the structure constants for each of these alternating triple systems and use computer algebra to determine their polynomial identities of degree ≤ 7. We discover a remarkable 14term identity in degree 7. The variety defined by this identity contains V (3), V (5) and V (7). Introduction An irreducible representation of a simple Lie algebra can be a direct summand of its own exterior cube. In this case, the representation admits the structure of This article is the sequel to the joint paper presented by one of us (M.R.B.) at the Fifth International Conference on Nonassociative Algebra and its Applications (NONAA-V) in Oaxtepec, Morelos, Mexico (July 27 to August 2, 2003). The results of that earlier work will appear in Experimental Mathematics. Email address: [email protected] Email address: [email protected] 1 an alternating triple system which is invariant in the sense that the Lie algebra acts as ternary derivations. This paper studies this situation in detail for the Lie algebra sl(2, C). In section 1 we review the basic representation theory of sl(2). In section 2 we prove a general formula for the multiplicity of an irreducible representation in its own exterior cube. From this we determine all representations for which the multiplicity equals 1; such representations admit an sl(2)-invariant alternating ternary structure which is unique up to a scalar multiple. In section 3 we review basic material about ternary operations. In the following six sections we describe computer searches for polynomial identities satisfied by the six representations which admit a unique alternating ternary structure; we determine all their identities of degree ≤ 7. A detailed discussion of our computational methods for discovering identities satisfied by nonassociative algebras may be found in three previous articles by the authors [3], [4], [5]. These methods involve expressing the identities as the nullspace of a large linear system, and then solving the system by using a computer algebra system to compute the row canonical form of the coefficient matrix. In section 10 we go beyond sl(2) and use the computer algebra package LiE [6] to determine all fundamental representations of simple Lie algebras of rank ≤ 8 which occur as summands of their own exterior cubes. This demonstrates the existence of a large number of new alternating triple systems, with simple Lie algebras in their derivation algebras, which deserve further study. 1 Representations of the Lie algebra sl(2) We first recall some standard facts about sl(2) and its representations. All vector spaces and tensor products are over F, an algebraically closed field of characteristic zero. Our basic reference is Humphreys [8], especially §II.7. 1.1 The Lie algebra sl(2) As an abstract Lie algebra, sl(2) has basis {E, H, F} and commutation relations [H, E] = 2E, [H, F ] = −2F, [E, F ] = H. All other relations between basis elements follow from anticommutativity. Since the Lie bracket is bilinear these relations determine the product [X, Y ] for all X, Y ∈ sl(2). 1.2 The irreducible representation V (n) For any nonnegative integer n, there is an irreducible representation of sl(2) containing a nonzero vector vn (called the highest weight vector) satisfying the conditions H.vn = nvn, E.vn = 0.