Abstract

AbstractThe space of tensors of metric curvature type on a Euclidean vector space carries a two-parameter family of orthogonally invariant commutative nonassociative multiplications invariant with respect to the symmetric bilinear form determined by the metric. For a particular choice of parameters these algebras recover the polarization of the quadratic map on metric curvature tensors that arises in the work of Hamilton on the Ricci flow. Here these algebras are studied as interesting examples of metrized commutative algebras and in low dimensions they are described concretely in terms of nonstandard commutative multiplications on self-adjoint endomorphisms. The algebra of curvature tensors on a 3-dimensional Euclidean vector space is shown isomorphic to an orthogonally invariant deformation of the standard Jordan product on$3 \times 3$symmetric matrices. This algebra is characterized up to isomorphism in terms of purely algebraic properties of its idempotents and the spectra of their multiplication operators. On a vector space of dimension at least 4, the subspace of Weyl (Ricci-flat) curvature tensors is a subalgebra for which the multiplication endomorphisms are trace-free and the Killing type trace-form is a multiple of the nondegenerate invariant metric. This subalgebra is simple when the Euclidean vector space has dimension greater than 4. In the presence of a compatible complex structure, the analogous result is obtained for the subalgebra of Kähler Weyl curvature tensors. It is shown that the anti-self-dual Weyl tensors on a 4-dimensional vector space form a simple 5-dimensional ideal isometrically isomorphic to the trace-free part of the Jordan product on trace-free$3 \times 3$symmetric matrices.

Highlights

  • Let V be an n-dimensional real vector space equipped with a metric hi j

  • Theorem 1.5 shows that each of these subalgebras is isometrically isomorphic to the deunitalization of the 6-dimensional rank 3 simple real Euclidean Jordan algebra of symmetric endomorphisms of a 3-dimensional vector space

  • By the invariance of h, L◦ (e) is a self-adjoint endomorphism of A that preserves the orthogonal complement of the span of e

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Summary

Introduction

Let V be an n-dimensional real vector space equipped with a metric hi j. An interesting class of examples of exact Killing metrized commutative nonassociative algebras, relevant here for the statement of Theorem 1.5, are the deunitalizations of the finite-dimensional simple real Euclidean Jordan algebras. Theorem 1.5 shows that each of these subalgebras is isometrically isomorphic to the deunitalization of the 6-dimensional rank 3 simple real Euclidean Jordan algebra of symmetric endomorphisms of a 3-dimensional vector space. Theorem 1.5 and the computations used to prove Theorem 7.12 In this direction, Lemma 10.4 shows that when dim V∗ = 2n ≥ 4, certain of the idempotents produced by Lemma 10.1 constitute an orbit of O (2n) acting in MCW (V∗) identified with the space SO (2n)/U (n) of orthogonal complex structures on V inducing a given orientation on V. The subspace 2V∗ is a subalgebra of (⊗2V∗, [ · , · ]) and ( 2V∗, [ · , · ]) is isomorphic to the Lie algebra so(V, h)

General results about commutative algebras
Action of metric curvature tensors on tensors
Algebra structures on the space of metric curvature tensors
10. Idempotents in the subalgebra of Weyl curvature tensors
There hold the relations
11. Subalgebra of Kähler–Weyl tensors and the 4-dimensional case revisited
12. Examples related with idempotents
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