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By Konrad Schöbel

Konrad Schöbel goals to put the rules for a consequent algebraic geometric remedy of variable Separation, that's one of many oldest and strongest how to build certain options for the elemental equations in classical and quantum physics. the current paintings finds a stunning algebraic geometric constitution in the back of the well-known record of separation coordinates, bringing jointly a very good diversity of arithmetic and mathematical physics, from the overdue nineteenth century idea of separation of variables to trendy moduli area concept, Stasheff polytopes and operads.

"I am rather inspired by means of his mastery of various strategies and his skill to teach basically how they have interaction to supply his results.” (Jim Stasheff)

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We can make use of the symmetries of Sa1 a2 b1 b2 to bring these indices to the first position: ¯ α = g¯b2 d1 Sb b a a Sd d c c xa1 xa2 xc1 ∇α xb1 ∇β xd2 ∇γ xc2 N βγ 2 1 1 2 1 2 1 2 + g¯b1 c2 Sb1 b2 a1 a2 Sc2 c1 d1 d2 xa1 xa2 xc1 ∇α xd1 ∇β xb2 ∇γ xd2 . Renaming, lowering and rising indices appropriately finally results in j j i ¯αβγ = g¯ij S i N a 2 b 1 b 2 S c 2 d 1 d 2 + S c2 b 1 b 2 S d 1 a 2 d 2 xb1 xb2 xd1 ∇α xa2 ∇β xc2 ∇γ xd2 . 3) and transform them into purely algebraic integrability conditions.

Again, the lemma could also be deduced from the symmetry classification of Riemann tensor polynomials [FKWC92]. We have shown the equivalence of the second integrability condition to g¯ij g¯kl S ikb1 b2 S jc2 d1 d2 + S ic2 b1 b2 S jd1 kd2 S lf2 e1 e2 b2 b1 d 1 e 1 e 2 c2 d2 f2 x b 1 x b 2 x d 1 x e 1 x e 2 u c2 v d 2 w f 2 = 0 ∀x ∈ M, ∀u, v, w ∈ Tx M. 3 The 2nd integrability condition 43 As before, the restrictions on the vectors u, v, w and x can be dropped, which allows us to write this condition independently of x, u, v, w ∈ V as b2 b1 d 1 e 1 e 2 c2 d2 f2 g¯ij g¯kl S ikb1 b2 S jc2 d1 d2 + S ic2 b1 b2 S jd1 kd2 S lf2 e1 e2 = 0.

The 2nd integrability condition . . . . . Redundancy of the 3rd integrability condition . . 30 . . 34 . . 40 . . Commuting Killing tensors . . . . . . . . 48 49 In this chapter we translate the Nijenhuis integrability conditions for a Killing tensor on a constant curvature manifold into algebraic conditions on the corresponding algebraic curvature tensors. 3) and then use the representation theory for general linear groups to get rid of the dependence on the base point in the manifold.

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