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By Konrad Schöbel

Konrad Schöbel goals to put the rules for a consequent algebraic geometric remedy of variable Separation, that is one of many oldest and strongest easy methods to build detailed strategies for the basic equations in classical and quantum physics. the current paintings finds a shocking algebraic geometric constitution at the back of the well-known checklist of separation coordinates, bringing jointly a superb variety of arithmetic and mathematical physics, from the overdue nineteenth century thought of separation of variables to fashionable moduli area conception, Stasheff polytopes and operads.

"I am fairly inspired via his mastery of numerous concepts and his skill to teach essentially how they have interaction to supply his results.” (Jim Stasheff)

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8) we get xa1 xa2 xc1 xc2 v [b1 wd2 ] 1 b1 1 = · a1 a2 c1 2! d2 4! = 4! · 3! 10368 c2 a1 a2 c 1 c 2 b1 d2 xa1 xa2 xc1 xc2 v [b1 wd2 ] + 2! · 5! 34560 b1 a 1 a 2 c 1 c 2 d2 x a 1 x a 2 x c1 x c 2 v b 1 w d 2 . 41), this yields ˜c d c d g b2 d 1 Ra1 b 1 a2 b2 R 1 1 2 2 1 72 a1 a2 c 1 c 2 b1 d2 + 1 144 b1 a 1 a 2 c 1 c 2 d2 x a 1 x a 2 x c 1 x c2 v b 1 w d 2 = 0. 52 1 The foundation: the algebraic integrability conditions As before, we can replace the Young symmetrisers acting on upper indices by their corresponding adjoints acting on the corresponding lower indices: 1 72 a1 a2 c 1 c 2 b1 d2 + 1 144 b1 a 1 a 2 c 1 c 2 d2 ˜c d c d g b2 d 1 Ra1 b 1 a2 b2 R 1 1 2 2 xa1 xa2 xc1 xc2 v b1 wd2 = 0.

5 Commuting Killing tensors 49 As before, the Young operators stand for a complete antisymmetrisation in the indices α, β and γ. 38c) 0= (λ2α + λ2β )(λα where “+cyclic” stands for a summation over the cyclic permutations of α, β, γ. 3). 39) 0 = Kαβ,γ + cyclic. 38c). To this end we write the first three equations for Kαβ,γ in matrix form as ⎤⎡ ⎤ ⎡ 1 1 1 Kαβ,γ ⎣λα − λβ λβ − λγ λγ − λα ⎦ ⎣Kβγ,α ⎦ = 0. λ2α − λ2β λ2β − λ2γ λ2γ − λ2α Kγα,β The determinant of the coefficient matrix is a multiple of the Vandermode determinant.

The 1st integrability condition . . . . . 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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