An Algebraic Geometric Approach to Separation of Variables by Konrad Schöbel

By Konrad Schöbel

Konrad Schöbel goals to put the principles for a consequent algebraic geometric therapy of variable Separation, that is one of many oldest and strongest how to build unique suggestions for the basic equations in classical and quantum physics. the current paintings finds a shocking algebraic geometric constitution in the back of the recognized checklist of separation coordinates, bringing jointly a very good variety of arithmetic and mathematical physics, from the overdue nineteenth century idea of separation of variables to fashionable moduli house thought, Stasheff polytopes and operads.

"I am really inspired through his mastery of quite a few recommendations and his skill to teach sincerely how they have interaction to provide his results.” (Jim Stasheff)

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Permutations of q indices, one can take the sum over q cyclic permutations, choose one index and then sum over all (q − 1)! permutations of the remaining (q − 1) indices. 25b) . For a better readability we underlined each antisymmetrised index. 25b) = b2 b1 d 1 c2 d2 a2 g¯ij S ia2 b1 b2 S jc 2 d1 d2 + S ic2 b1 d2 S ja 2 d 1 b2 + 12 S ic2 b1 b2 S jd 2 d 1 a2 + 12 S id2 b1 a2 S jc 2 d1 b2 . 25c) = b2 b 1 d 1 c2 d2 a2 g¯ij S ia2 b1 b2 S jc 2 d1 d2 + S ia2 d1 b2 S jc 2 b1 d2 + 12 S ic2 b1 b2 S jd 2 d 1 a2 + 12 S ic2 d1 b2 S jd 2 b1 a2 .

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.

5. 1 q! a1 .. aq 1 · p! 2 + s1 · · · sp a1 s1 · · · sp a1 aq aq .. .. 2 a1 s1 · · · sp a1 s 1 · · · s p aq aq .. .. 11) In particular, for p = q = 3: 1 3! c2 d2 a2 1 · 3! b2 b1 d 1 1 = 7 4 2 3 b2 b1 d 1 c2 d2 a2 b2 b 1 d 1 c2 d2 a2 + 1 7 2 34 c 2 b 2 b1 d 1 d2 a2 c 2 b2 b1 d 1 d2 a2 . 12) Proof. 11) as P = P1 + P2 . 9), one easily checks that P , P1 and P2 are orthogonal projectors verifying P1 P2 = 0 = P2 P1 , P P1 = P1 and P P2 = P2 . Therefore P1 + P2 is an orthogonal projector with image im P1 ⊕ im P2 ⊆ im P .

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