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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 you can build specified ideas for the basic equations in classical and quantum physics. the current paintings unearths a stunning 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 concept of separation of variables to fashionable moduli house conception, Stasheff polytopes and operads.

"I am relatively inspired by means of his mastery of quite a few recommendations and his skill to teach essentially how they have interaction to provide his results.” (Jim Stasheff)

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Consequently, the previous equation is equivalent to ⎛⎛ ⎞ ⎞ ⎝⎝ b 2 b1 d 1 c2 d2 a2 + c 2 b2 b1 d 1 d2 a2 ⎠ g¯ij S ia 2 b1 b2 S jc2 d1 d2 ⎠ xb1 xb2 xd1 u[a2 v c2 wd2 ] = 0. 14) vanishes identically. 7. 17) S ia1 b1 b2 + S ib1 b2 a1 + S ib2 a1 b1 = 0 in b1 , b2 : b1 b2 S ia2 b1 b2 = −2 b1 b2 S ib1 b2 a2 . 18) We refer to this identity as symmetrised Bianchi identity. Proof. 16) is c 2 b2 b 1 d 1 d2 a2 = g¯ij S ia2 b1 b2 S jc2 d1 d2 = c 2 b2 b1 d 1 c 2 b2 b1 d 1 c2 d2 a2 g¯ij S ia2 b1 b2 S jc2 d1 d2 gij S ia2 b1 b2 S jc2 d1 d2 −S id2 b1 b2 S jc2 d1 a2 +S id2 b1 b2 S ja2 d1 c2 −S ia2 b1 b2 S jd2 d1 c2 +S ic2 b1 b2 S jd2 d1 a2 −S ic2 b1 b2 S ja2 d1 d2 Regard the parenthesis under complete symmetrisation in c2 , b2 , b1 and d1 .

16). 21d) is trivial. 21c) through a stepwise manipulation of b2 b1 d 1 c2 d2 a2 g¯ij S ia2 b1 b2 S jc2 d1 d2 = b2 b1 d 1 b2 c2 d2 a2 g¯ij S ia2 b1 b2 S jc2 d1 d2 . 25a) In order to sum over all q! 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 .

This already solves part (ii) of Problem I. Exploiting the algebraic geometric structure of the Killing-St¨ ackel variety then allows us to deduce the following results for the sphere S3 . 42: An algebraic geometric description of the space I(S3 ) of integrable Killing tensors on S3 . This solves part (iii) of Problem I. 12: A set of polynomial isometry invariants characterising the integrability of an arbitrary Killing tensor on S3 . This solves part (iv) of Problem I. 4: A juxtaposition between algebraic geometric properties of the KS variety on one side (such as singularities, projective lines and projective planes on the variety) and geometric properties of the corresponding Killing tensors on the other side.

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