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 remedy of variable Separation, that's one of many oldest and strongest ways to build particular options for the elemental equations in classical and quantum physics. the current paintings unearths a stunning algebraic geometric constitution at the back of the recognized record of separation coordinates, bringing jointly an exceptional variety of arithmetic and mathematical physics, from the past due nineteenth century concept of separation of variables to trendy moduli house concept, Stasheff polytopes and operads.

"I am fairly inspired by means of his mastery of numerous ideas and his skill to teach truly how they have interaction to provide his results.” (Jim Stasheff)

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To continue, antisymmetrise 0= c2 d2 a2 =2 c2 d2 a2 b2 b1 d 1 g¯ij S ia2 b1 b2 S jc g¯ij S ia2 b1 b2 S jc 2 d1 d2 2 d1 d2 + S ia2 b2 d1 S jc 2 b1 d2 + S ia2 d1 b1 S jc 2 b2 d2 in a2 , b2 , c2 , d2 . Then the last term vanishes by the symmetry of S j c2 b2 d2 in b2 , d2 and yields 0= a2 b2 c2 d2 g¯ij S ia2 b1 b2 S jc 2 d1 d2 + S ia2 d1 b2 S jc 2 b1 d2 . Both sum terms are equal under antisymmetrisation in a2 , b2 , c2 , d2 and contraction with g¯ij . Indeed, exchanging b1 and d1 is tantamount to exchanging a2 with c2 and b2 with d2 and renaming i, j as j, i.

3 Isokernel planes and integrable Killing tensors from S2 K.

1 Decomposition . . . . . . . . . . . 2 The action of the isometry group . . . . . . 3 Aligned algebraic curvature tensors . . . . . . 4 Diagonal algebraic curvature tensors . . . . . 5 The residual action of the isometry group . . . . 2 Solution of the algebraic integrability conditions . 1 Reformulation of the first integrability condition . . 2 Integrability implies diagonalisability . . . . . 3 Solution of the second integrability condition . . . 4 Interpretation of the Killing-St¨ ackel variety .

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