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 is one of many oldest and strongest the right way to build particular options for the elemental equations in classical and quantum physics. the current paintings finds a shocking algebraic geometric constitution at the back of the recognized record of separation coordinates, bringing jointly an outstanding variety of arithmetic and mathematical physics, from the overdue nineteenth century conception of separation of variables to fashionable moduli house concept, Stasheff polytopes and operads.

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

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S p V ⊗Λq V → S p+1 V ⊗Λq−1 V → . . → S d V → 0, known as the Koszul complex. The permutation group Sd acts on d-fold covariant or contravariant tensors by permuting indices. This action extends linearly to an action 34 1 The foundation: the algebraic integrability conditions of the entire group algebra. In particular, any Young tableau acts on tensors with corresponding indices. For example, b1 a2 c2 T b 1 a 2 c 2 = Tb 1 a 2 c 2 − T a 2 b 1 c 2 − T b 1 c 2 a 2 − T c 1 a 2 b 2 + T a 1 c 2 b 2 + T c 1 b 2 a 2 .

For compactness of notation, let us denote just in this proof the covariant derivative of Kαβ by Kαβ,γ instead of ∇γ Kαβ . A Killing tensor is symmetric by definition. e. K α β = λα δ α β (no sum). 2) reads Nαβγ = (λα − λβ )Kα[β,γ] . 3) we get 0 = N δ[βγ gα]δ = 0 = N δ[βγ Kα]δ = α β γ α β γ Nαβγ = α β γ λα Nαβγ = 0 = N δ[βγ Kα]ε K εδ = α β γ (λα − λβ )Kαβ,γ α β γ λα (λα − λβ )Kαβ,γ λ2α Nαβγ = α β γ λ2α (λα − λβ )Kαβ,γ . 5 Commuting Killing tensors 49 As before, the Young operators stand for a complete antisymmetrisation in the indices α, β and γ.

3) is redundant. Proof. For compactness of notation, let us denote just in this proof the covariant derivative of Kαβ by Kαβ,γ instead of ∇γ Kαβ . A Killing tensor is symmetric by definition. e. K α β = λα δ α β (no sum). 2) reads Nαβγ = (λα − λβ )Kα[β,γ] . 3) we get 0 = N δ[βγ gα]δ = 0 = N δ[βγ Kα]δ = α β γ α β γ Nαβγ = α β γ λα Nαβγ = 0 = N δ[βγ Kα]ε K εδ = α β γ (λα − λβ )Kαβ,γ α β γ λα (λα − λβ )Kαβ,γ λ2α Nαβγ = α β γ λ2α (λα − λβ )Kαβ,γ . 5 Commuting Killing tensors 49 As before, the Young operators stand for a complete antisymmetrisation in the indices α, β and γ.

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