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 designated options for the basic 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 outstanding variety of arithmetic and mathematical physics, from the past due nineteenth century concept of separation of variables to fashionable moduli area conception, Stasheff polytopes and operads.
"I am rather inspired through his mastery of various strategies and his skill to teach sincerely how they have interaction to provide his results.” (Jim Stasheff)
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Additional resources for An Algebraic Geometric Approach to Separation of Variables
13), we get g¯ij S ia2 b1 b2 S jc2 d1 d2 ⎛⎛ ⎝⎝ b2 b1 d 1 c2 d2 a2 ⎞ + c 2 b2 b1 d 1 d2 a2 ⎞ ⎠ xb1 xb2 xd1 u[a2 v c2 wd2 ] ⎠ = 0. 14) Symmetrising the upper indices in a contraction is equivalent to symmetrising the corresponding lower indices and the same is true for antisymmetrisation. Hence we can replace a Young tableau acting on upper indices in a contraction by its dual acting on the corresponding lower indices. 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.
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. If the eigenvalues λα , λβ , λγ are pairwise different, this implies that Kαβ,γ = Kβγ,α = Kγα,β = 0. If exactly two of the eigenvalues are equal, say λα = λβ = λγ , then we have Kαβ,γ = − 12 Kβγ,α = Kγα,β . For three equal eigenvalues the only restriction on Kαβ,γ is the Killing equation Kαβ,γ + Kβγ,α + Kγα,β = 0.
38 1 The foundation: the algebraic integrability conditions We will now give a number of equivalent formulations. 9. 3a) for a Killing tensor on a constant curvature manifold M . 21) where P is any of the following pseudo-projectors. 22). 24) where the wedge product is deﬁned by taking the exterior product in the Λ2 V -component and usual matrix multiplication in the End(V )-component. Proof. 20). 21a). 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 .
An Algebraic Geometric Approach to Separation of Variables by Konrad Schöbel