Capelli's Identity - Generalizations - The Mukhin–Tarasov–Varchenko Identity and The Gaudin Model - Statement

Statement

Consider matrices X and D as in Capelli's identity, i.e. with elements and at position (ij).

Let z be another formal variable (commuting with x). Let A and B be some matrices which elements are complex numbers.

detleft( frac{partial}{partial_z} - A - X frac{1}{z-B} D^t right)
={det}^text{calculate as if all commute}_{text{Put all }xtext{ and }ztext{ on the left, while all derivations on the right}}
left( frac{partial}{partial_z} - A - X frac{1}{z-B} D^t right)

Here the first determinant is understood (as always) as column-determinant of a matrix with non-commutative entries. The determinant on the right is calculated as if all the elements commute, and putting all x and z on the left, while derivations on the right. (Such recipe is called a Wick ordering in the quantum mechanics).

Read more about this topic:  Capelli's Identity, Generalizations, The Mukhin–Tarasov–Varchenko Identity and The Gaudin Model

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