Capelli identity

The Capelli identity equates a determinant of differential operators with a shifted determinant built from products of coordinate and derivative operators. Its correction terms account for the noncommutativity of the Weyl algebra.

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Weyl algebra: An associative algebra generated by coordinates and derivatives with commutation relations encoding differentiation. The identity lives in this algebra, where coordinate and derivative operators do not commute.
Differential operator: An operator defined using derivatives that acts on functions or other mathematical objects. The left-hand determinant is formed from differential operators.
Alfred Capelli: An Italian mathematician known for work in invariant theory and algebra. Capelli published the identity that now bears his name.
Representation theory: The study of algebraic structures through their actions by linear transformations on vector spaces. The identity supplies operator relations useful in studying representations of general linear groups.
Commutative determinant: The usual determinant of a matrix whose entries belong to a commutative ring. It obeys familiar determinant rules that cannot be applied directly to the identity’s operator entries.
Capelli determinant: A determinant-like expression in noncommuting generators of the general linear Lie algebra, with prescribed operator ordering. It is the shifted operator determinant appearing on the identity’s right-hand side.
General linear Lie algebra: The Lie algebra of all square matrices under the commutator bracket. Coordinate-derivative products furnish operators representing its standard generators.
Invariant theory: The study of quantities unchanged under transformations of algebraic objects. The identity arose in calculations involving invariants under linear transformations.
Howe duality: A correspondence between representations of two commuting groups acting on the same vector space. Capelli-type operators help describe commuting actions and their invariant operators.
Cayley–Hamilton theorem: The theorem that every square matrix satisfies its own characteristic polynomial. It is a classical matrix identity, unlike Capelli’s correction for noncommuting differential operators.