The classical Capelli identity is an equality of differential operators written in a form roughly resembling the classical identity $det(A B) = det(A) det(B)$ where however at the right hand side are true determinants of commutative entries and at the left side is a column determinant whose entries are (corrections of) non-commuting vector fields, so called polarization operators.
There are numerous generalizations.
Define the polarization operators $E_{i j} = \sum_{a=1}^n x_{i a}\frac{\partial}{\partial x_{j a}}$ for $i,j\in\{1,\ldots n\}$. These operators satisfy the commutation relations of the Lie algebra $gl(n)$.
Define the column determinant of a matrix $A = (a_{i j})_{i,,j=1,\ldots,n}$ with not necessarily commuting entries
This is in generally an ill-behaved object of comparing to the case of the determinant of a matrix with commuting entries; in particular the other formulas for a determinant give different results in noncommutative case. Then
A treatment in terms of quasideterminants is in 3.7 of
Israel Gelfand, Sergei Gelfand, Vladimir Retakh, Robert Lee Wilson, Quasideterminants, Advances in Mathematics 193 (2005) 56–141 doi
Bertram Kostant, Siddharta Sahi, Jordan algebras and Capelli identities, Invent. math. 112 (1993) 657–664
The purpose of this paper is to establish a connection between semisimple Jordan algebras and certain invariant differential operators on symmetric spaces; and to prove an identity for such operators which generalizes the classical Capelli identity.
Last revised on July 31, 2024 at 09:25:18. See the history of this page for a list of all contributions to it.