nLab Capelli identity

Idea

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.

Statement

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

$$det_c(A) = \sum_{\sigma\in\Sigma(n)}sign(\sigma) a_{\sigma(1)1}a_{\sigma(2)2}\cdots a_{\sigma(n)n}$$

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

$$det_c((E_{i j} - (n-i)\delta_{i j})_{i j}) = det((x_{i j})_{i j}) det((\frac{\partial}{\partial x_{i j}})_{i j})$$

Literature

• wikipedia Capelli’s identity

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.

• Roger Howe, T. Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann. 290 (1991) 565–619
• E. Mukhin, V. Tarasov, A. Varchenko, A generalization of the Capelli identity, pdf
• Maxim Nazarov, Yangians and Capelli identities, in: Kirillov’s seminar on representation theory, Amer. Math. Soc. Translations: 181 (Series 2) 1998 doi
• Dimitri Gurevich, Varvara Petrova, Pavel Saponov, Matrix Capelli identities related to reflection equation algebra, J. Geom. Phys. 179 (2022) 104606 doi

category: algebra