Capelli identity

In mathematics, Capelli's identity, named after Alfredo Capelli (1887), is an analogue of the formula det(AB) = det(A) det(B), for certain matrices with noncommuting entries, related to the representation theory of the Lie algebra ${\displaystyle {\mathfrak {gl}}_{n}}$. It can be used to relate an invariant ƒ to the invariant Ωƒ, where Ω is Cayley's Ω process.

Suppose that x_ij for i,j = 1,...,n are commuting variables. Write E_ij for the polarization operator

The Capelli identity states that the following differential operators, expressed as determinants, are equal:

Both sides are differential operators. The determinant on the left has non-commuting entries, and is expanded with all terms preserving their "left to right" order. Such a determinant is often called a column-determinant, since it can be obtained by the column expansion of the determinant starting from the first column. It can be formally written as

where in the product first come the elements from the first column, then from the second and so on. The determinant on the far right is Cayley's omega process, and the one on the left is the Capelli determinant.

The operators E_ij can be written in a matrix form:

where ${\displaystyle E,X,D}$ are matrices with elements E_ij, x_ij, ${\displaystyle {\frac {\partial }{\partial x_{ij}}}}$ respectively. If all elements in these matrices would be commutative then clearly ${\displaystyle \det(E)=\det(X)\det(D^{t})}$. The Capelli identity shows that despite noncommutativity there exists a "quantization" of the formula above. The only price for the noncommutativity is a small correction: ${\displaystyle (n-i)\delta _{ij}}$ on the left hand side. For generic noncommutative matrices formulas like

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