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In mathematics, Capelli's identity, named after , 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 . It can be used to relate an invariant ''ƒ'' to the invariant Ω''ƒ'', where Ω is Cayley's Ω process. ==Statement== Suppose that ''x''''ij'' for ''i'',''j'' = 1,...,''n'' are commuting variables. Write ''E''ij for the polarization operator : 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 are matrices with elements ''E''ij, ''x''ij, on the left hand side. For generic noncommutative matrices formulas like : do not exist, and the notion of the 'determinant' itself does not make sense for generic noncommutative matrices. That is why the Capelli identity still holds some mystery, despite many proofs offered for it. A very short proof does not seem to exist. Direct verification of the statement can be given as an exercise for ''n' = 2, but is already long for ''n'' = 3. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Capelli's identity」の詳細全文を読む スポンサード リンク
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