For any Lie algebra extension, satisfying suitable conditions, a simple method for the construction of “constant invariant” forms of all degrees (and hence for the construction of characteristic classes of all even dimensions) is given. Taking a Lie algebra extension, associated with a smooth real vector bundle, one obtains a well known system of generators of the classical characteristic ring of the vector bundle.
In this Note the Author continues the investigations of [2] on the construction of “constant invariant” forms (and hence of characteristic classes) for any Lie algebra extension, satisfying suitable conditions.
The multilinear forms, obtained by polarizing the coefficients of the characteristic polynomial of a matrix, are considered. A general relation (formula A) between such forms is proved. It follows in particular a rational expression for the above-mentioned coefficients (formula C), which is in a sense analogous to Newton's formulas, but with the use of the determinant function instead of the trace function.
A classification theorem for (R, F)-extensions of Lie algebras is proved, which generalizes a classical result of C. Chevalley and S. Eilenberg.
The multilinear forms, obtained by polarizing the coefficients of the characteristic polynomial of a matrix, are considered. A general relation (formula A) between such forms is proved. It follows in particular a rational expression for the above-mentioned coefficients (formula C), which is in a sense analogous to Newton's formulas, but with the use of the determinant function instead of the trace function.
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