Laplace Operator and Polynomial Invariants

AbstractLetAbe a finite dimensional simple algebra (not necessarily associative) over the field of complex numbersC, and letGdenote the automorphism group Aut(A). Suppose thatAhas a symmetric nondegenerate associativeG-invariant bilinear form 〈x,y〉 and a compact real form, i.e., a subalgebraBoverRof dimension dimRB=dimCA, whereAis equal to the span ofBoverCand the restriction of 〈x,y〉 toBis positive definite. We describe generators of the algebra of polynomialG-invariants of a system of several vectors fromAin terms of 〈x,y〉 and Laplace operators. In particular, we give generators of the algebra of polynomial invariants of the adjoint representation of a simple linear algebraic group of any exceptional type ≠E6. As a consequence, we get the First Main Theorem on matrix invariants, invariants of minimal representation ofG2andF4