Polynomial Lie algebras
Let \(P\) be an algebra of complex polynomials in \(\lambda_0, \ldots, \lambda_n\) and \(L_P\) the free left \(P\)-module with a basis \(1, l_0, \ldots, l_n\). Define the structure of a Lie algebra \(\mathfrak a(C,V)\) on \(L_P\) by setting \[ [l_i,l_j]=\sum c_{i,j}^k(\lambda)l_k,\quad [l_i,\lambda_q]=v_{i,q}(\lambda), \quad [\lambda_i,\lambda_j]=0, \] where \(c_{i,j}^k, v_{i,q}(\lambda)\in P\) and \(C=\{c_{i,j}^k\}\), \(V=\{v_{i,q}\}\). It is easy to see that the linear span of \( l_0, \ldots, l_n\) is a Lie subalgebra and the matrix \(V\) induces a representation of \(\mathfrak a(C,V)\) into the Lie algebra \(\text{Der}\, P\) of derivations of the algebra \(P\). A Lie algebra \(\mathfrak a(C,V)\) is non-degenerate if \(\Delta=\det V\neq 0\). The algebra \(\mathfrak a(0,1)\) is isomorphic to the canonical Lie algebra on \(\mathcal L=P \oplus P\partial_1\oplus \cdots \oplus P\partial_n.\) A homomorphism of \(P\)-modules \(L_P\to \mathcal L\) defined by \[ l_j\mapsto L_j=\sum_qv_{j,q}(\lambda)\partial_q \] is a homomorphism of Lie algebras. This representation is faithful if and only if \(\mathfrak a(C,V)\) is non-degenerate. One of the main results of the paper shows that if \(\mathfrak a(C,V)\) is non-degenerate then \[ L_i(\Delta)=\phi_i\Delta,\quad \phi_i\in P, \] and the vector fields \(L_j\) are tangent to the hyperspace \(\{(\lambda_0, \ldots, \lambda_n)\in \mathbb C^{n+1}\mid \Delta=0\}\). The authors study changes of \(C\) under actions of invertible linear operators on \(V\) and gradings on \(\mathfrak a(C,V)\). In terms of an operator of a symmetrization a structure of a coassociative coalgebra on an affine commutative algebra is constructed. This construction is applied to a study of convolution matrices of invariants for isolated singularities of functions.
scientific article; zbMATH DE number 1930674