Formally-radical Functions in Elements of a Nilpotent Lie Algebra and Noncommutative Localizations

In the present paper, we introduce the sheaf 𝔗𝔤 of germs of non-commutative holomorphic functions in elements of a finite-dimensional nilpotent Lie algebra 𝔤, which is a sheaf of non-commutative Fréchet algebras over the character space of 𝔤. We prove that 𝔗𝔤(D) is a localization over the universal enveloping algebra [Formula: see text] whenever D is a polydisk, which in turn allows to describe the Taylor spectrum of a supernilpotent Lie algebra of operators in terms of the transversality.

PROJECTIVE SPECTRUM IN BANACH ALGEBRAS

For a tuple A = (A1, A2, …, An) of elements in a unital algebra [Formula: see text] over ℂ, its projective spectrumP(A) or p(A) is the collection of z ∈ ℂn, or respectively z ∈ ℙn-1 such that A(z) = z1A1 + z2A2 + ⋯ + znAn is not invertible in [Formula: see text]. In finite dimensional case, projective spectrum is a projective hypersurface. When A is commuting, P(A) looks like a bundle over the Taylor spectrum of A. In the case [Formula: see text] is reflexive or is a C*-algebra, the projective resolvent setPc(A) := ℂn \ P(A) is shown to be a disjoint union of domains of holomorphy. [Formula: see text]-valued 1-form A-1(z)dA(z) reveals the topology of Pc(A), and a Chern–Weil type homomorphism from invariant multilinear functionals to the de Rham cohomology [Formula: see text] is established.