Nonassociative gradings on infinite-dimensional triangular algebras

Abstract

We study group gradings on triangular algebras relative to a well-ordered basis, with multiplication $x{\circ }_{\lambda }y=xy+\lambda yx$. For $\lambda \notin \{0,\pm 1\}$, every grading on a finite-dimensional triangular algebra is isomorphic to an elementary grading. In infinite dimension, the same conclusion holds for the algebra of finite-range maps, for the finite-map algebra on a basis of order type $\omega$, and for closed gradings on the full triangular algebra. The infinite-dimensional results also hold for $\lambda =1$ in characteristic different from $2$. For intermediate algebras whose finite-range ideal is graded, we obtain an elementary description relative to a unitriangular basis. The proofs construct neutral associative idempotents with singleton diagonal supports. We also prove that the radical of the full triangular algebra is graded for every nonzero $\lambda$, and that gradings on the finite-range algebra and closed gradings on the full algebra have finite support under the classification hypotheses.