Left-symmetric algebra structures on the W-algebra W(2, 2)

We study the compatible left-symmetric algebra structures on the W-algebra W(2, 2) with some natural grading conditions. The results of earlier work on left-symmetric algebra structures on the Virasoro algebra play an essential role in determining these compatible structures. As a corollary, any such left-symmetric algebra contains an infinite-dimensional trivial subalgebra that is also a submodule of the regular module.