We continue the study of the lower central series and its associated graded components for a free associative algebra with n generators, as initiated in Feigin and Shoikhet (2007) [FS]. We establish a linear bound on the degree of tensor field modules appearing in the Jordan-Hölder series of each graded component. We also bound the leading coefficient of the Hilbert polynomial of each graded component. As applications, we confirm conjectures of P. Etingof and B. Shoikhet concerning the structure of the third graded component.

Integrals of characteristic classes of tautological sheaves on the Hilbert scheme of points on a surface frequently arise in enumerative problems. We use the K-theoretic Donaldson-Thomas theory of certain toric Calabi-Yau threefolds to study K-theoretic variants of such expressions.We study limits of the K-theoretic Donaldson-Thomas partition function of a toric Calabi-Yau threefold under certain one-parameter subgroups called slopes and formulate a condition under which two such limits coincide. We then explicitly compute the limits of components of the partition function under so-called preferred slopes, obtaining explicit combinatorial expressions related to the refined topological vertex of Iqbal, Kosçaz and Vafa.Applying these results to specific Calabi-Yau threefolds, we deduce dualities satisfied by a generating function built from tautological bundles on the Hilbert scheme of points on C 2 . We then use this duality to study holomorphic Euler characteristics of exterior and symmetric powers of tautological bundles on the Hilbert scheme of points on a general surface.

We provide a generators and relation description of the deformed W 1+∞ -algebra introduced in previous joint work of E. Vasserot and the second author. This gives a presentation of the (spherical) cohomological Hall algebra of the one-loop quiver, or alternatively of the spherical degenerate double affine Hecke algebra of GL(∞).

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