We continue the study of the lower central series of a free associative
algebra, initiated by B. Feigin and B. Shoikhet (arXiv:math/0610410). We
generalize via Schur functors the constructions of the lower central series to
any symmetric tensor category; specifically we compute the modified first
quotient \bar{B}_1, and second and third quotients B_2, and B_3 of the series
for a free algebra T(V) in any symmetric tensor category, generalizing the main
results of (arXiv:math/0610410) and (arXiv:0902.4899). In the case
A_{m|n}:=T(\CC^{m|n}), we use these results to compute the explicit Hilbert
series. Finally, we prove a result relating the lower central series to the
corresponding filtration by two-sided associative ideals, confirming a
conjecture from (arXiv:0805.1909), and another one from (arXiv:0902.4899), as
corollaries.Comment: Corrected an error in the proof of Lemma 6.1 pointed out by Alexei
Krasilniko
We describe a family of compactifications of the space of Bridgeland stability conditions of any triangulated category following earlier work by Bapat, Deopurkar, and Licata. We particularly consider the case of the 2-Calabi-Yau category of the A 2 quiver. The compactification is the closure of an embedding (depending on q) of the stability space into an infinite-dimensional projective space.In the A 2 case, the three-strand braid group B 3 acts on this closure. We describe two distinguished braid group orbits in the boundary, points of which can be identified with certain rational functions in q. Points in one of the orbits are exactly the q-deformed rational numbers recently introduced by Morier-Genoud and Ovsienko, while the other orbit gives a new q-deformation of the rational numbers. Specialising q to a positive real number, we obtain a complete description of the boundary of the compactification.
Given certain intersection cohomology sheaves on a projective variety with a
torus action, we relate the cohomology groups of their tensor product to the
cohomology groups of the individual sheaves. We also prove a similar result in
the case of equivariant cohomology.Comment: 8 page
The Bernstein-Sato polynomial, or the b-function, is an important invariant of hypersurface singularities. The local topological zeta function is also an invariant of hypersurface singularities that has a combinatorial description in terms of a resolution of singularities. The Strong Topological Monodromy Conjecture of Denef and Loeser states that poles of the local topological zeta function are also roots of the b-function.We use a result of Opdam to produce a lower bound for the b-function of hyperplane arrangements of Weyl type. This bound proves the "n/d conjecture", by Budur, Mustaţȃ, and Teitler for this class of arrangements, which implies the Strong Monodromy Conjecture for this class of arrangements.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.