Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0511571

The Algebra $P_n$ is Koszul

David Nacin
View original source ↗Read source document ↗Export metadata ↓

Abstract

The algebras $Q_n$ describe the relationship between the roots and coefficients of a non-commutative polynomial. I.Gelfand, S.Gelfand, and V. Retakh have defined quotients of these algebras corresponding to graphs. In this work we find the Hilbert series of the class of algebras corresponding to the $n$-vertex path, $P_n$. We also show this algebra is Koszul. We do this by first looking at class of quadratic algebras we call Partially Generator Commuting. We then find a sufficient condition for a PGC-Algebra to be Koszul and use this to show a similar class of PGC algebras, which we call ch$P_n$, is Koszul. Then we show it is possible to extend what we did to the algebras $P_n$ although they are not PGC. Finally we examine the Hilbert Series of the algebras $P_n$

Explore related subjects

math.QA

Keep this discovery

BibTeXRIS

David Nacin. 2005-11-22. The Algebra $P_n$ is Koszul. https://arxiv.org/abs/math/0511571

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On Diagrammatic Categorification of Verma Modules I: Braiding

In this paper, we study the extensions of KLRW algebras to tensor products of Verma module representations of $\mathfrak{sl}_2$. Our motivation is to construct a theory of Khovanov homology for knot complements in $S^3$ (and which also categorifies the Gukov-Manolescu two-variable series for knot complements), which will be done in the second part of this work. We construct the categorification of R-matrices for Verma modules as functors given by derived tensor products with diagrammatic bimodules and explicitly compute their projective resolutions. We also prove these braiding functors induce an action of the braid group on the relevant categories. Then, we describe how to incorporate strands in finite-dimensional representations of $\mathfrak{sl}_2$, thereby establishing functors that serve as the Khovanov homology on a braid complement. In the case of the unknot, this gives knot homologies in $S^1\times D^2$, which we compare to Annular Khovanov Homology through several examples and show they are very closely related, conjecturing they are of the same dimension. We conclude with a proposal for the categorification of the cups and caps of Verma module colored strands, which we build upon in the next paper.

math.QA↗
arXiv2026

Some finite dimensional representations of shifted quantum affine algebras of type A

In this paper, we study finite dimensional representations of shifted quantum affine algebras of type A. We give an explicit description of the tensor product of simple evaluation modules of the quantum loop algebra and a one-dimensional representation of the shifted quantum affine algebra under the separation condition. As a consequence, we give the q-characters of some finite dimensional simple modules of the shifted quantum affine algebra.

math.QA↗