Categorifications of the polynomial ring. (English) Zbl 1335.16016
The authors categorify the ring \(\mathbb Z[x]\) by interpreting it as the Grothendieck ring of the monoidal category of finitely generated projective modules over a certain idempotented diagrammatically defined ring \(A\). It is also shown that the ring \(A\) has a version of the BGG reciprocity, where monomials \(x^n\) become indecomposable projective modules while the polynomials \((x-1)^n\) become standard modules. The ring \(A\) has a purely topological definition and all involved modules have clear diagrammatic description.
Reviewer: Volodymyr Mazorchuk (Uppsala)
MSC:
| 16S36 | Ordinary and skew polynomial rings and semigroup rings |
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
| 16E20 | Grothendieck groups, \(K\)-theory, etc. |
References:
| [1] | [1]D. J. Benson, Representations and Cohomology I, Cambridge Stud. Adv. Math. 30, Cambridge Univ. Press, Cambridge, 1991. 280M. Khovanov and R. Sazdanovic |
| [2] | [2]J. Bernstein, I. Gelfand, and S. Gelfand, A category of g-modules, Functional Anal. Appl. 10 (1976), 87–92. · Zbl 0353.18013 |
| [3] | [3]S. Gelfand and Yu. Manin, Methods of Homological Algebra, Springer, 1996. · Zbl 0855.18001 |
| [4] | [4]M. Khovanov and R. Sazdanovic, Categorification of orthogonal polynomials, in preparation. |
| [5] | [5]D. Mili\check{}ci\'{}c, Lectures on Derived Categories, http://www.math.utah.edu/\tilde{}milicic/ Eprints/dercat.pdf. |
| [6] | [6]R. Sazdanovi\'{}c, Categorification of knot and graph polynomials and the polynomial ring, GWU electronic dissertation published by ProQuest (2010); http://surveyor. gelman.gwu.edu/. |
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