Framed moduli and Grassmannians of submodules

Fedotov, Stanislav

doi:10.1090/S0002-9947-2013-05764-9

Framed moduli and Grassmannians of submodules
HTML articles powered by AMS MathViewer

by Stanislav Fedotov PDF

Trans. Amer. Math. Soc. 365 (2013), 4153-4179 Request permission

Abstract:

In this work we study a realization of moduli spaces of framed quiver representations as Grassmannians of submodules devised by Markus Reineke. Obtained is a generalization of this construction to finite dimensional associative algebras and for quivers with oriented cycles over an arbitrary infinite field. As an application we get an explicit realization of fibers for the moduli space bundle over the categorical quotient for the quiver $A_{n-1}^{(1)}$ and the ground fields $\mathbb {C}$ and $\mathbb {R}$.

References

Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995. MR 1314422, DOI 10.1017/CBO9780511623608
Klaus Bongartz, A geometric version of the Morita equivalence, J. Algebra 139 (1991), no. 1, 159–171. MR 1106345, DOI 10.1016/0021-8693(91)90288-J
Philippe Caldero and Markus Reineke, On the quiver Grassmannian in the acyclic case, J. Pure Appl. Algebra 212 (2008), no. 11, 2369–2380. MR 2440252, DOI 10.1016/j.jpaa.2008.03.025
Stephen Donkin, Polynomial invariants of representations of quivers, Comment. Math. Helv. 69 (1994), no. 1, 137–141. MR 1259609, DOI 10.1007/BF02564477
Yurij A. Drozd and Vladimir V. Kirichenko, Finite-dimensional algebras, Springer-Verlag, Berlin, 1994. Translated from the 1980 Russian original and with an appendix by Vlastimil Dlab. MR 1284468, DOI 10.1007/978-3-642-76244-4
Johannes Engel and Markus Reineke, Smooth models of quiver moduli, Math. Z. 262 (2009), no. 4, 817–848. MR 2511752, DOI 10.1007/s00209-008-0401-y
Birge Huisgen-Zimmermann, Classifying representations by way of Grassmannians, Trans. Amer. Math. Soc. 359 (2007), no. 6, 2687–2719. MR 2286052, DOI 10.1090/S0002-9947-07-03997-9
B. Huisgen-Zimmermann, A hierarchy of parametrizing varieties for representations, Rings, modules and representations, Contemp. Math., vol. 480, Amer. Math. Soc., Providence, RI, 2009, pp. 207–239. MR 2508153, DOI 10.1090/conm/480/09376
A. D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 515–530. MR 1315461, DOI 10.1093/qmath/45.4.515
Hanspeter Kraft, Geometrische Methoden in der Invariantentheorie, Aspects of Mathematics, D1, Friedr. Vieweg & Sohn, Braunschweig, 1984 (German). MR 768181, DOI 10.1007/978-3-322-83813-1
Lieven Le Bruyn and Claudio Procesi, Semisimple representations of quivers, Trans. Amer. Math. Soc. 317 (1990), no. 2, 585–598. MR 958897, DOI 10.1090/S0002-9947-1990-0958897-0
Hiraku Nakajima, Varieties associated with quivers, Representation theory of algebras and related topics (Mexico City, 1994) CMS Conf. Proc., vol. 19, Amer. Math. Soc., Providence, RI, 1996, pp. 139–157. MR 1388562
Markus Reineke, Framed quiver moduli, cohomology, and quantum groups, J. Algebra 320 (2008), no. 1, 94–115. MR 2417980, DOI 10.1016/j.jalgebra.2008.01.025
Alexei Rudakov, Stability for an abelian category, J. Algebra 197 (1997), no. 1, 231–245. MR 1480783, DOI 10.1006/jabr.1997.7093
A. N. Zubkov, The Razmyslov-Procesi theorem for quiver representations, Fundam. Prikl. Mat. 7 (2001), no. 2, 387–421 (Russian, with English and Russian summaries). MR 1866464