Classifying representations by way of Grassmannians
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- by Birge Huisgen-Zimmermann PDF
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Abstract:
Let $\Lambda$ be a finite-dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine or coarse moduli space classifying, up to isomorphism, the representations of $\Lambda$ with fixed dimension $d$ and fixed squarefree top $T$. Next to providing a complete theoretical picture, some of these equivalent conditions are readily checkable from quiver and relations of $\Lambda$. In the case of existence of a moduli space—unexpectedly frequent in light of the stringency of fine classification—this space is always projective and, in fact, arises as a closed subvariety $\operatorname {\mathfrak {Grass}}^T_d$ of a classical Grassmannian. Even when the full moduli problem fails to be solvable, the variety $\operatorname {\mathfrak {Grass}}^T_d$ is seen to have distinctive properties recommending it as a substitute for a moduli space. As an application, a characterization of the algebras having only finitely many representations with fixed simple top is obtained; in this case of ‘finite local representation type at a given simple $T$’, the radical layering $\bigl ( J^{l}M/ J^{l+1}M \bigr )_{l \ge 0}$ is shown to be a classifying invariant for the modules with top $T$. This relies on the following general fact obtained as a byproduct: proper degenerations of a local module $M$ never have the same radical layering as $M$.References
- E. Babson, B. Huisgen-Zimmermann, and R. Thomas, Generic representation theory of quivers with relations, in preparation.
- Klaus Bongartz, On degenerations and extensions of finite-dimensional modules, Adv. Math. 121 (1996), no. 2, 245–287. MR 1402728, DOI 10.1006/aima.1996.0053
- Klaus Bongartz, A note on algebras of finite uniserial type, J. Algebra 188 (1997), no. 2, 513–515. MR 1435371, DOI 10.1006/jabr.1996.6845
- Klaus Bongartz, Some geometric aspects of representation theory, Algebras and modules, I (Trondheim, 1996) CMS Conf. Proc., vol. 23, Amer. Math. Soc., Providence, RI, 1998, pp. 1–27. MR 1648601
- Klaus Bongartz and Birge Huisgen-Zimmermann, The geometry of uniserial representations of algebras. II. Alternate viewpoints and uniqueness, J. Pure Appl. Algebra 157 (2001), no. 1, 23–32. MR 1809214, DOI 10.1016/S0022-4049(00)00031-1
- Klaus Bongartz and Birge Huisgen-Zimmermann, Varieties of uniserial representations. IV. Kinship to geometric quotients, Trans. Amer. Math. Soc. 353 (2001), no. 5, 2091–2113. MR 1813609, DOI 10.1090/S0002-9947-01-02712-X
- Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
- Lutz Hille, Tilting line bundles and moduli of thin sincere representations of quivers, An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 4 (1996), no. 2, 76–82. Representation theory of groups, algebras, and orders (Constanţa, 1995). MR 1428456
- Birge Huisgen-Zimmermann, The geometry of uniserial representations of finite-dimensional algebra. I, J. Pure Appl. Algebra 127 (1998), no. 1, 39–72. MR 1609508, DOI 10.1016/S0022-4049(96)00184-3
- —, Top-stable degenerations of finite dimensional representations I, posted at www.math.ucsb.edu/$\sim$birge/papers.html.
- James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773, DOI 10.1007/978-1-4684-9443-3
- 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 Aidan Schofield, Rational invariants of quivers and the ring of matrixinvariants, Perspectives in ring theory (Antwerp, 1987) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 233, Kluwer Acad. Publ., Dordrecht, 1988, pp. 21–29. MR 1048393
- David Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 34, Springer-Verlag, Berlin-New York, 1965. MR 0214602, DOI 10.1007/978-3-662-00095-3
- P. E. Newstead, Introduction to moduli problems and orbit spaces, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 51, Tata Institute of Fundamental Research, Bombay; Narosa Publishing House, New Delhi, 1978. MR 546290
- Christine Riedtmann, Degenerations for representations of quivers with relations, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 2, 275–301. MR 868301, DOI 10.24033/asens.1508
- Maxwell Rosenlicht, On quotient varieties and the affine embedding of certain homogeneous spaces, Trans. Amer. Math. Soc. 101 (1961), 211–223. MR 130878, DOI 10.1090/S0002-9947-1961-0130878-0
- Maxwell Rosenlicht, Questions of rationality for solvable algebraic groups over nonperfect fields, Ann. Mat. Pura Appl. (4) 61 (1963), 97–120 (English, with Italian summary). MR 158891, DOI 10.1007/BF02412850
- Aidan Schofield, Birational classification of moduli spaces of representations of quivers, Indag. Math. (N.S.) 12 (2001), no. 3, 407–432. MR 1914089, DOI 10.1016/S0019-3577(01)80019-7
- Grzegorz Zwara, Degenerations for modules over representation-finite algebras, Proc. Amer. Math. Soc. 127 (1999), no. 5, 1313–1322. MR 1476404, DOI 10.1090/S0002-9939-99-04714-0
Additional Information
- Birge Huisgen-Zimmermann
- Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
- MR Author ID: 187325
- Email: birge@math.ucsb.edu
- Received by editor(s): April 20, 2004
- Received by editor(s) in revised form: March 21, 2005
- Published electronically: January 25, 2007
- Additional Notes: This research was partially supported by a grant from the National Science Foundation.
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 2687-2719
- MSC (2000): Primary 16G10, 16G20, 16G60, 14D20, 14D22
- DOI: https://doi.org/10.1090/S0002-9947-07-03997-9
- MathSciNet review: 2286052
Dedicated: Dedicated to the memory of Sheila Brenner