[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.5666/KMJ.2017.57.4.601

On Semisimple Representations of the Framed g-loop Quiver

Choy, Jaeyoo (Department of Mathematics, Kyungpook National University)

Publication Information

Kyungpook Mathematical Journal / v.57, no.4, 2017 , pp. 601-612 More about this Journal

Abstract

Let Q be the frame g-loop quiver, i.e. a generalized ADHM quiver obtained by replacing the two loops into g loops. The vector space M of representations of Q admits an involution ${\ast}$ if orthogonal and symplectic structures on the representation spaces are endowed. We prove equivalence between semisimplicity of representations of the ${\ast}-invariant$ subspace N of M and the orbit-closedness with respect to the natural adjoint action on N. We also explain this equivalence in terms of King's stability [8] and orthogonal decomposition of representations.

Keywords

instanton; quiver; quiver representation; semisimple;

Citations & Related Records

Reference

1	M. F. Atiyah, V. G. Drinfeld, N. J. Hitchin and Y. I. Manin, Construction of instantons, Phys. Lett. A, 65(3)(1978), 185-187. DOI
2	J. Choy, Moduli spaces of framed symplectic and orthogonal bundles on ${\mathbb{P}}^2$ and the K-theoretic Nekrasov partition functions, Ph.D. thesis, 2015, Kyoto Univ.
3	J. Choy, Moduli spaces of framed symplectic and orthogonal bundles on ${\mathbb{P}}^2$ and the K-theoretic Nekrasov partition functions, J. Geom. Phys., 106(2016), 284-304. DOI
4	W. Crawley-Boevey, Normality of Marsden-Weinstein reductions for representations of quivers, Math. Ann., 325(1)(2003), 55-79. DOI
5	W. Crawley-Boevey and P. Shaw, Multiplicative preprojective algebras, middle convolution and the Deligne-Simpson problem, Adv. Math., 201(2006), 180-208. DOI
6	S. K. Donaldson, Instantons and geometric invariant theory, Comm. Math. Phys., 93(4)(1984), 453-460. DOI
7	N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups, Inst. Hautes Etudes Sci. Publ. Math., 25(1965), 5-48. DOI
8	A. King, Moduli of representations of finite dimensional algebras, Quart. J. Math. Oxford Ser. (2), 45(1994), 515-530. DOI
9	L. Le Bruyn and C. Procesi, Semisimple representations of quivers, Trans. Amer. Math. Soc., 317(2)(1990), 585-598. DOI
10	H. Nakajima, Lectures on Hilbert schemes of points on surfaces, Univ. Lect. Ser. 18, AMS, 1999.
11	Y. Tachikawa, A pseudo-mathematical pseudo-review on 4d N=2 supersymmetric QFTs, available at "http://member.ipmu.jp/yuji.tachikawa/tmp/review-rebooted7.pdf"