﻿ Algebra Seminar: Reprentation Theory and Geometric Topology-清华大学求真书院

### Algebra Seminar: Reprentation Theory and Geometric Topology

Upcoming Talks

The notion of weighted projective lineswas inventedbyGeigle and Lenzing [2], motivatedbygiving a geometric treatment for Ringel’s canonical algebras. By a deep theorem of Happel [4], up to derived equivalence, a hereditary cat- egory with a tiltingobjectis either equivalent to the module category of apath algebra, or equivalent to the category of coherent sheaves on a weighted projective line.

The study of weighted projective lines is related to many mathematical areas, for example, noncommutative algebraicgeometry,Riemann surface, Lietheory,and singularitytheory.

In this series of talks, we will introduce the fundamental context on weighted projective lines, including its geometry counterpart, the category of coherent sheaves, tilting theory, Auslander-Reiten quiver, Grothendieck group and the auto-equivalence group.

The outline is as follows.

§1 Coherent sheavesoverprojective line (Xiao-wu Chen)

§2 From projective line to weighted projective lines (Xiao-wu Chen)

§3 Canonical tilting sheaves and canonical algebras (Jianmin Chen)

§4 Trichotomy: domestic, tubular and wild cases (Jianmin Chen)

§5 K-theoretic study of weighted projective lines (ShiquanRuan)

§6 Auto-equivalence group of the bounded derived category (Shiquan Ruan)

References

[1]D. Baer, W. Geigle and H. Lenzing,Thepreprojectivealgebra of a tamehereditaryArtin algebra.Comm. Algebra 15 (1987), no. 1–2, 425–457.

[2]W. Geigle and H. Lenzing,A class of weightedprojectivecurves arising in representation theory of finite dimensional algebras. Singularities, representations of algebras, andVectorbundles, Springer Lect. Notes Math.1273(1987), 265–297.

[3]W. Geigle and H. Lenzing,Perpendicularcategorieswith applications to representations and sheaves.J. Algebra 144 (1991), no. 2, 273–343.

[4]D. Happel,A characterization ofhereditary categorieswith tilting object.Invent.Math.144(2001), 381–398.

[5] D. Kussin, H. Lenzing and H. Meltzer, Triangle singularities, ADE-chains and weighted projective lines. Adv. Math. 237 (2013), 194–251.

[6] H. Lenzing, Curve singularities arising from the representation theory of tame hereditary algebras. Representation theory, I (Ottawa, Ont., 1984), 199–231, Lecture Notes in Math., 1177, Springer, Berlin, 1986.

[7] H. Lenzing, Wild canonical algebras and rings of automorphic forms. Finite-dimensional algebras and related topics (Ottawa, ON, 1992), 191–212, NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 424, Kluwer Acad. Publ., Dordrecht, 1994.

[8] H. Lenzing, Representations of finite-dimensional algebras and singularity theory. Trends in ring theory (Miskolc, 1996), 71–97, CMS Conf. Proc., 22, Amer. Math. Soc., Provi- dence, RI, 1998.

[9] H. Lenzing, Hereditary categories. Handbook of tilting theory, 105–146, London Math. Soc. Lecture Note Ser., 332, Cambridge Univ. Press, Cambridge, 2007.

[10] H. Lenzing, Weighted projective lines and applications. In: Representations of algebras and related topics, EMS Ser. Congr. Rep., pages 153–187. Eur. Math. Soc., Zu¨rich, 2011.

[11] H. Lenzing, Weighted projective lines and Riemann surfaces. In: Proc. of the 49th Sympo- sium on Ring Theory and Representation Theory, 67–79, Symp. Ring Theory Represent. Theory Organ. Comm., Shimane, 2017.

[12] H. Lenzing, On the K-theory of weighted projective curves. Representations of algebras, 131–153, Contemp. Math., 705, Amer. Math. Soc., [Providence], RI, 2018.

[13] H. Lenzing and J.A. de la Pen˜a, Spectral analysis of finite dimensional algebras and singularities. In: Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep., pages 541–588. Eur. Math. Soc., Zu¨rich, 2008.

[14] H. Lenzing and J.A. de la Pen˜a, A Chebysheff recursion formula for Coxeter polynomials. Linear Algebra Appl. 430 (2009), no. 4, 947–956.

[15] H. Lenzing and H. Meltzer, Sheaves on a weighted projective line of genus one, and representations of a tubular algebra. Representations of algebras (Ottawa, ON, 1992), 313–337, CMS Conf. Proc., 14, Amer. Math. Soc., Providence, RI, 1993.

[16] H. Lenzing and H. Meltzer, The automorphism group of the derived category for a weight- ed projective line. Comm. Algebra 28 (2000), no. 4, 1685–1700.

## Past Talks

In this talk, we combine the above theories to understand the cell structures of the moduli space of quadratic differentials on a fixed Riemann surface. We show the relations between the stratification and wall crossing phenomenon in some concrete moduli spaces. This is a joint work with A. King and Y. Qiu in progress.

• ### Geometric Representation Theory Seminar | The FLE and the W-algebra

Abstract：The FLE is a basic assertion in the quantum geometric Langlands program, proposed by Gaitsgory-Lurie, which provides a deformation of the geometric Satake equivalence to all Kac-Moody levels. We will report on a proof via the representation theory of the affine W-algebra, which is joint work in progress with Gaitsgory

• ### Geometric Representation Theory Seminar | Higher representation theory of gl(1|1)

AbstractThe notion of representations of Lie algebras on categories ("2-representations") has proven useful in representation theory. I will discuss joint work with Andrew Manion for the case of the super Lie algebra gl(1|1). A motivation is the reconstruction of Heegaard-Floer theory, a 4-dimensional topological field theory, and its extension down to dimension 1.About the speakerRaphaël Alexi...