Bilinear Forms on Grothendieck Groups of Triangulated Categories

Peter Webb

doi:10.1007/978-3-319-94033-5_19

Bilinear Forms on Grothendieck Groups of Triangulated Categories

Peter Webb

School of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

2 Scopus citations

Abstract

We extend the theory of bilinear forms on the Green ring of a finite group developed by Benson and Parker to the context of the Grothendieck group of a triangulated category with Auslander–Reiten triangles, taking only relations given by direct sum decompositions. We examine the non-degeneracy of the bilinear form given by dimensions of homomorphisms, and show that the form may be modified to give a Hermitian form for which the standard basis given by indecomposable objects has a dual basis given by Auslander–Reiten triangles. An application is given to the homotopy category of perfect complexes over a symmetric algebra, with a consequence analogous to a result of Erdmann and Kerner.

Original language	English (US)
Title of host publication	Geometric and Topological Aspects of the Representation Theory of Finite Groups - PIMS Summer School and Workshop, 2016
Editors	Srikanth B. Iyengar, Julia Pevtsova, Jon F. Carlson
Publisher	Springer New York LLC
Pages	465-480
Number of pages	16
ISBN (Print)	9783319940328
DOIs	https://doi.org/10.1007/978-3-319-94033-5_19
State	Published - 2018
Event	PIMS Summer School and Workshop on Geometric Methods in the Representation Theory of Finite Groups, 2016 - Vancouver, Canada Duration: Jul 27 2016 → Aug 5 2016

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	242
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Other

Other	PIMS Summer School and Workshop on Geometric Methods in the Representation Theory of Finite Groups, 2016
Country/Territory	Canada
City	Vancouver
Period	7/27/16 → 8/5/16

Bibliographical note

Funding Information:
The author was supported by Simons Foundation award 282425.

Publisher Copyright:
© 2018, Springer International Publishing AG.

Keywords

Auslander–Reiten triangle
Green ring
Perfect complex
Symmetric algebra

Publisher link

10.1007/978-3-319-94033-5_19

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Webb, P. (2018). Bilinear Forms on Grothendieck Groups of Triangulated Categories. In S. B. Iyengar, J. Pevtsova, & J. F. Carlson (Eds.), Geometric and Topological Aspects of the Representation Theory of Finite Groups - PIMS Summer School and Workshop, 2016 (pp. 465-480). (Springer Proceedings in Mathematics and Statistics; Vol. 242). Springer New York LLC. https://doi.org/10.1007/978-3-319-94033-5_19

Bilinear Forms on Grothendieck Groups of Triangulated Categories. / Webb, Peter.
Geometric and Topological Aspects of the Representation Theory of Finite Groups - PIMS Summer School and Workshop, 2016. ed. / Srikanth B. Iyengar; Julia Pevtsova; Jon F. Carlson. Springer New York LLC, 2018. p. 465-480 (Springer Proceedings in Mathematics and Statistics; Vol. 242).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Webb, P 2018, Bilinear Forms on Grothendieck Groups of Triangulated Categories. in SB Iyengar, J Pevtsova & JF Carlson (eds), Geometric and Topological Aspects of the Representation Theory of Finite Groups - PIMS Summer School and Workshop, 2016. Springer Proceedings in Mathematics and Statistics, vol. 242, Springer New York LLC, pp. 465-480, PIMS Summer School and Workshop on Geometric Methods in the Representation Theory of Finite Groups, 2016, Vancouver, Canada, 7/27/16. https://doi.org/10.1007/978-3-319-94033-5_19

Webb P. Bilinear Forms on Grothendieck Groups of Triangulated Categories. In Iyengar SB, Pevtsova J, Carlson JF, editors, Geometric and Topological Aspects of the Representation Theory of Finite Groups - PIMS Summer School and Workshop, 2016. Springer New York LLC. 2018. p. 465-480. (Springer Proceedings in Mathematics and Statistics). doi: 10.1007/978-3-319-94033-5_19

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abstract = "We extend the theory of bilinear forms on the Green ring of a finite group developed by Benson and Parker to the context of the Grothendieck group of a triangulated category with Auslander–Reiten triangles, taking only relations given by direct sum decompositions. We examine the non-degeneracy of the bilinear form given by dimensions of homomorphisms, and show that the form may be modified to give a Hermitian form for which the standard basis given by indecomposable objects has a dual basis given by Auslander–Reiten triangles. An application is given to the homotopy category of perfect complexes over a symmetric algebra, with a consequence analogous to a result of Erdmann and Kerner.",

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AB - We extend the theory of bilinear forms on the Green ring of a finite group developed by Benson and Parker to the context of the Grothendieck group of a triangulated category with Auslander–Reiten triangles, taking only relations given by direct sum decompositions. We examine the non-degeneracy of the bilinear form given by dimensions of homomorphisms, and show that the form may be modified to give a Hermitian form for which the standard basis given by indecomposable objects has a dual basis given by Auslander–Reiten triangles. An application is given to the homotopy category of perfect complexes over a symmetric algebra, with a consequence analogous to a result of Erdmann and Kerner.

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Bilinear Forms on Grothendieck Groups of Triangulated Categories

Abstract

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