An algorithmic approach to integral quadratic and bilinear forms Bibliography, related topics, historical remarks
Andrzej Mróz
Nicolaus Copernicus University, Toru«, Poland, e-mail: amroz@mat.umk.pl, andrzej@cimat.mx
The main ideas of the course are based on the works of D. Simson (and his collaborators and PhD students), mainly [Si3], but also [Si][Si6], [Ko, FS, FS2], [KS]-[KS3], [MPS]. They are closely related to older results of M. Barot, H. J. von Höhne, S. A. Ovsienko and J. A. de la Peña [B, B2, BP, BP2, Ho, HoP, O]. In fact (to the best of my knowledge), S. A. Ovsienko [O] was chronologically rst author considering in ations/de ations and classi cations of (weakly positive/critical) unit integral quadratic forms, in the 1970's (probably also A.V. Roiter). Twenty years later these ideas were generalized in [B][BP2] and [Ho, HoP] (among others, for semi-unit non-negative integral quadratic forms of arbitrary corank [BP]). And in the recent few years, D. Simson et al. are considering yet more general ideas: see e.g. [Si3] (slightly more general in ations for unit quadratic forms) and [KS]-[KS3] (in ations for so-called Cox-regular quadratic forms/bigraphs, admitting dotted loops). Additionally, D. Simson et al. study also bilinear (strong) Z-congruence of bigraphs (in contrast to weak Z-congruence studied by the previous authors), also they study related topics for posets [GSZ, GSZ2, GZ], Diophantine equations induced by integral quadratic forms [Si, Si2, Si6], and the huge topic of Coxeter spectral analysis of bigraphs/posets, mesh geometries of roots, morsi cations of quadratic forms and isotropy groups [Si][Si6], [FS, FS2], [KS]-[KS3].
See [Si3, Si5] for more complete bibliography and remarks on motivations coming from representation theory of algebras, Lie theory, derived categories, cluster algebras and various problems of discrete mathematics and computer algebra.
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Below I collect other related topics and important references, including these not men-tioned during the course:
1. Theorem of Ovsienko (cf. also 8. below): original paper: [O], handbook presentation and more facts on integral quadratic forms: [R, Chapter 1].
2. Root systems in the sense of Bourbaki, applications in Lie theory: [Bou] or [Hu, Chapter III].
3. General handbook on applied linear algebra: [Me]. It contains, among others, nice presentation of all the basic facts concerning: real-symmetric positive (semi)de nite matrices, Sylvester's criteria, matrix decompositions and orthogonal diagonalization, eigenvalues and Jordan form [Me, Ch. 7], Gram-Schmidt orthogonalization [Me, Ch. 5]. 4. General handbooks on quadratic and bilinear forms over rings and elds: [Lam, MH, Sz]. 5. Representation theory of quivers, algebras, homological algebra, Auslander-Reiten
the-ory: [ASS, SS] and [R].
6. Triangulated and derived categories in representation theory of algebras: [H].
7. Quadratic forms of quivers (algebras), Gabriel's theorem (on bijection between the set of roots of quadratic form of Dynkin quiver and the set of isoclasses of indecomposable representations): [ASS, Chapter VII], [R].
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8. Discussion on interrelations between critical integral quadratic forms (in the sense of Ovsienko), P-critical and principal ones: [MPS]. Actually the following inclusions hold:
critical ⊆ P-critical ⊆ principal.
One can nd in [MPS] also a construction of P-critical forms out from positive ones (see also [Si3, Section 4]), which is an abstraction of one-point extension operation for algebras ([R, SS]), and according to Kasjan-Simson [KS3, Proposition 6.7(b)], after some work, can provide an alternative, elementary proof of Ovsienko's theorem stating that the roots of positive integral quadratic forms have coe cients bounded by 6.
9. Nice survey articles on Coxeter spectral analysis in representation theory of algebras, containing rich collection of facts and bibliography: [LP, P, L2]. Another articles from the area of Coxeter spectral analysis in representation theory: [L, MP, MP2, P2, P4, R2, X, Z].
10. Bilinear lattices (in the sense of Lenzing), introduced in: [L] (cf. survey [L2]); [MP2]. 11. Recognition for Grothendieck groups problem formulation and solution for canonical
bilinear lattices (= bilinear lattices of piecewise hereditary algebras of projective type): [L2].
12. Cyclotomic algebras: [P2], cyclotomic factors of the Coxeter polynomials of algebras and bilinear lattices: [MP, MP2].
13. Mahler measure of (the Coxeter polynomial of) an algebra: [P3, P4] cf. also [L2]. Results concerning the spectrum of algebras (hence related to Mahler measure): [R2, X, Z], cf. also the survey [LP]. See [P4] for references to the studies on Mahler measure of polynomials, Lehmer problem etc.
14. Quivers and Path Algebras package for GAP: [QPA]. It is an open source o cial GAP package for computations with quotients of path algebras, modules, including Coxeter formalism and quadratic forms of algebras and much more...
15. Various packages of software related to representation theory of algebras and matrix problems, including ginflations program, presented during the course: [AM].
References
[ASS] I. Assem, D. Simson and A. Skowro«ski, Elements of Representation Theory of Asso-ciative Algebras, Volume 1, Techniques of Representation Theory, London Math. Soc. Student Texts 65, Cambridge Univ. Press, Cambridge - New York, 2006.
[B] M. Barot, A characterization of positive unit forms, Bol. Soc. Mat. Mexicana (3), 5 (1999), 8793.
[B2] M. Barot, A characterization of positive unit forms. II, Bol. Soc. Mat. Mexicana (3), 7 (2001), 1322.
[BP] M. Barot and J. A. de la Peña, The Dynkin type of a non-negative unit form, Expo. Math., 17 (1999), 339348.
[BP2] M. Barot and J. A. de la Peña, Root-induced integral quadratic forms, Linear Algebra Appl., 412 (2006), 291302.
[Bou] N. Bourbaki, Groupes et algébres de Lie. IV-VI. Hermann & Co, Paris 1960.
[FS] M. Felisiak and D. Simson, On combinatorial algorithms computing mesh root systems and matrix morsi cations for the Dynkin diagram An, Discrete Math. 313 (2013),
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[FS2] M. Felisiak and D. Simson, Applications of matrix morsi cations to Coxeter spectral study of loop-free edge-bipartite graphs, Discrete Appl. Math. (2015), in press. [GSZ] M. G asiorek, D. Simson and K. Zaj ac, Structure and a Coxeter-Dynkin type
classi -cation of corank two non-negative posets, Linear Algebra Appl. 469 (2015), 76-113. [GSZ2] M. G asiorek, D. Simson and K. Zaj ac, On Coxeter type study of non-negative posets
using matrix morsi cations and isotropy groups of Dynkin and Euclidean diagrams, Europ. J. Comb. 2015, 48 (2015), 127142.
[GZ] M. G asiorek and K. Zaj ac, On algorithmic study of non-negative posets of corank at most two and their Coxeter-Dynkin types, Fund. Inform. (2015), in press.
[H] D. Happel, Triangulated categories in the representation theory of nite dimensional algebras, in: London Mathematical Society LNM, vol. 119. Cambridge University Press (1988).
[Ho] H. J. von Höhne, On weakly positive unit forms, Comment. Math. Helv. 63 (1988), 312336.
[HoP] H. J. von Höhne and J. A. de la Peña, Isotropic vectors of non-negative integral quadratic forms, European J. Combin. 19 (1998), 621638.
[Hu] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics 9, Springer-Verlag, New York Heidelberg, Berlin, 1972.
[KS] S. Kasjan and D. Simson, Mesh Algorithms for Coxeter Spectral Classi cation of Cox-regular Edge-bipartite Graphs with Loops, I. Mesh Root Systems, Fund. Inform. 139 (2015), 153184.
[KS2] S. Kasjan and D. Simson, Mesh Algorithms for Coxeter Spectral Classi cation of Cox-regular Edge-bipartite Graphs with Loops, II. Application to Coxeter Spectral Analysis, Fund. Inform. 139 (2015), 185209.
[KS3] S. Kasjan and D. Simson, Algorithms for Isotropy Groups of Cox-regular Edge-bipartite Graphs, Fund. Inform. 139 (2015), 249-275.
[Ko] J. Kosakowska, In ation algorithms for positive and principal edge-bipartite graphs and unit quadratic forms, Fund. Inform. 119 (2012), 149162.
[Lam] T. Y. Lam, Introduction to Quadratic Forms over Fields. 2005.
[L] H. Lenzing, A K-theoretical study of canonical algebras, Representations of algebras, Seventh International Conference, Cocoyoc (Mexico) 1994 (eds R. Bautista et al.), CMS Conference Proceedings 18 (American Mathematical Society, Providence, R.I., 1996), 433454.
[L2] H. Lenzing, Coxeter transformations associated with nite dimensional algebras. In Computional Methods for Representations of Groups and Algebras. Progress in Math-ematics 173, Birkhäser-Verlag, Basel-Boston, 1999, 287308.
[LP] H. Lenzing and J. A. de la Peña, Spectral analysis of nite dimensional algebras and singularities, Trends in Representation Theory of Algebras and Related Topics, ed. A. Skowro«ski, EMS Publishing House, Zürich (2008) 541588.
[MPS] G. Marczak, A. Polak and D. Simson, P-critical integral quadratic forms and positive unit forms. An algorithmic approach, Linear Algebra Appl. 433 (2010), 18731888. [Me] C. D. Meyer, Matrix Analysis and Applied Linear Algebra, Philadelphia, SIAM (2000). [MH] John Milnor, Dale Husemoller, Symmetric Bilinear Forms. 1973.
[MP] A. Mróz and J. A. de la Peña, Tubes in derived categories and cyclotomic factors of the Coxeter polynomial of an algebra, J. Algebra 420 (2014) 242-260.
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[MP2] A. Mróz and J. A. de la Peña, Periodicity in bilinear lattices and the Coxeter formal-ism, preprint (2015), in preparation.
[O] S. A. Ovsienko, Integral weakly positive forms, in Schur Matrix Problems and Quadratic Forms, preprint 78.25, Inst. Mat. Akad. Nauk USSR, Kiev 1978, 317 (in Russian).
[P] J. A. de la Peña, Coxeter transformations and the representation theory of algebras. In Finite Dimensional Algebras and Related Topics, NATO ASI Series C: Mathematical and Physical Sciences 424, Kluwer Academic Publishers, Dordrecht, 1994, 223253. [P2] J. A. de la Peña, Algebras whose Coxeter polynomials are products of cyclotomic
polynomials, Algebr.Represent.Theory (2014) 17:905930.
[P3] J. A. de la Peña, On the Mahler measure of the Coxeter polynomial of tensor products of algebras, Bol. Soc. Mat. Mex. (2014) 20:257275.
[P4] J. A. de la Peña, On the Mahler measure of the Coxeter polynomial of algebra, Adv. Math. 270 (2015), 375399.
[R] C. M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Math., 1099, Springer, Berlin, 1984.
[R2] C. M. Ringel, The spectral radius of the Coxeter transformations for a generalized Cartan matrix, Math. Ann. 300, (1994) 331339.
[Si] D. Simson, Mesh algorithms for solving principal Diophantine equations, sand-glass tubes and tori of roots, Fund. Inform., 109 (2011), 425462.
[Si2] D. Simson, Mesh geometries of root orbits of integral quadratic forms, J. Pure Appl. Algebra 215 (2011), 1334.
[Si3] D. Simson, A Coxeter-Gram classi cation of positive simply laced edge-bipartite graphs, SIAM J. Discrete Math., 27, No. 2 (2013), 827854.
[Si4] D. Simson, Algorithms determining matrix morsi cations, Weyl orbits, Coxeter poly-nomials and mesh geometries of roots for Dynkin diagrams, Fund. Inform. 123 (2013), 447490.
[Si5] D. Simson, A framework for Coxeter spectral analysis of edge-bipartite graphs, their rational morsi cations and mesh geometries of root orbits, Fund. Inform. 124 (2013), 309338.
[Si6] D. Simson, Toroidal algorithms for mesh geometries of root orbits of the Dynkin diagram D4, Fund. Inform. 124 (2013), 339364.
[SS] D. Simson and A. Skowro«ski, Elements of Representation Theory of Associative Al-gebras, Vol.2, Tubes and Concealed Algebras of Euclidean type, London Math. Soc. Student Texts, 70 Cambridge Univ. Press, Cambridge-New York, 2007.
[Sz] K. Szymiczek, Bilinear Algebra: An Introduction to the Algebraic Theory of Quadratic Forms, Algebra, Logic and Applications 7, Gordon & Breach Science Publishers, Am-sterdam, 1997.
[X] Ch. Xi, On wild algebras with the small growth number, Comm. Algebra 18 (1990), 34133422.
[Z] Y. Zhang, Eigenvalues of Coxeter transformations and the structure of regular com-ponents of an Auslander-Reiten quiver, Comm. Algebra 17 (1989), 23472362.
[QPA] QPA (Quivers and Path Algebras), o cial GAP package, direct download: http://sourceforge.net/projects/quiverspathalg/
or https://github.com/gap-system/qpa. [AM] http://www.mat.umk.pl/~amroz/projen.html.
