Tilting theory

It turns out that there are applications of our functors which make use of the analogous transformations which we like to think of as a change of basis for a fixed root- system - a tilting of the axes relative to the roots which results in a different subset of roots lying in the positive cone. ... For this reason, and because the word 'tilt' inflects easily, we call our functors tilting functors or simply tilts.

Brenner & Butler (1980, p.103) explaining why they introduced the word "tilt"

In algebra, tilting theory uses a tilting module T over an algebra A to construct tilting functors relating modules over A to modules over the tilted algebra End_A(T) of endomorphisms of T.

Tilting theory was motivated by the introduction of Coxeter functors by Bernšteĭn, Gelfand & Ponomarev (1973), which were reformulated by Auslander, Platzeck & Reiten (1979), and generalized by Brenner & Butler (1980) who introduced tilting functors. Happel & Ringel (1982) defined tilted algebras and tilting modules as further generalizations of this.

Definitions

Happel & Ringel (1982) defined tilting modules and tilted algebras as follows. Suppose that A is a finite-dimensional algebra over a field. Then a right A-module T is called a tilting module if it has the following 3 properties:

• T has projective dimension at most 1, in other words it is a quotient of a projective module by a projective submodule.
• Ext¹(T,T) = 0
• The right A-module A is the kernel of a surjective morphism between finite sums of summands of T.

A tilted algebra B is an algebra of endomorphisms of a tilting module T over a hereditary finite dimensional algebra A.

The tilting functors are the 4 functors Hom_A(T,*), Ext1
A(T,*), *⊗_BT, and TorB
1(*,T), where T is considered as a right A module and a left B module. Brenner & Butler (1980) showed that tilting functors give equivalences between certain subcategories of right A-modules and right B-modules.

See also

• Cluster algebra

References

• Angeleri Hügel, Lidia; Happel, Dieter; Krause, Henning, eds. (2007), Handbook of tilting theory, London Mathematical Society Lecture Note Series, 332, Cambridge University Press, doi:10.1017/CBO9780511735134, ISBN 978-0-521-68045-5; 978-0-521-68045-5, MR2385175, http://books.google.com/books?isbn=052168045X 
• Assem, Ibrahim (1990), Tilting theory---an introduction, in Balcerzyk, Stanisław; Józefiak, Tadeusz; Krempa, Jan et al., "Topics in algebra, Part 1 (Warsaw, 1988)", Banach Center Publications, Banach Center Publ. (Warszawa: PWN) 26: 127–180, MR1171230, http://journals.impan.gov.pl/BC/oldindex.html 
• Auslander, Maurice; Platzeck, María Inés; Reiten, Idun (1979), "Coxeter functors without diagrams", Transactions of the American Mathematical Society 250: 1–46, doi:10.2307/1998978, ISSN 0002-9947, MR530043 
• Bernšteĭn, I. N.; Gelfand, I. M.; Ponomarev, V. A. (1973), "Coxeter functors, and Gabriel's theorem", Russian mathematical surveys 28 (2): 17–32, doi:10.1070/RM1973v028n02ABEH001526, ISSN 0042-1316, MR0393065 
• Brenner, Sheila; Butler, M. C. R. (1980), "Generalizations of the Bernstein-Gel'fand-Ponomarev reflection functors", Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., 832, Berlin, New York: Springer-Verlag, pp. 103–169, doi:10.1007/BFb0088461, MR607151 
• Coelho, Flávio Ulhoa (2001), "Tilting functor", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/t/t130110.htm 
• Coelho, Flávio Ulhoa (2001), "Tilting module", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/t/t130120.htm 
• Happel, Dieter; Ringel, Claus Michael (1982), "Tilted algebras", Transactions of the American Mathematical Society 274 (2): 399–443, doi:10.2307/1999116, ISSN 0002-9947, MR675063 
• Kerner, O. (2001), "Tilted algebra", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/t/t130100.htm 
• Unger, L. (2001), "Tilting theory", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/T/t130130.htm