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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.I. Auslander,   I. Reiten,   S. Smalø,   "Representation theory of Artin algebras" , ''Studies Adv. Math.'' , '''36''' , Cambridge Univ. Press (1995)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> I.N. Bernstein,   I.M. Gelfand,   V.A. Ponomarev,   "Coxeter functors and Gabriel's theorem" ''Russian Math. Surveys'' , '''28''' (1973) pp. 17–32 ''Uspekhi Mat. Nauk.'' , '''28''' (1973) pp. 19–33</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> K. Bongartz,   "Algebras and quadratic forms" ''J. London Math. Soc.'' , '''28''' (1983) pp. 461–469</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> V. Dlab,   C.M. Ringel,   "Indecomposable representations of graphs and algebras" , ''Memoirs'' , '''173''' , Amer. Math. Soc. (1976)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> Yu.A. Drozd,   "Coxeter transformations and representations of partially ordered sets" ''Funkts. Anal. Prilozhen.'' , '''8''' (1974) pp. 34–42 (In Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> Yu.A. Drozd,   "On tame and wild matrix problems" , ''Matrix Problems'' , Akad. Nauk. Ukr. SSR., Inst. Mat. Kiev (1977) pp. 104–114 (In Russian)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> Yu.A. Drozd,   "Tame and wild matrix problems" , ''Representations and Quadratic Forms'' (1979) pp. 39–74 (In Russian)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> P. Gabriel,   "Unzerlegbare Darstellungen 1" ''Manuscripta Math.'' , '''6''' (1972) pp. 71–103 (Also: Berichtigungen 6 (1972), 309)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> P. Gabriel,   "Représentations indécomposables" , ''Séminaire Bourbaki (1973/74)'' , ''Lecture Notes in Mathematics'' , '''431''' , Springer (1975) pp. 143–169</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> P. Gabriel,   A.V. Roiter,   "Representations of finite dimensional algebras" , ''Algebra VIII'' , ''Encycl. Math. Stud.'' , '''73''' , Springer (1992)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> S. Kasjan,   D. Simson,   "Tame prinjective type and Tits form of two-peak posets II" ''J. Algebra'' , '''187''' (1997) pp. 71–96</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> L.A. Nazarova,   "Representations of quivers of infinite type" ''Izv. Akad. Nauk. SSSR'' , '''37''' (1973) pp. 752–791 (In Russian)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> J.A. de la Peña,   "Algebras with hypercritical Tits form" , ''Topics in Algebra'' , ''Banach Center Publ.'' , '''26: 1''' , PWN (1990) pp. 353–369</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> J.A. de la Peña,   "On the dimension of the module-varieties of tame and wild algebras" ''Commun. Algebra'' , '''19''' (1991) pp. 1795–1807</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> J.A. de la Peña,   D. Simson,   "Prinjective modules, reflection functors, quadratic forms and Auslander–Reiten sequences" ''Trans. Amer. Math. Soc.'' , '''329''' (1992) pp. 733–753</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> J.A. de la Peña,   A. Skowroński,   "The Euler and Tits forms of a tame algebra" ''Math. Ann.'' , '''315''' (2000) pp. 37–59</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> C.M. Ringel,   "Tame algebras and integral quadratic forms" , ''Lecture Notes in Mathematics'' , '''1099''' , Springer (1984)</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> A.V. Roiter,   M.M. Kleiner,   "Representations of differential graded categories" , ''Lecture Notes in Mathematics'' , '''488''' , Springer (1975) pp. 316–339</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> D. Simson,   "Linear representations of partially ordered sets and vector space categories" , ''Algebra, Logic Appl.'' , '''4''' , Gordon &amp; Breach (1992)</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> D. Simson,   "Posets of finite prinjective type and a class of orders" ''J. Pure Appl. Algebra'' , '''90''' (1993) pp. 77–103</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top"> D. Simson,   "Representation types, Tits reduced quadratic forms and orbit problems for lattices over orders" ''Contemp. Math.'' , '''229''' (1998) pp. 307–342</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top"> D. Simson,   "Coalgebras, comodules, pseudocompact algebras and tame comodule type" ''Colloq. Math.'' , '''in press''' (2001)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.I. Auslander, I. Reiten, S. Smalø, "Representation theory of Artin algebras" , ''Studies Adv. Math.'' , '''36''' , Cambridge Univ. Press (1995) {{MR|1314422}} {{ZBL|0834.16001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> I.N. Bernstein, I.M. Gelfand, V.A. Ponomarev, "Coxeter functors and Gabriel's theorem" ''Russian Math. Surveys'' , '''28''' (1973) pp. 17–32 ''Uspekhi Mat. Nauk.'' , '''28''' (1973) pp. 19–33</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> K. Bongartz, "Algebras and quadratic forms" ''J. London Math. Soc.'' , '''28''' (1983) pp. 461–469 {{MR|0724715}} {{ZBL|0532.16020}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> V. Dlab, C.M. Ringel, "Indecomposable representations of graphs and algebras" , ''Memoirs'' , '''173''' , Amer. Math. Soc. (1976) {{MR|0447344}} {{ZBL|0332.16015}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> Yu.A. Drozd, "Coxeter transformations and representations of partially ordered sets" ''Funkts. Anal. Prilozhen.'' , '''8''' (1974) pp. 34–42 (In Russian) {{MR|0351924}} {{ZBL|0356.06003}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> Yu.A. Drozd, "On tame and wild matrix problems" , ''Matrix Problems'' , Akad. Nauk. Ukr. SSR., Inst. Mat. Kiev (1977) pp. 104–114 (In Russian) {{MR|498704}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> Yu.A. Drozd, "Tame and wild matrix problems" , ''Representations and Quadratic Forms'' (1979) pp. 39–74 (In Russian) {{MR|0600111}} {{ZBL|0454.16014}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> P. Gabriel, "Unzerlegbare Darstellungen 1" ''Manuscripta Math.'' , '''6''' (1972) pp. 71–103 (Also: Berichtigungen 6 (1972), 309) {{MR|332887}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> P. Gabriel, "Représentations indécomposables" , ''Séminaire Bourbaki (1973/74)'' , ''Lecture Notes in Mathematics'' , '''431''' , Springer (1975) pp. 143–169 {{MR|0485996}} {{ZBL|0335.17005}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> P. Gabriel, A.V. Roiter, "Representations of finite dimensional algebras" , ''Algebra VIII'' , ''Encycl. Math. Stud.'' , '''73''' , Springer (1992) {{MR|1239446}} {{MR|1239447}} {{ZBL|0839.16001}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> S. Kasjan, D. Simson, "Tame prinjective type and Tits form of two-peak posets II" ''J. Algebra'' , '''187''' (1997) pp. 71–96 {{MR|1425560}} {{ZBL|0944.16013}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> L.A. Nazarova, "Representations of quivers of infinite type" ''Izv. Akad. Nauk. SSSR'' , '''37''' (1973) pp. 752–791 (In Russian) {{MR|0338018}} {{ZBL|0298.15012}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> J.A. de la Peña, "Algebras with hypercritical Tits form" , ''Topics in Algebra'' , ''Banach Center Publ.'' , '''26: 1''' , PWN (1990) pp. 353–369 {{MR|}} {{ZBL|0731.16008}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> J.A. de la Peña, "On the dimension of the module-varieties of tame and wild algebras" ''Commun. Algebra'' , '''19''' (1991) pp. 1795–1807 {{MR|}} {{ZBL|0818.16013}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> J.A. de la Peña, D. Simson, "Prinjective modules, reflection functors, quadratic forms and Auslander–Reiten sequences" ''Trans. Amer. Math. Soc.'' , '''329''' (1992) pp. 733–753 {{MR|1025753}} {{ZBL|0789.16010}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> J.A. de la Peña, A. Skowroński, "The Euler and Tits forms of a tame algebra" ''Math. Ann.'' , '''315''' (2000) pp. 37–59 {{MR|}} {{ZBL|0941.16010}} </TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> C.M. Ringel, "Tame algebras and integral quadratic forms" , ''Lecture Notes in Mathematics'' , '''1099''' , Springer (1984) {{MR|0774589}} {{ZBL|0546.16013}} </TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> A.V. Roiter, M.M. Kleiner, "Representations of differential graded categories" , ''Lecture Notes in Mathematics'' , '''488''' , Springer (1975) pp. 316–339 {{MR|0435145}} {{ZBL|0356.16011}} </TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> D. Simson, "Linear representations of partially ordered sets and vector space categories" , ''Algebra, Logic Appl.'' , '''4''' , Gordon &amp; Breach (1992) {{MR|1241646}} {{ZBL|0818.16009}} </TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> D. Simson, "Posets of finite prinjective type and a class of orders" ''J. Pure Appl. Algebra'' , '''90''' (1993) pp. 77–103 {{MR|1246276}} {{ZBL|0815.16006}} </TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top"> D. Simson, "Representation types, Tits reduced quadratic forms and orbit problems for lattices over orders" ''Contemp. Math.'' , '''229''' (1998) pp. 307–342 {{MR|1676228}} {{ZBL|0921.16007}} </TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top"> D. Simson, "Coalgebras, comodules, pseudocompact algebras and tame comodule type" ''Colloq. Math.'' , '''in press''' (2001) {{MR|1874368}} {{ZBL|1055.16038}} </TD></TR></table>

Revision as of 14:52, 24 March 2012

Let be a finite quiver (see [a8]), that is, an oriented graph with vertex set and set of arrows (oriented edges; cf. also Graph, oriented; Quiver). Following P. Gabriel [a8], [a9], the Tits quadratic form of is defined by the formula

where and is the number of arrows from to in .

There are important applications of the Tits form in representation theory. One easily proves that if is connected, then is positive definite if and only if (viewed as a non-oriented graph) is any of the Dynkin diagrams , , , , or (cf. also Dynkin diagram). On the other hand, the Gabriel theorem [a8] asserts that this is the case if and only if has only finitely many isomorphism classes of indecomposable -linear representations, where is an algebraically closed field (see also [a2]). Let be the Abelian category of finite-dimensional -linear representations of formed by the systems of finite-dimensional vector -spaces , connected by -linear mappings corresponding to arrows of . By a theorem of L.A. Nazarova [a12], given a connected quiver the category is of tame representation type (see [a7], [a10], [a19] and Quiver) if and only if is positive semi-definite, or equivalently, if and only if (viewed as a non-oriented graph) is any of the extended Dynkin diagrams , , , , or (see [a1], [a10], [a19]; and [a4] for a generalization).

Let be the Grothendieck group of the category . By the Jordan–Hölder theorem, the correspondence defines a group isomorphism . One shows that the Tits form coincides with the Euler characteristic , , along the isomorphism , that is, for any in (see [a10], [a17]).

The Tits quadratic form is related with an algebraic geometry context defined as follows (see [a9], [a10], [a19]).

For any vector , consider the affine irreducible -variety of -representations of of the dimension type (in the Zariski topology), where is the space of -matrices for any arrow of . Consider the algebraic group and the algebraic group action defined by the formula , where is an arrow of , , , and . An important role in applications is played by the Tits-type equality , , where denotes the dimension of the algebraic variety (see [a8]).

Following the above ideas, Yu.A. Drozd [a5] introduced and successfully applied a Tits quadratic form in the study of finite representation type of the Krull–Schmidt category of matrix -representations of partially ordered sets with a unique maximal element (see [a10], [a19]). In [a6] and [a7] he also studied bimodule matrix problems and the representation type of boxes by means of an associated Tits quadratic form (see also [a18]). In particular, he showed [a6] that if is of tame representation type, then is weakly non-negative, that is, for all .

K. Bongartz [a3] associated with any finite-dimensional basic -algebra a Tits quadratic form as follows. Let be a complete set of primitive pairwise non-isomorphic orthogonal idempotents of the algebra . Fix a finite quiver with and a -algebra isomorphism , where is the path -algebra of the quiver (see [a1], [a10], [a19]) and is an ideal of contained in the square of the Jacobson radical of and containing a power of . Assume that has no oriented cycles (and hence the global dimension of is finite). The Tits quadratic form of is defined by the formula

where , for a minimal set of generators of contained in . One checks that , where is the simple -module associated to the vertex . Then the definition of depends only on , and when is of global dimension at most two, the form coincides with the Euler characteristic , , under a group isomorphism , where is the Grothendieck group of the category of finite-dimensional right -modules (see [a17]). Note that if .

By applying a Tits-type equality as above, Bongartz [a3] proved that if is of finite representation type, then is weakly positive, that is, for all non-zero vectors . The converse implication does not hold in general, but it has been established if the Auslander–Reiten quiver of (see Riedtmann classification) has a post-projective component (see [a10]), by applying an idea of Drozd [a5]. J.A. de la Peña [a14] proved that if is of tame representation type, then is weakly non-negative. The converse implication does not hold in general, but it has been proved under a suitable assumption on (see [a13] and [a16] for a discussion of this problem and relations between the Tits quadratic form and the Euler quadratic form of ).

Let be a partially ordered set with partial order relation and let be the set of all maximal elements of . Following [a5] and [a15], D. Simson [a20] defined the Tits quadratic form of by the formula

and applied it in the study of prinjective -modules, that is, finite-dimensional right modules over the incidence -algebra of such that there is an exact sequence , where is a projective -module and is a direct sum of simple projectives. The additive Krull–Schmidt category of prinjective -modules is equivalent to the category of matrix -representations of [a20]. Under an identification , the Tits form is equal to the Euler characteristic . A Tits-type equality is also valid for [a15]. It has been proved in [a20] that is weakly positive if and only if has only a finite number of iso-classes of indecomposable modules. By [a15], if is of tame representation type, then is weakly non-negative. The converse implication does not hold in general, but it has been proved under some assumption on (see [a11]).

A Tits quadratic form for a class of classical -orders , where is a complete discrete valuation domain, has been defined in [a21]. Criteria for the finite lattice type and tame lattice type of are given in [a21] by means of .

For a class of -co-algebras , a Tits quadratic form is defined in [a22], and the co-module types of are studied by means of , where is a complete set of pairwise non-isomorphic simple left -co-modules and is a free Abelian group of rank .

References

[a1] V.I. Auslander, I. Reiten, S. Smalø, "Representation theory of Artin algebras" , Studies Adv. Math. , 36 , Cambridge Univ. Press (1995) MR1314422 Zbl 0834.16001
[a2] I.N. Bernstein, I.M. Gelfand, V.A. Ponomarev, "Coxeter functors and Gabriel's theorem" Russian Math. Surveys , 28 (1973) pp. 17–32 Uspekhi Mat. Nauk. , 28 (1973) pp. 19–33
[a3] K. Bongartz, "Algebras and quadratic forms" J. London Math. Soc. , 28 (1983) pp. 461–469 MR0724715 Zbl 0532.16020
[a4] V. Dlab, C.M. Ringel, "Indecomposable representations of graphs and algebras" , Memoirs , 173 , Amer. Math. Soc. (1976) MR0447344 Zbl 0332.16015
[a5] Yu.A. Drozd, "Coxeter transformations and representations of partially ordered sets" Funkts. Anal. Prilozhen. , 8 (1974) pp. 34–42 (In Russian) MR0351924 Zbl 0356.06003
[a6] Yu.A. Drozd, "On tame and wild matrix problems" , Matrix Problems , Akad. Nauk. Ukr. SSR., Inst. Mat. Kiev (1977) pp. 104–114 (In Russian) MR498704
[a7] Yu.A. Drozd, "Tame and wild matrix problems" , Representations and Quadratic Forms (1979) pp. 39–74 (In Russian) MR0600111 Zbl 0454.16014
[a8] P. Gabriel, "Unzerlegbare Darstellungen 1" Manuscripta Math. , 6 (1972) pp. 71–103 (Also: Berichtigungen 6 (1972), 309) MR332887
[a9] P. Gabriel, "Représentations indécomposables" , Séminaire Bourbaki (1973/74) , Lecture Notes in Mathematics , 431 , Springer (1975) pp. 143–169 MR0485996 Zbl 0335.17005
[a10] P. Gabriel, A.V. Roiter, "Representations of finite dimensional algebras" , Algebra VIII , Encycl. Math. Stud. , 73 , Springer (1992) MR1239446 MR1239447 Zbl 0839.16001
[a11] S. Kasjan, D. Simson, "Tame prinjective type and Tits form of two-peak posets II" J. Algebra , 187 (1997) pp. 71–96 MR1425560 Zbl 0944.16013
[a12] L.A. Nazarova, "Representations of quivers of infinite type" Izv. Akad. Nauk. SSSR , 37 (1973) pp. 752–791 (In Russian) MR0338018 Zbl 0298.15012
[a13] J.A. de la Peña, "Algebras with hypercritical Tits form" , Topics in Algebra , Banach Center Publ. , 26: 1 , PWN (1990) pp. 353–369 Zbl 0731.16008
[a14] J.A. de la Peña, "On the dimension of the module-varieties of tame and wild algebras" Commun. Algebra , 19 (1991) pp. 1795–1807 Zbl 0818.16013
[a15] J.A. de la Peña, D. Simson, "Prinjective modules, reflection functors, quadratic forms and Auslander–Reiten sequences" Trans. Amer. Math. Soc. , 329 (1992) pp. 733–753 MR1025753 Zbl 0789.16010
[a16] J.A. de la Peña, A. Skowroński, "The Euler and Tits forms of a tame algebra" Math. Ann. , 315 (2000) pp. 37–59 Zbl 0941.16010
[a17] C.M. Ringel, "Tame algebras and integral quadratic forms" , Lecture Notes in Mathematics , 1099 , Springer (1984) MR0774589 Zbl 0546.16013
[a18] A.V. Roiter, M.M. Kleiner, "Representations of differential graded categories" , Lecture Notes in Mathematics , 488 , Springer (1975) pp. 316–339 MR0435145 Zbl 0356.16011
[a19] D. Simson, "Linear representations of partially ordered sets and vector space categories" , Algebra, Logic Appl. , 4 , Gordon & Breach (1992) MR1241646 Zbl 0818.16009
[a20] D. Simson, "Posets of finite prinjective type and a class of orders" J. Pure Appl. Algebra , 90 (1993) pp. 77–103 MR1246276 Zbl 0815.16006
[a21] D. Simson, "Representation types, Tits reduced quadratic forms and orbit problems for lattices over orders" Contemp. Math. , 229 (1998) pp. 307–342 MR1676228 Zbl 0921.16007
[a22] D. Simson, "Coalgebras, comodules, pseudocompact algebras and tame comodule type" Colloq. Math. , in press (2001) MR1874368 Zbl 1055.16038
How to Cite This Entry:
Tits quadratic form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tits_quadratic_form&oldid=21950
This article was adapted from an original article by Daniel Simson (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article