Auslander - Reiten sequences and intuition

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New Pathways between Group Theory and Model Theory - A conference in memory of Rüdiger Göbel (1940-2014)

Mülheim an der Ruhr (Germany), Feb 1 - 4, 2016

Thank you very much (Herzlichen Dank !) to the Organizers of this Conference dedicated with great affection and gratitude to the memory of Rüdiger Göbel , to celebrate his life, work and legacy.

I met him for the first time at the Rome Conference (last millennium), where he gave a lecture on slender groups. The title of his joint work with Burkhard Wald in the conference proceedings is “Wachtumstypen (= the growth of types) und schlanke Gruppen”.

I met him for the second time in Essen. I remember with gratitude his invitation to give a talk during my first year in Bielefeld and the great hospitality (from my arrival at the train station to my departure after a joint dinner with many students).

The team I met in Essen was a very active and enthousiastic group, working with a great passion, ready and happy to collaborate with colleagues and/or future colleagues of any age and country. This attitude did not vanished over the years, as this conference proves.

My first talk in Essen (= very long talk with many questions of the interested audience!) dealt with endomorphism rings of reduced torsion free abelian groups. What I also remember: many black-boards, the difficulty to wipe them out (in German style) and the precious help of many participants.

Outline of the talk

(1) Preliminary definitions & properties (2) An Auslander - Reiten sequence with 3 middle terms ...... (3) An Auslander - Reiten quiver ( Abbildung 3 in the home page of C.M. RINGEL www.mathematik.uni- bielefeld.de …..) = a complicated topological object with 28 vertices (4) The prehistory of (3) [ 7 , 14 ] and what comes next.

Why AR - sequences and quivers ?

1st reason: AR - quivers are beautiful (for mathematicians and not) 2nd reason: I used AR - quivers many times, but I showed only a few of them at official meetings.

Personal experience with quite different people (children, …)

AR - quivers & sequences have so many aspects that the best definitions and theorems cannot describe them completely. Their beauty is somehow independent of the knowledge of the whole theory behind them. Perhaps, this is the reason why the word ”intuition” shows up at the end ( pages 70 - 71 ) of Gabriel’s paper ….

GABRIEL , “ Auslander - Reiten sequences and repres. - finite algebras”, LMN 831 ]

The name is a dedication of Ringel to the authors of “almost split sequences”. He introduced Auslander - Reiten quivers in his Brandeis lectures (1975) and determined their structure for tame and wild quivers [31] . Since then, various specialists like Bautista, Brenner, Butler, Riedtmann . . . . have hoarded a few hundread examples in their dossiers, thus getting an intuition which no theoretical argument can replace.

The whole is greater than the sum of its parts, !

BUT

My belief/experience: also small peaces may be useful to understand the whole. E.g. : the 2 “almost split” maps in an AR - sequence are more important than the irreducible maps (between indecomposable modules), the small ingredients they are made of, but more complicated to compute and/or guess.

Why guess ?

- ONLY FEW IRREDUCIBLE maps ( between indecomp. modules ) are well - know. !- Symmetric considerations + “topology” ( = shape of irred. maps in the area) + ??? suggest the form of the maps τ(M) X and X M !possible meaning of ??? = intuition

1st “definition”

An Auslander - Reiten sequence f g 0 ---! τ(M) ---! X ---! M ---! 0 is a short exact sequence which does NOT split, with M & τ(M) indecomposable s.t. any L --! M non split epi (resp. τ(M) --! L non split mono) factors through g (resp. f ).

Consequences

- Any possible “candidate” factors through f and g . !

- Any possible “candidate” obtained from an Auslander - Reiten sequence (by means of non split morphisms, pushouts and pullbacks) splits.

2nd “definition”

An Auslander - Reiten quiver is a oriented graph s.t. there are 1 - 1 maps vertices "-! indecomposable f.g. modules arrows "-! “irreducible” maps between…. (“irreducible” maps have only obvious factorizations & they are the components of the two non - zero maps of some Auslander - Reiten sequence)

1st toy example of AR - sequence

0 --! P(2) --! P(1) = I(2) --! I(1) --! 0 !with two projective & two injective modules defined over the algebra given by the Dynkin diagram A : o -----! o 2 1 2 !

1st toy example of AR - quiver

I(1) is the unique indecomposable non projective module. Hence the previous sequence is the unique AR - sequence. Consequence: The AR - quiver has 3 vertices: P(2), P(1) = I(2) , I(1) 2 arrows: P(2) --! P(1) , I(2) --! I(1)

2nd toy example of AR - sequence

0 ---! S ---! A ---! S ---! 0 with 2 A = K[x] / (x ) and S simple

2nd toy example of AR - quiver

S is the unique indecomposable non projective module. Hence the previous sequence is the unique AR - sequence. Consequence: The AR - quiver has 2 vertices: S and A 2 arrows: S ---! A , A ---! S

Remarks

- The 1st AR - quiver does not have oriented cycles. - The 2nd AR - quiver has an oriented cycle ( S --! A --! S and S = τ(S) ) .

- In both cases, the middle term is indecomposable (& projective - injective) .

Two AR sequences with decomposable middle term

2 3 2 3 0 ---! 4 ---! 4 + 4 ---! 4 ---! 0 , ! 2 3 1 1 0 --! 4 --! 2 + 3 + 2 3 --! 2 3 --! 0 4 with 2 3 = rad P(1) , 1 = P(1)/soc P(1) 4 2 3

Two well - known irreducible maps

(between indecomposable modules) !

X -----! P with P projective & X summand of rad P = max. submodule of P I -----! Y with I injective & Y summand of I / soc I

Examples (already seen)

2 3 1 4 ---! 4 , 4 ---! 4 , 2 3 ---! 2 3 , 4 4 2 S ---! A = K[x]/(x ) , P(2) ---! P(1) ; 1 1 I(2) ---! I(1), A ---! S, 2 3 --! 2 3 . 4

A more complicated AR sequence

….. of representations of the form

( V(1), V(2) ; f , g ) with !

V(1) and V(2) vector spaces / K

f : V(1) ----! V(2) linear map

g : V(2) ----! V(2) endomorphism s.t. 6 2 g = 0 and g f = 0

( 6, 24 ) , ( 7 , 27 ) , ( 5 , 18 ) , ( 3, 13 ) , (5 , 20 ) :

the dimension types ( a , b ) = ( dim V(1) , dim V(2) ) of the indecomp. modules M , τ(M) and

of the 3 indecomp. summands of the

middle term ? in the AR - sequence

0 τ(M) ? M 0

!I want to show you.

CONVENTION

Arrows of different shape denote more or less visible irreducible maps !

[ if we use only one choice for the basis of the 2 vector spaces, the support of the indecomposable modules of dimension type (6,24) and (7,27) respectively ].

First remarks

The injective map τ(M) X

is of the form (f, g , h) with f, g and h irreducible & surjective. !

The surjective map X M is of the form (f , g, h) with f , g and h irreducible & injective.

The 2 obvious morphisms :

The epimorphism (7, 27) (5, 20) is a kind of LEFT cancellation. The embedding (5, 18) (6, 24) is a kind of LEFT addition.

BUT with respect to a different basis

of one of the 2 modules of dimension - type (7,27) and (6,24) , also the remaning 4 irreducible maps become obvious epimorphisms or embeddings (RIGHT cancellation, LEFT cancellation, LEFT addition, RIGHT addition). !

Ringel’s quiver with 28 vertices

describing the indec. representations ( V(1), V(2) ; f , g ) with

V(1) and V(2) vector spaces / K

f : V(1) ----! V(2) linear map

g : V(2) ----! V(2) endomorphism s.t. 4 2 g = 0 and g f = 0

!Abbildung 3 !Diskrete Methoden in der Darstellungstheorie. Bielefelder Universitätszeitung. (1992). http://www.math.uni-bielefeld.de/~ringel/opus/diskret/picture.html

C.M. RINGEL Unzerlegbare Darstellungen endlich - dimensionaler Algebren (page 93 …. )

Position of the 8 instable modules

Position of 12 “visible” stable modules

Orbit of the length 8

(Picture 1)

Orbit of the length 8

(Picture 2)

Two less complicated AR quivers

There exist 66 (and not 56) representations of the form

( V(1), V(2) ; f , g ) with !

V(1) and V(2) vector spaces / K

f : V(1) ----! V(2) linear map

g : V(2) ----! V(2) endomorphism s.t. 5 2 g = 0 and g f = 0

THANK YOU ! DANKE !