Cluster-tilted algebras of Dynkin type E

This page contains supplementary material to the paper Derived equivalence classification of cluster-tilted algebras of Dynkin type E by Janine Bastian, Thorsten Holm and Sefi Ladkani.

The derived equivalence classes of the cluster-tilted algebras of Dynkin type E are characterized by their so-called associated polynomials.

In the tables below, click on a polynomial to see the list of cluster-tilted algebras belonging to the corresponding class. For each algebra we write its quiver, defining relations, Cartan matrix and the complete list of good mutations.

Type E6

There are 67 cluster-tilted algebras of Dynkin type E6 which fall into 6 derived equivalence classes.

Associated polynomial Size
x6 - x5 + x3 - x + 1 20
2( x6 - 2x4 + 4x3 - 2x2 + 1) 3
2( x6 - x4 + 2x3 - x2 + 1) 16
3( x6 + x3 + 1) 19
4( x6 + x4 + x2 + 1) 7
4( x6 + x5 - x4 + 2x3 - x2 + x + 1) 2

Type E7

There are 416 cluster-tilted algebras of Dynkin type E7 which fall into 14 derived equivalence classes.

Associated polynomial Size
x7 - x6 + x4 - x3 + x - 1 64
2( x7 - 2x5 + 4x4 - 4x3 + 2x2 - 1) 8
2( x7 - x5 + x4 - x3 + x2 - 1) 72
2( x7 - x5 + 2x4 - 2x3 + x2 - 1) 32
3( x7 - 1) 124
4( x7 + x5 - 2x4 + 2x3 - x2 - 1) 8
4( x7 + x5 - x4 + x3 - x2 - 1) 56
4( x7 + x6 - 2x5 + 2x4 - 2x3 + 2x2 - x - 1) 2
4( x7 + x6 - x5 - x4 + x3 + x2 - x - 1) 4
4( x7 + x6 - x5 + x4 - x3 + x2 - x - 1) 16
5( x7 + x5 - x4 + x3 - x2 - 1) 17
6( x7 + x5 - x2 - 1) 1
6( x7 + x6 - x4 + x3 - x - 1) 11
8( x7 + x6 + x5 - x4 + x3 - x2 - x - 1) 1

Type E8

There are 1574 cluster-tilted algebras of Dynkin type E8 which fall into 15 derived equivalence classes.

Associated polynomial Size
x8 - x7 + x5 - x4 + x3 - x + 1 128
2( x8 - 2x6 + 4x5 - 4x4 + 4x3 - 2x2 + 1) 16
2( x8 - x6 + x5 + x3 - x2 + 1) 256
2( x8 - x6 + 2x5 - 2x4 + 2x3 - x2 + 1) 64
3( x8 + x4 + 1) 384
4( x8 + x6 - 2x5 + 4x4 - 2x3 + x2 + 1) 22
4( x8 + x6 - x5 + 2x4 - x3 + x2 + 1) 221
4( x8 + x7 - 2x6 + 2x5 + 2x3 - 2x2 + x + 1) 12
4( x8 + x7 - x6 + 2x4 - x2 + x + 1) 48
4( x8 + x7 - x6 + x5 + x3 - x2 + x + 1) 72
5( x8 + x6 + x4 + x2 + 1) 167
6( x8 + x6 + x5 + x3 + x2 + 1) 38
6( x8 + x7 + 2x4 + x + 1) 118
8( x8 + x7 + x6 + 2x4 + x2 + x + 1) 24
8( x8 + 2x7 + 2x4 + 2x + 1) 4