geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
A Dynkin quiver is a quiver whose underlying undirected graph is a one of the Dynkin diagrams in the ADE series.
graphics grabbed from Qiu 15
See e.g. Qiu 15, Def. 2.1
Gabriel's theorem (Gabriel 72) says that connected quivers with a finite number of indecomposable quiver representations over an algebraically closed field are precisely the Dynkin quivers: those whose underlying undirected graph is a Dynkin diagram in the ADE series
Moreover, the indecomposable quiver representations in this case are bijection with the positive roots in the root system of the Dynkin diagram.
ADE classification and McKay correspondence
Dynkin diagram/ Dynkin quiver | Platonic solid | finite subgroups of SO(3) | finite subgroups of SU(2) | simple Lie group |
---|---|---|---|---|
$A_{n \geq 1}$ | cyclic group $\mathbb{Z}_{n+1}$ | cyclic group $\mathbb{Z}_{n+1}$ | special unitary group $SU(n+1)$ | |
D4 | Klein four-group $D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$ | quaternion group $2 D_4 \simeq$ Q8 | SO(8) | |
$D_{n \geq 4}$ | dihedron, hosohedron | dihedral group $D_{2(n-2)}$ | binary dihedral group $2 D_{2(n-2)}$ | special orthogonal group $SO(2n)$ |
$E_6$ | tetrahedron | tetrahedral group $T$ | binary tetrahedral group $2T$ | E6 |
$E_7$ | cube, octahedron | octahedral group $O$ | binary octahedral group $2O$ | E7 |
$E_8$ | dodecahedron, icosahedron | icosahedral group $I$ | binary icosahedral group $2I$ | E8 |
Gabriel's theorem is due to
Discussion of Bridgeland stability conditions on quiver representations over Dynkin Quivers includes
Yu Qiu, Stability conditions and quantum dilogarithm identities for Dynkin quivers, Adv. Math., 269 (2015), pp 220-264 (arXiv:1111.1010)
Tom Bridgeland, Yu Qiu, Tom Sutherland, Stability conditions and the $A_2$ quiver (arXiv:1406.2566)
Last revised on October 2, 2018 at 11:47:04. See the history of this page for a list of all contributions to it.