Frobenius Degenerations of Preprojective Algebras

TL;DR

This paper introduces a generalized preprojective algebra for decorated quivers, explores its moduli space, and studies flat degenerations of ordinary preprojective algebras, with proven results in the Dynkin case.

Contribution

It generalizes preprojective algebras to decorated quivers, analyzes their moduli spaces, and proves flatness of degenerations in the Dynkin case.

Findings

Moduli space of representations recovers Hamiltonian reduction.

Degenerations are flat in the Dynkin case.

Conjecture extends flatness to all decorated quivers.

Abstract

In this paper, we study a preprojective algebra for quivers decorated with $k$-algebras and bimodules, which generalizes work of Gabriel for ordinary quivers, work of Dlab and Ringel for $k$-species, and recent work of de Thanhoffer de V\"olcsey and Presotto, which has recently appeared from a different perspective in work of K\"ulshammer. As for undecorated quivers, we show that its moduli space of representations recovers the Hamiltonian reduction of the cotangent bundle over the space of representations of the decorated quiver. These algebras yield degenerations of ordinary preprojective algebras, by folding the quiver and then degenerating the decorations. We prove that these degenerations are flat in the Dynkin case, and conjecture, based on computer results, that this extends to arbitrary decorated quivers.

Equations

$${\mathcal{M}}_d:=T^{\ast }{\text{Rep}}_d(Q)///G{L}_d(k):={\mu }^{-1}(0)//G{L}_d(k)$$

$$\Pi (Q):=P(\overline{Q})/⟨\sum _{\alpha \in Q_1}\alpha {\alpha }^{\ast }-{\alpha }^{\ast }\alpha ⟩$$

$${\mathcal{M}}_d\cong {\text{Rep}}_d\Pi (Q)//G{L}_d(k).$$

$${\text{Rep}}_d\Pi (Q,D_Q)//G_d\cong {\mu }^{-1}(0)//G_d.$$

$$M^{{\alpha }^{\ast }}:={\text{Hom}}_{A_i\otimes A_j}(M^{\alpha },A_i\otimes A_j).$$

$${\mu }_d:T^{\ast }{\text{Rep}}_d(Q,D_Q)\to {\mathfrak{g}}_d^{\ast }.$$

$$(Q,D_Q)=k\stackrel{k}{⟵}k\stackrel{k}{⟶}k.$$

$$k\overset{k}{\longleftarrow}k\overset{k}{\longrightarrow}k\hskip 28.45274pt\overset{\text{fold}}{\leavevmode\resizebox{28.45274pt}{}{$\Longrightarrow$}}\hskip 28.45274ptk\overset{k\oplus k}{\longrightarrow}k\oplus k.$$

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$$(k[[t]]\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$B_j:=\underset{i\in \text{Aut}(Q)\cdot j}{⨁}{A}_i\text{and}{N}^{\beta }:=\underset{\alpha \in \text{Aut}(Q)\cdot \beta }{⨁}{M}^{\alpha }.$$

$$\text{Rep}(Q,D_Q)\cong \text{Rep}(Q/\text{Aut}(Q),D_Q/\text{Aut}(Q))$$

$$\Pi (Q,D_Q)\cong \Pi (Q/\text{Aut}(Q),D_Q/\text{Aut}(Q)).\text{vspace}\ast .3cm$$

$$h_{\Pi (Q,D_Q)}(t)=h_{\Pi (Q,D_Q^{\prime })}(t).$$

$$k\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$a_{ij}:=\sum _{\alpha :i\to j}\frac{{\dim }_k(M^{\alpha })}{{\dim }_k(A_i)}=\sum _{\alpha :i\to j}{\dim }_{A_i}(M^{\alpha })\in \mathbb{Z}.$$

$$(DA{)}_{ij}=\sum _{\alpha :i\to j}{\dim }_k(M^{\alpha })=\sum _{{\alpha }^{\ast }:j\to i}{\dim }_k(M^{{\alpha }^{\ast }})=(DA{)}_{ji}.$$

$$h_{\Pi (Q,D_Q)}(t)=\{\begin{array}{ll}(I+t^{{\text{max}}_t+2}P)D(1-At+t^2{)}^{-1}& \text{for }(Q,D_Q)\text{ Dynkin}\\ D(1-At+t^2{)}^{-1}& \text{otherwise}\end{array}$$

$$h_{\Pi (Q,D_Q)}(t,s)=\{\begin{array}{l}(1+s^{{\max }_s}{t}^{{\max }_t+2}P)D(1-At+B\end{array}$$