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Twisted Gelfand-Ponomarev modules

Joseph Muller, Chia-Fu Yu

Abstract

In this expository paper, given a field $K$ and an automorphism $σ\in \mathrm{Aut}(K)$, we give a self-contained proof of the classification of finite dimensional $K$-vector spaces equipped with two operators $F$ and $V$, respectively $σ$-linear and $σ^{-1}$-linear, such that $FV = VF = 0$. This classification was originally due to the combined results of Gelfand and Ponomarev (1968), and of Kraft (1975). Following a recent suggestion of Chai (2025), we reworked their classification in light of the notion of Kraft quivers.

Twisted Gelfand-Ponomarev modules

Abstract

In this expository paper, given a field and an automorphism , we give a self-contained proof of the classification of finite dimensional -vector spaces equipped with two operators and , respectively -linear and -linear, such that . This classification was originally due to the combined results of Gelfand and Ponomarev (1968), and of Kraft (1975). Following a recent suggestion of Chai (2025), we reworked their classification in light of the notion of Kraft quivers.
Paper Structure (13 sections, 44 theorems, 170 equations, 5 figures)

This paper contains 13 sections, 44 theorems, 170 equations, 5 figures.

Table of Contents

  1. $\sigma$-linear relations
  2. Basic definitions
  3. Stable domain, kernel, image and indeterminacy
  4. The decomposition theorem
  5. Kraft quivers and $\sigma$-linear representations
  6. Kraft quivers
  7. Connected Kraft quivers and words
  8. $\sigma$-linear representations of Kraft quivers
  9. Twisted Gelfand-Ponomarev modules of the first and second kinds
  10. Stabilized sequence and elementary intervals
  11. Submodules of the first kind and linear Kraft quivers
  12. Submodules of the second kind and circular Kraft quivers
  13. The classification of twisted Gelfand-Ponomarev modules

Key Result

Theorem 1

Let $M$ be a twisted Gelfand-Ponomarev module. There exist such that $M \simeq M(\Gamma,U,\rho)$ as $K[F,V]_\sigma$-modules. If $\Gamma'$ and $(U',\rho')$ are another Kraft quiver and strict $\sigma$-linear representation meeting the same conditions, then there exists an isomorphism $\Gamma \simeq \Gamma'$ making $(U,\rho)$ isomorphic to $(U',\rho')$.

Figures (5)

  • Figure 1: Two examples of Kraft quivers.
  • Figure 2: A directed graph labeled by $\{F,V\}$ which is not a Kraft quiver.
  • Figure 3: The converse graphs of the Kraft quivers of Figure \ref{['Figure1']}.
  • Figure 4: Connected circular Kraft quivers with repetitions (left), and their reductions (right).
  • Figure 5: An example of graph $\Gamma(M,1\mathrm{st})$.

Theorems & Definitions (131)

  • Theorem 1: Theorem \ref{['AllTwistedGFModulesComeFromKraftQuiver']}
  • Theorem 2: Theorem \ref{['TheoremClassificationIndecomposableGFModules']}
  • Definition 1.1
  • Example 1.2
  • Definition 1.3
  • Proposition 1.4
  • proof
  • Definition 1.5
  • Definition 1.6
  • Example 1.7
  • ...and 121 more