Self-Injective Quivers with Potential

• Self-injective QPs are quivers with potential whose finite-dimensional Jacobian algebras exhibit self-injectivity and rich homological properties.
• The framework unifies mutation theory, silting/Okuyama–Rickard complexes, and truncated Jacobian algebras to establish derived equivalences.
• Planar constructions and symmetric Postnikov diagrams provide concrete examples for generating 2-representation-finite algebras through explicit Nakayama permutations.

Searching arXiv for the cited papers and closely related work on self-injective quivers with potential. Self-injective quivers with potential are quivers with potential $(Q,W)$ whose Jacobian algebra $J(Q,W)$ is finite-dimensional and self-injective. In the completed-path-algebra formulation, a potential is a possibly infinite linear combination of cyclic paths, the Jacobian ideal is generated by cyclic derivatives $\partial_a W$, and the Jacobian algebra is the corresponding quotient. Within this framework, self-injective QPs form a distinguished class linking mutation theory, silting and tilting constructions, truncated Jacobian algebras, and $2$-representation-finite algebras. The subject was developed systematically in work showing that $2$-representation-finite algebras arise as truncated Jacobian algebras of selfinjective QPs (Herschend et al., 2010), that silting mutation for self-injective Jacobian algebras is compatible with quiver-with-potential mutation (Mizuno, 2012), and that symmetric Postnikov diagrams furnish a geometric source of self-injective Jacobian algebras and new $2$-representation-finite algebras (Pasquali, 2017).

1. Definition and Jacobian-algebra framework

Let $Q=(Q_0,Q_1)$ be a finite connected quiver without loops. In one standard formulation, the complete path algebra is

$KQ=\prod_{i\ge 0} KQ_i,$

the $J$-adic completion, and $$KQ_{\mathrm{cyc}}=\bigoplus_{i\ge 2} KQ_i^{\mathrm{cyc}}$$ is the subspace generated by oriented cycles (Mizuno, 2012). Equivalently, for a finite quiver $J(Q,W)$0 with arrow ideal $J(Q,W)$1, the completed path algebra may be written as

$J(Q,W)$2

(Herschend et al., 2010). A potential $J(Q,W)$3 is an element of the cyclic subspace, or, in the completed setting, a possibly infinite $J(Q,W)$4-linear combination of cyclic paths considered up to cyclic rotation (Herschend et al., 2010).

For each arrow $J(Q,W)$5, the cyclic derivative is defined by cutting each occurrence of $J(Q,W)$6 in a cyclic monomial. In the notation of Mizuno,

$J(Q,W)$7

for a cycle $J(Q,W)$8, extended continuously (Mizuno, 2012). The Jacobian ideal is the closure of the ideal generated by all cyclic derivatives,

$J(Q,W)$9

and the Jacobian algebra is

$\partial_a W$0

(Mizuno, 2012). In the completed notation this is written

$\partial_a W$1

(Herschend et al., 2010).

A QP $\partial_a W$2 is called reduced if $\partial_a W$3 has no summand in $\partial_a W$4, and trivial if $\partial_a W$5 is a linear combination of $\partial_a W$6-cycles in such a way that $\partial_a W$7 (Mizuno, 2012). It is called self-injective if its Jacobian algebra is a finite-dimensional self-injective algebra (Mizuno, 2012). The exposition in Herschend and Iyama also distinguishes the finite-dimensional condition and then imposes selfinjectivity as an additional requirement; they note that whenever $\partial_a W$8 is finite dimensional it is Iwanaga–Gorenstein of dimension at most one, so selfinjectivity is a strong extra condition (Herschend et al., 2010).

This setup places self-injective QPs at the intersection of combinatorial quiver data and homological algebra. A plausible implication is that the completed-path-algebra formalism is not merely technical: it is the natural ambient setting in which cyclic derivatives, mutation, and finite-dimensional Jacobian quotients coexist.

2. Mutation and preservation of selfinjectivity

Mutation of quivers with potential in the sense of Derksen–Weyman–Zelevinsky is defined at a vertex $\partial_a W$9 not incident with any $2$0-cycle. The pre-mutation $2$1 is constructed by reversing arrows incident with $2$2, adding composite arrows $2$3 for each path $2$4, and replacing subpaths $2$5 in the potential by the new arrows. Explicitly,

$2$6

before reduction (Mizuno, 2012). Reduction then cancels direct $2$7-cycles and discards trivial QP summands, yielding the DWZ mutation $2$8 (Mizuno, 2012). Herschend and Iyama summarize the same process as the three-step recipe of adding composite arrows, reversing incident arrows, and adjusting the potential (Herschend et al., 2010).

For a subset $2$9 such that no two vertices of $2$0 are joined by an arrow or lie on a $2$1-cycle, mutations can be performed successively in any order to define

$2$2

(Mizuno, 2012). In the self-injective setting, the Nakayama permutation becomes central. If $2$3 is self-injective, the Nakayama permutation $2$4 is characterized by $2$5, and for any vertex $2$6 one may consider its orbit

$2$7

provided that no arrow connects distinct vertices of the orbit and none lies on a $2$8-cycle (Herschend et al., 2010). Under these conditions, the successive orbit mutation

$2$9

is well defined up to right equivalence and independent of the order (Herschend et al., 2010).

A fundamental preservation theorem states that if $2$0 is a selfinjective QP and $2$1 satisfies the orbit conditions above, then $2$2 is again a finite-dimensional selfinjective QP, and its Nakayama permutation remains $2$3 (Herschend et al., 2010). The proof sketch given there proceeds through the $2$4-CY cluster category of the Ginzburg DG algebra, where an $2$5-stable summand $2$6 with $2$7 is mutated via Iyama–Yoshino mutation; Amiot–Keller–Yang identify the endomorphism algebra of the mutated cluster-tilting object with the Jacobian algebra of the mutated QP (Herschend et al., 2010).

This preservation result establishes self-injective QPs as a mutation-stable class under orbit mutation. It also frames the Nakayama permutation as the structure controlling which simultaneous or successive mutations remain internal to the class.

3. Silting mutation, Okuyama–Rickard complexes, and compatibility

Let $2$8 be a finite-dimensional algebra and $2$9 its homotopy category of perfect complexes. A silting object $Q=(Q_0,Q_1)$0 satisfies $Q=(Q_0,Q_1)$1 for all $Q=(Q_0,Q_1)$2 and generates the category as a thick subcategory; if moreover $Q=(Q_0,Q_1)$3 for all $Q=(Q_0,Q_1)$4, then $Q=(Q_0,Q_1)$5 is tilting (Mizuno, 2012). Given a basic silting object $Q=(Q_0,Q_1)$6, a minimal left $Q=(Q_0,Q_1)$7-approximation $Q=(Q_0,Q_1)$8 extends to a triangle

$Q=(Q_0,Q_1)$9

and the left silting mutation is $KQ=\prod_{i\ge 0} KQ_i,$0 (Mizuno, 2012).

When $KQ=\prod_{i\ge 0} KQ_i,$1 is self-injective and $KQ=\prod_{i\ge 0} KQ_i,$2 is stable under the Nakayama permutation, the associated silting mutation is the Okuyama–Rickard complex. Writing $KQ=\prod_{i\ge 0} KQ_i,$3, one takes a minimal left $KQ=\prod_{i\ge 0} KQ_i,$4-approximation $KQ=\prod_{i\ge 0} KQ_i,$5, completes it to a triangle, and obtains

$KQ=\prod_{i\ge 0} KQ_i,$6

as a tilting complex whenever $KQ=\prod_{i\ge 0} KQ_i,$7 is Nakayama-stable (Mizuno, 2012).

Mizuno’s central theorem identifies this homological mutation with QP mutation. If $KQ=\prod_{i\ge 0} KQ_i,$8 is a self-injective QP, $KQ=\prod_{i\ge 0} KQ_i,$9, and $J$0 satisfies:

1. no vertex of $J$1 lies on a $J$2-cycle,
2. there are no arrows between vertices of $J$3,

then there is an algebra isomorphism

$J$4

(Mizuno, 2012). If in addition $J$5 is invariant under the Nakayama permutation of $J$6, then $J$7 is a tilting complex, and hence

$J$8

are derived equivalent (Mizuno, 2012).

The proof is based on an explicit map $J$9, where $$KQ_{\mathrm{cyc}}=\bigoplus_{i\ge 2} KQ_i^{\mathrm{cyc}}$$0 denotes the mutated complex, sending vertices to idempotent summands, old arrows to induced maps, new composite arrows to compositions through the cone, and reversed arrows to morphisms arising from the defining triangles (Mizuno, 2012). Using the Buan–Iyama–Reiten–Smith presentation criterion, one shows that $$KQ_{\mathrm{cyc}}=\bigoplus_{i\ge 2} KQ_i^{\mathrm{cyc}}$$1 is surjective and that its kernel is the closure of the ideal generated by the cyclic derivatives $$KQ_{\mathrm{cyc}}=\bigoplus_{i\ge 2} KQ_i^{\mathrm{cyc}}$$2, yielding $$KQ_{\mathrm{cyc}}=\bigoplus_{i\ge 2} KQ_i^{\mathrm{cyc}}$$3 (Mizuno, 2012). The homological core of the argument is the construction of right and left $$KQ_{\mathrm{cyc}}=\bigoplus_{i\ge 2} KQ_i^{\mathrm{cyc}}$$4-almost split sequences in $$KQ_{\mathrm{cyc}}=\bigoplus_{i\ge 2} KQ_i^{\mathrm{cyc}}$$5 mirroring the Jacobian relations (Mizuno, 2012).

This compatibility theorem is a precise bridge between combinatorial mutation of QPs and derived-category mutation of self-injective algebras. It explains why QP mutation can produce derived-equivalent Jacobian algebras in the self-injective case, rather than only derived-equivalent Ginzburg DG algebras.

4. Truncated Jacobian algebras and $$KQ_{\mathrm{cyc}}=\bigoplus_{i\ge 2} KQ_i^{\mathrm{cyc}}$$6-representation-finite algebras

A cut of a QP $$KQ_{\mathrm{cyc}}=\bigoplus_{i\ge 2} KQ_i^{\mathrm{cyc}}$$7 is a subset $$KQ_{\mathrm{cyc}}=\bigoplus_{i\ge 2} KQ_i^{\mathrm{cyc}}$$8 such that every cyclic monomial appearing in $$KQ_{\mathrm{cyc}}=\bigoplus_{i\ge 2} KQ_i^{\mathrm{cyc}}$$9 contains exactly one arrow of $J(Q,W)$00 (Herschend et al., 2010). One defines a grading by

$J(Q,W)$01

so that $J(Q,W)$02 has total degree $J(Q,W)$03, and forms the truncated Jacobian algebra

$J(Q,W)$04

(Herschend et al., 2010). Equivalently,

$J(Q,W)$05

where $J(Q,W)$06 is the subquiver with arrow set $J(Q,W)$07 (Herschend et al., 2010).

The connection with higher Auslander–Reiten theory is given by the structure theorem of Herschend and Iyama. A finite-dimensional algebra $J(Q,W)$08 is $J(Q,W)$09-representation-finite if $J(Q,W)$10 and there exists a cluster-tilting $J(Q,W)$11-module $J(Q,W)$12 with $J(Q,W)$13 non-isomorphic indecomposable summands (Herschend et al., 2010). They show:

• if $J(Q,W)$14 is a selfinjective QP and $J(Q,W)$15 is any cut, then $J(Q,W)$16 is $J(Q,W)$17-representation-finite;
• conversely every basic $J(Q,W)$18-representation-finite algebra arises as $J(Q,W)$19 for some selfinjective QP $J(Q,W)$20 and algebraic cut $J(Q,W)$21 (Herschend et al., 2010).

They also state that if $J(Q,W)$22 is a selfinjective QP and $J(Q,W)$23 is any cut, then the truncated Jacobian algebra has global dimension at most $J(Q,W)$24 and admits a cluster-tilting module; equivalently, it is $J(Q,W)$25-representation-finite, and its $J(Q,W)$26-preprojective algebra recovers $J(Q,W)$27 (Pasquali, 2017).

The converse direction is organized via the complete $J(Q,W)$28-preprojective algebra $J(Q,W)$29. It is shown in [IO2], as summarized by Herschend and Iyama, that $J(Q,W)$30 and $J(Q,W)$31 is $J(Q,W)$32-representation-finite if and only if $J(Q,W)$33 is finite-dimensional and selfinjective; moreover one constructs a QP $J(Q,W)$34 with a canonical cut $J(Q,W)$35 such that

$J(Q,W)$36

(Herschend et al., 2010).

This identifies self-injective QPs as a source and organizing principle for the theory of $J(Q,W)$37-representation-finite algebras. A plausible implication is that the self-injective Jacobian algebra should be viewed as the ambient object, while its cuts encode finite-global-dimension shadows.

5. Derived equivalence of cuts and planar self-injective QPs

All algebras $J(Q,W)$38 arising from cuts of a fixed QP are cluster equivalent, in the sense that their generalized cluster categories coincide, but they need not be derived equivalent in general (Herschend et al., 2010). To address this, Herschend and Iyama relate cut mutation to $J(Q,W)$39-APR tilting. If $J(Q,W)$40 is a source of $J(Q,W)$41 (or dually a sink), then mutating the cut at $J(Q,W)$42 produces a new cut $J(Q,W)$43, obtained by removing all arrows starting at $J(Q,W)$44 from $J(Q,W)$45 and adding all arrows ending at $J(Q,W)$46, and the corresponding algebra is the $J(Q,W)$47-APR tilt or cotilt of $J(Q,W)$48 (Herschend et al., 2010).

They introduce two combinatorial conditions on a QP:

• fully compatible: any two cuts $J(Q,W)$49 induce the same $J(Q,W)$50-degree on every cyclic walk;
• has enough cuts: every arrow lies in at least one cut (Herschend et al., 2010).

Under these conditions, the set of all cuts is connected by successive cut mutations, so any two truncated Jacobian algebras are iterated $J(Q,W)$51-APR tilts of each other. Consequently, if $J(Q,W)$52 is selfinjective, fully compatible, and has enough cuts, then for any two algebraic cuts $J(Q,W)$53, the algebras $J(Q,W)$54 and $J(Q,W)$55 are derived equivalent (Herschend et al., 2010).

A major class where these hypotheses are available is given by planar QPs. The associated CW-complex $J(Q,W)$56, called the canvas, has $J(Q,W)$57-cells $J(Q,W)$58, $J(Q,W)$59-cells corresponding to arrows, and $J(Q,W)$60-cells corresponding to cyclic monomials in $J(Q,W)$61 (Herschend et al., 2010). A QP is planar if $J(Q,W)$62 is simply connected and admits an embedding into $J(Q,W)$63 (Herschend et al., 2010). By van Kampen, every simply connected QP is fully compatible, and the examples considered are chosen to have enough cuts; hence every truncated Jacobian algebra of a selfinjective planar QP is derived equivalent (Herschend et al., 2010).

The emphasis on planarity should not be conflated with mere visualizability of a quiver. Here planarity is a property of the canvas attached to the pair $J(Q,W)$64, and it interacts with compatibility of cuts and derived equivalence of truncated Jacobian algebras.

6. Constructions, examples, and the Postnikov-diagram realization

Several concrete families illustrate the theory. Mizuno’s examples include a $J(Q,W)$65-cycle

$J(Q,W)$66

for which $J(Q,W)$67 is self-injective (Mizuno, 2012). Mutation at vertex $J(Q,W)$68 reverses $J(Q,W)$69 and $J(Q,W)$70, adds the composition arrow $J(Q,W)$71, and produces the potential

$J(Q,W)$72

(Mizuno, 2012). The corresponding Okuyama–Rickard complex has endomorphism algebra isomorphic to $J(Q,W)$73, but since $J(Q,W)$74 is not Nakayama-stable, the mutation need not yield a derived equivalence, and in this example it does not (Mizuno, 2012).

Further examples in (Mizuno, 2012) include a $J(Q,W)$75 mesh with nine vertices and potential equal to the sum of all small squares, where the Nakayama permutation is $J(Q,W)$76; mutating at an orbit such as $J(Q,W)$77 or $J(Q,W)$78 gives new self-injective QPs whose Jacobian algebras are derived equivalent to the original (Mizuno, 2012). Another example of tubular type $J(Q,W)$79 has identity Nakayama permutation, so any vertex may be mutated, yielding a large family of derived-equivalent self-injective Jacobian algebras (Mizuno, 2012).

Herschend and Iyama give planar families: triangular lattice examples $J(Q,W)$80 with potential given by the sum of all little $J(Q,W)$81-cycles, square-shaped QPs on the $J(Q,W)$82 grid with potential the sum of all $J(Q,W)$83-cycles with alternating signs, and the $J(Q,W)$84-gon family where $J(Q,W)$85 is the oriented $J(Q,W)$86-cycle and

$J(Q,W)$87

In the $J(Q,W)$88-gon case, $J(Q,W)$89 is selfinjective, the canvas is an $J(Q,W)$90-gon, any cut is any single arrow, and all cuts differ by a single $J(Q,W)$91-APR tilt (Herschend et al., 2010).

A geometric realization comes from Postnikov diagrams. For a reduced $J(Q,W)$92-Postnikov diagram $J(Q,W)$93, Baur–King–Marsh associate an ice quiver with potential $J(Q,W)$94, and quotienting the corresponding frozen Jacobian algebra $J(Q,W)$95 by the ideal generated by the boundary idempotents yields a finite-dimensional algebra

$J(Q,W)$96

where $J(Q,W)$97 is the unfrozen QP obtained by deleting the frozen vertices and all incident arrows (Pasquali, 2017). Jensen–King–Su construct a cluster-tilting object

$J(Q,W)$98

in $J(Q,W)$99, and there is an isomorphism $\partial_a W$00; passing to stable endomorphisms gives

$\partial_a W$01

(Pasquali, 2017).

Self-injectivity is characterized by rotational symmetry: $\partial_a W$02 is self-injective if and only if $\partial_a W$03 admits the rotation symmetry by angle $\partial_a W$04, equivalently $\partial_a W$05 (Pasquali, 2017). In that case, the Nakayama permutation acts by

$\partial_a W$06

and the induced automorphism of the quiver is the geometric rotation by $\partial_a W$07 (Pasquali, 2017). When this symmetry holds, $\partial_a W$08 is precisely the Jacobian algebra of the unfrozen QP, making $\partial_a W$09 a planar self-injective QP (Pasquali, 2017).

This Postnikov-diagram construction shows that self-injective QPs arise not only from abstract mutation theory but also from the geometry of $\partial_a W$10-related combinatorics.

7. Significance, scope, and limitations

One central significance of the theory is the passage from mutation statements about dg or cluster categories to statements about ordinary finite-dimensional algebras. Keller–Yang had shown earlier that for any QP mutation the associated $\partial_a W$11-Calabi–Yau Ginzburg dg algebras are derived equivalent; Mizuno’s result is stronger in the self-injective case because it shows that already the Jacobian algebras are derived equivalent under the special silting or Okuyama–Rickard mutation (Mizuno, 2012).

A second major significance is structural: self-injective QPs provide a model for $\partial_a W$12-representation-finite algebras through cuts, and conversely every basic $\partial_a W$13-representation-finite algebra arises from a selfinjective QP with an algebraic cut (Herschend et al., 2010). In the Postnikov setting, this yields new $\partial_a W$14-representation-finite algebras from symmetric diagrams, and mutations at Nakayama orbits produce further examples (Pasquali, 2017).

Several restrictions are intrinsic to the theory. Mutation at a vertex requires exclusion of $\partial_a W$15-cycles at that vertex, and simultaneous or orbit mutation requires the absence of arrows among the chosen vertices (Mizuno, 2012, Herschend et al., 2010). Derived equivalence after mutation is not automatic: it depends on Nakayama stability of the mutated summand or orbit, as the $\partial_a W$16-cycle example demonstrates (Mizuno, 2012). Likewise, cluster equivalence of truncated Jacobian algebras from different cuts does not in general imply derived equivalence; additional assumptions such as full compatibility and enough cuts are required (Herschend et al., 2010).

These results suggest a coherent picture in which self-injective QPs are a mutation-stable class of Jacobian presentations whose homological behavior is controlled by the Nakayama permutation, whose cut algebras encode $\partial_a W$17-representation-finite structures, and whose geometric realizations—particularly planar QPs and symmetric Postnikov diagrams—supply concrete families with explicit derived-equivalence mechanisms (Herschend et al., 2010, Mizuno, 2012, Pasquali, 2017).