Semilinear Biquivers

• Semilinear biquivers are quivers with arrows partitioned into linear and semilinear types, where semilinear maps incorporate automorphisms like complex conjugation into the representation framework.
• They enable the construction of semilinear path and clannish algebras, linking the combinatorics of quivers with the classification of indecomposable modules via strings and bands.
• Their theory connects algebraic representations with geometric surface models, using nodal covers and seams to track semilinearity and derive practical classification results.

A semilinear biquiver is a quiver whose arrows are partitioned into linear and semilinear arrows relative to a fixed involution $\sigma$ on a base field $k$, or more generally into arrows carrying prescribed automorphisms $\sigma_a \in \operatorname{Aut}(K)$ of a division ring $K$. A representation assigns a vector space or module to each vertex and, along each arrow, either a linear map or a $\sigma$-semilinear map. In the recent literature, semilinear biquivers are treated via semilinear path algebras, semilinear clannish algebras, and semilinear locally gentle algebras; these frameworks connect the combinatorics of quivers with nodal embeddings, strings and bands, and a geometric model based on marked surfaces with seams (Banaian et al., 2024, Bennett-Tennenhaus et al., 2022).

1. Basic notion and representation-theoretic setup

Let $k$ be a field and let $\sigma:k \to k$ be an involutive automorphism, so $\sigma^2=\operatorname{id}$. A map $f:V \to W$ between $k$-vector spaces is $k$0-semilinear if

$k$1

for all $k$2 and $k$3. More generally, if $k$4 is a division ring and $k$5, a map $k$6 between left $k$7-modules is $k$8-semilinear iff $k$9. Equivalently, for a $\sigma_a \in \operatorname{Aut}(K)$0-module $\sigma_a \in \operatorname{Aut}(K)$1, the $\sigma_a \in \operatorname{Aut}(K)$2-twist $\sigma_a \in \operatorname{Aut}(K)$3 is the same abelian group with scalar action $\sigma_a \in \operatorname{Aut}(K)$4, and a $\sigma_a \in \operatorname{Aut}(K)$5-semilinear map $\sigma_a \in \operatorname{Aut}(K)$6 is the same as a $\sigma_a \in \operatorname{Aut}(K)$7-linear map $\sigma_a \in \operatorname{Aut}(K)$8 (Bennett-Tennenhaus et al., 2022).

A semilinear biquiver $\sigma_a \in \operatorname{Aut}(K)$9 consists of a finite set of vertices $K$0, a finite set of arrows $K$1, and a partition

$K$2

With a fixed involution $K$3 on $K$4, a representation of $K$5 assigns a $K$6-vector space $K$7 to each vertex $K$8, a $K$9-linear map $\sigma$0 to each $\sigma$1, and a $\sigma$2-semilinear map $\sigma$3 to each $\sigma$4 (Banaian et al., 2024).

The behavior of paths is governed by the number of semilinear arrows they contain. If $\sigma$5 contains $\sigma$6 semilinear arrows, then the induced action of $\sigma$7 is $\sigma$8-linear. When $\sigma$9 is an involution, $k$0 for even $k$1 and $k$2 for odd $k$3. This parity rule is one of the basic structural features distinguishing semilinear biquivers from ordinary quivers (Banaian et al., 2024).

2. Semilinear path algebras and algebraic presentations

The algebraic realization of a semilinear biquiver uses a semilinear path algebra. For a division ring $k$4 and a function $k$5 with $k$6, the semilinear path algebra $k$7 is the $k$8-ring generated by the trivial paths $k$9 and arrows $\sigma:k \to k$0 subject to

$\sigma:k \to k$1

$\sigma:k \to k$2

for $\sigma:k \to k$3, $\sigma:k \to k$4, and $\sigma:k \to k$5. For a path $\sigma:k \to k$6, one has twisted scalar multiplication $\sigma:k \to k$7, where $\sigma:k \to k$8 is the product of the arrow automorphisms along $\sigma:k \to k$9 (Bennett-Tennenhaus et al., 2022).

In the fixed-involution biquiver setting, one puts $\sigma^2=\operatorname{id}$0 on linear arrows and $\sigma^2=\operatorname{id}$1 on semilinear arrows. If $\sigma^2=\operatorname{id}$2 is generated by quadratic zero-relations, then the associated bound path algebra is

$\sigma^2=\operatorname{id}$3

Representations of the semilinear biquiver with those relations are equivalent to left $\sigma^2=\operatorname{id}$4-modules, because left multiplication by an arrow $\sigma^2=\operatorname{id}$5 satisfies

$\sigma^2=\operatorname{id}$6

so each arrow acts by the appropriate semilinear map (Banaian et al., 2024).

A broader class is given by semilinear clannish algebras. These are quotients of $\sigma^2=\operatorname{id}$7 by a monomial ideal of zero-relations together with quadratic relations at designated special loops. If $\sigma^2=\operatorname{id}$8 is a special loop, its relation is

$\sigma^2=\operatorname{id}$9

where the skew polynomial ring $f:V \to W$0 is defined by $f:V \to W$1. The cited classification theory imposes normality, non-singularity, and semisimplicity conditions on these quadratics; specifically, non-singularity is equivalent to $f:V \to W$2, and semisimplicity means that $f:V \to W$3 is semisimple artinian (Bennett-Tennenhaus et al., 2022).

This algebraic presentation places semilinearity directly into the multiplication rule. As a consequence, semilinear quiver representations are not an auxiliary decoration on an ordinary path algebra; they are modules over a ring in which the scalar action and path action are already intertwined.

3. Locally gentle conditions, semilinear gentle algebras, and nodality

A bound quiver $f:V \to W$4 is locally gentle if two combinatorial constraints hold. First, each vertex is the head of at most two arrows and the tail of at most two arrows. Second, for each arrow $f:V \to W$5, there is at most one admissible and at most one inadmissible path of length two of the form $f:V \to W$6, and at most one admissible and at most one inadmissible path of the form $f:V \to W$7. A gentle pair is a locally gentle pair with only finitely many admissible paths. Semilinear locally gentle algebras are precisely algebras of the form

$f:V \to W$8

with $f:V \to W$9 locally gentle; they form a subclass of semilinear clannish algebras (Banaian et al., 2024).

Within this framework, semilinear gentle algebras admit a nodal description. A finite-dimensional $k$0-ring $k$1 is nodal if it is a subring of a finite-dimensional hereditary $k$2-ring $k$3 such that the embeddings $k$4 and $k$5 compose to the embedding $k$6, one has $k$7, and for every simple left $k$8-module $k$9, the module $k$00 has length at most $k$01 (Banaian et al., 2024).

The relevant hereditary cover is obtained by adapting Zembyk’s excision procedure. A vertex is relational if it occurs in a zero-relation $k$02 with $k$03. Zembyk’s algorithm iteratively splits relational vertices by taking the levee at each such vertex—quadtributaries, tributaries, distributaries, and streams are the local configurations named in the procedure—and deletes the corresponding length-two zero-relations. The resulting quiver is denoted $k$04; the process terminates and is independent of the order of choices (Banaian et al., 2024).

The main theorem in this direction is:

Any finite-dimensional semilinear gentle algebra $k$05 is nodal, connected with $k$06.

The proof strategy recorded in the source has four steps. One constructs an embedding $k$07 compatible with the base ring; one proves that $k$08 is acyclic, hence $k$09 is hereditary; one compares radicals using quotient calculations for length-two relations; and one checks the bound on $k$10 for simples by tracking idempotents under $k$11 (Banaian et al., 2024).

This nodality theorem places semilinear gentle algebras in the same structural orbit as classical nodal algebras, but now with arrowwise semilinearity retained throughout the hereditary cover.

4. Surface models and seams

For a locally gentle pair $k$12, Palu–Pilaud–Plamondon associate a marked surface $k$13 together with dual cellular dissections $k$14. In this model, arcs in the dissection encode the quiver data and faces encode relations. The complement of $k$15 among length-two paths is represented by interchanging the roles of $k$16 and $k$17 (Banaian et al., 2024).

The semilinear refinement introduces seams. One selects a subset $k$18 consisting of arcs dual to relational vertices, cuts the surface along $k$19, and obtains simpler components whose associated quivers are exactly the connected components of $k$20. The seams correspond to the split vertices $k$21 and $k$22 arising in the levee construction. The source states this precisely as:

The quivers associated to the connected components of the split of $k$23 along $k$24 bijectively correspond to the connected components of $k$25 (Banaian et al., 2024).

The module category is then described by curves on the surface. Indecomposable modules correspond to permissible arcs, which model strings, and permissible closed curves, which model bands. A permissible arc $k$26 crosses a sequence of arcs $k$27 in $k$28 and traverses faces $k$29 in $k$30, subject to local non-kissing and non-puncture rules. A permissible closed curve is non-contractible, avoids $k$31-punctures, and obeys analogous local constraints (Banaian et al., 2024).

Semilinearity is encoded by face labels. For a labeled tiling $k$32, where $k$33 assigns automorphisms of $k$34 to type-2 faces, the cumulative semilinearity along a permissible arc is defined inductively by

$k$35

and, for $k$36,

$k$37

if the common endpoint of $k$38 lies to the right of $k$39, while

$k$40

if it lies to the left. For a permissible closed curve, the same rule determines $k$41 at the final step. When the labels lie in $k$42, the cumulative automorphism is $k$43 for the net signed count of semilinear faces (Banaian et al., 2024).

The geometric and algebraic descriptions coincide: the automorphisms governing the right $k$44-action on the string and band bimodules are exactly the semilinearities accumulated by the corresponding arcs and closed curves. This makes seams the geometric locus where semilinear twisting is recorded rather than merely inferred.

5. Strings, bands, and the classification of indecomposables

The classification of finite-dimensional indecomposable modules proceeds through strings and bands. In the semilinear locally gentle setting, a finite admissible word $k$45 gives a string module $k$46, and a doubly-infinite periodic admissible word gives a band module. In the semilinear clannish setting, the word combinatorics is extended to include ordinary direct letters, ordinary inverse letters, and special $k$47-letters attached to special loops; relation-admissibility and end-admissibility exclude forbidden subwords and inappropriate endpoint behavior (Banaian et al., 2024, Bennett-Tennenhaus et al., 2022).

The fundamental classification statement is that, as $k$48 runs through representatives of equivalence classes of strings and bands and $k$49 runs through a complete set of pairwise non-isomorphic finite-dimensional indecomposable modules over the associated parameter ring $k$50 or $k$51, the tensor products

$k$52

exhaust the finite-dimensional indecomposable modules. In the semilinear locally gentle formulation, the parameter ring is $k$53 for strings and $k$54 for bands; in the semilinear clannish formulation, four explicit parameter-ring cases occur (Banaian et al., 2024, Bennett-Tennenhaus et al., 2022).

Case	Parameter ring	Free right-basis
Asymmetric string	$k$55	$k$56
Symmetric string	$k$57	$k$58
Asymmetric band	$k$59	$k$60
Symmetric band	$k$61	$k$62

For strings and bands, the right-module structure is governed by automorphisms $k$63 or $k$64, obtained as products of arrow automorphisms along the canonical walk. In the geometric model, these are identified with the semilinearity accumulated along the corresponding permissible arc or closed curve. In the semilinear clannish theory, this produces skew polynomial, skew Laurent, or free-product parameter rings; for strings these rings are semisimple artinian, while for bands they are hereditary noetherian prime under the stated hypotheses (Banaian et al., 2024, Bennett-Tennenhaus et al., 2022).

This classification retains the canonical strings-and-bands paradigm of special biserial and clannish representation theory, but semilinearity alters the coefficient rings and the right-module transport. A plausible implication is that the discrete-versus-family dichotomy of strings and bands survives semilinear twisting, while the parameter spaces are reorganized by automorphisms such as complex conjugation or Frobenius.

6. Worked configurations and broader connections

A basic example takes $k$65 with $k$66 equal to complex conjugation, vertices $k$67, a linear arrow $k$68, and a $k$69-semilinear arrow $k$70, with $k$71. The resulting algebra $k$72 is finite-dimensional and hereditary because the quiver is acyclic. For the string $k$73, the associated module has basis elements $k$74 with

$k$75

and right action determined by

$k$76

Hence

$k$77

In representation-theoretic terms, $k$78 is linear and $k$79 is conjugate-linear (Banaian et al., 2024).

A band configuration is obtained by adding a linear arrow $k$80 and relations $k$81 so that the only admissible cycle is $k$82 and every length-two subpath in $k$83 is admissible. The corresponding closed curve has face labels $k$84, $k$85, and $k$86, so the semilinearity over one period is $k$87, and the band parameter ring becomes $k$88. The finite-dimensional band modules are then classified by indecomposables over $k$89 (Banaian et al., 2024).

Beyond the locally gentle case, semilinear clannish algebras provide the broader ambient class. They generalize clannish algebras by incorporating semilinear structure through arrowwise automorphisms and special-loop quadratics. The cited literature also records examples such as Dieudonné modules mod $k$90, realized as semilinear string algebras with one vertex and two loops $k$91 and $k$92 satisfying $k$93, where $k$94 is Frobenius and $k$95. Semilinear Kronecker-type configurations and doubles of affine $k$96-graphs with $k$97 or complex conjugation are also mentioned as studied cases (Bennett-Tennenhaus et al., 2022).

These developments place semilinear biquivers at the intersection of quiver representations, skew polynomial methods, string-and-band combinatorics, and surface models. The available theory shows that semilinearity can be encoded simultaneously in the path algebra, in the hereditary cover of a nodal algebra, and in the seams and face labels of a marked surface (Banaian et al., 2024).