Isomorphisms of skew-zigzag algebras up to graph automorphisms #
Two skew-zigzag parameters c and c' of a finite simple graph G are gauge equivalent,
equivalently have the same class in H¹(G, kˣ), exactly when their relation quotients are
isomorphic by an isomorphism fixing every vertex idempotent
(TauCeti.SkewZigzagParameter.cohomologyClass_eq_iff_exists_vertexFixing_algEquiv). This file
drops the requirement that the vertex idempotents be fixed: isomorphisms which only permute them are
classified by the gauge classes modulo the automorphisms of G, that is by the orbits of
Aut(G) on H¹(G, kˣ).
An algebra isomorphism φ : Z_k(G, c) ≃ Z_k(G, c') with φ(e_i) = e_{σ i} for a permutation σ of
the vertices maps the corner e_j Z e_i onto the corner e_{σ j} Z e_{σ i}. For distinct vertices
this corner is nonzero exactly when i and j are adjacent: it contains the arrow of an edge, and
it vanishes otherwise. So σ is a graph automorphism, and composing φ with the inverse of the
relabelling isomorphism Z_k(G, c) ≃ Z_k(G, c.relabel σ) gives a vertex-fixing isomorphism
Z_k(G, c.relabel σ) ≃ Z_k(G, c'), which forces c.relabel σ and c' to be gauge equivalent.
When the coefficient ring has only trivial idempotents, an isomorphism which maps every vertex
idempotent into their span automatically permutes them. The images φ(e_i) are nonzero orthogonal
idempotents in that span. Their vertex coordinates lie in the product algebra V → k, where an
idempotent is the indicator function of a set of vertices. The sets attached to the e_i are then
nonempty, pairwise disjoint and as many as the vertices, hence singletons.
These results classify vertex-permuting and vertex-span-preserving algebra isomorphisms; they do
not assert a classification of arbitrary graded isomorphisms.
Main results #
AlgEquiv.adj_iff_of_vertexPermuting: the permutation of the vertices induced by an isomorphism of skew-zigzag relation quotients is a graph automorphism.AlgEquiv.exists_iso_isGaugeEquivalent_relabel_of_vertexPermuting: an isomorphism permuting the vertex idempotents byσmakesσa graph automorphism along which the relabelled parameter is gauge equivalent to the target parameter.AlgEquiv.exists_equiv_apply_vertexIdempotent_of_mem_span: over a ring with only trivial idempotents, an isomorphism mapping the vertex idempotents into their span permutes them.TauCeti.SkewZigzagParameter.exists_isGaugeEquivalent_relabel_iff_exists_vertexPermuting_algEquivandexists_isGaugeEquivalent_relabel_iff_exists_algEquiv_mem_span: the relation quotients are isomorphic by an isomorphism permuting the vertex idempotents, or over a ring with only trivial idempotents by one preserving their span, exactly when the parameters are gauge equivalent up to a graph automorphism.exists_isGaugeEquivalent_relabel_iff_exists_algEquiv_vertexImages_mem_grade_zero, in the same namespace: the span criterion expressed using the quotient's degree-zero piece.exists_firstCohomologyRelabel_eq_iff_exists_vertexPermuting_algEquivandexists_firstCohomologyRelabel_eq_iff_exists_algEquiv_mem_span, in the same namespace: the same classification by the orbits of the graph automorphisms onH¹(G, kˣ).
References #
C. Couture, Skew-Zigzag Algebras, Section 4, Theorem 4.12, https://arxiv.org/abs/1509.08405,
which identifies the graded isomorphism classes of skew-zigzag algebras with the orbits of the
graph automorphisms on the graph cohomology H¹(G, kˣ).
Vertex coefficients #
Isomorphisms permuting the vertex idempotents #
The vertex permutation of an isomorphism is a graph automorphism. If an algebra isomorphism
between skew-zigzag relation quotients sends each vertex idempotent e_i to e_{σ i}, then σ
preserves and reflects adjacency.
An isomorphism permuting the vertex idempotents is a relabelling up to gauge. If an algebra
isomorphism between the skew-zigzag relation quotients of c and c' sends each vertex idempotent
e_i to e_{σ i}, then σ is a graph automorphism, and relabelling c along it gives a parameter
gauge equivalent to c'.
An isomorphism preserving the span of the vertex idempotents permutes them. If the
coefficient ring has only trivial idempotents and an algebra isomorphism between skew-zigzag
relation quotients maps each vertex idempotent into
the span of the vertex idempotents, then it sends each e_i to e_{σ i} for a permutation σ of
the vertices.
Gauge classes up to graph automorphisms are the isomorphism classes permuting the vertex
idempotents. Over a nontrivial commutative ring, a relabelling of c along a graph automorphism
is gauge equivalent to c' exactly when the relation quotients of c and c' are isomorphic by an
algebra isomorphism sending each vertex idempotent e_i to e_{σ i} for a permutation σ.
Gauge classes up to graph automorphisms are the isomorphism classes preserving vertex span.
If the coefficient ring has only trivial idempotents, a relabelling of c along a graph
automorphism is gauge equivalent to c' exactly when the relation quotients of c and c' are
isomorphic by an algebra isomorphism mapping every
vertex idempotent into the span of the vertex idempotents.
Gauge equivalence up to graph automorphisms is equivalent to the existence of an algebra isomorphism taking each vertex idempotent into the degree-zero piece of the target skew-zigzag quotient. This reformulates the vertex-span criterion using the induced grading.
Couture's classification up to graph automorphisms, permuting form. Over a nontrivial
commutative ring, the cohomology classes of c and c' lie in one orbit of the graph automorphisms
on H¹(G, kˣ) exactly when the relation quotients of c and c' are isomorphic by an algebra
isomorphism permuting the vertex idempotents.
Couture's classification up to graph automorphisms. If the coefficient ring has only
trivial idempotents, the cohomology classes of c and c' lie in one orbit of the graph
automorphisms on H¹(G, kˣ) exactly when the
relation quotients of c and c' are isomorphic by an algebra isomorphism mapping every vertex
idempotent into the span of the vertex idempotents.