Coxeter-Dynkin Algebras of Canonical Type

• Coxeter-Dynkin algebras of canonical type are finite-dimensional algebras defined via species, weights, and bimodules, generalizing quiver path algebras.
• They employ tilting theory through both generalized APR methods and one-point extensions to establish derived equivalences with canonical and squid algebras.
• Their block-matrix structure, encoding division algebras and exceptional points, connects to Saito's marked extended affine root systems in representation and singularity studies.

Coxeter-Dynkin algebras of canonical type are a class of finite-dimensional $k$-algebras that generalize path algebras of quivers in such a way as to provide alternative, often more "canonical," representatives for derived equivalence classes associated with canonical algebras, squid algebras, and related species. These algebras are constructed by augmenting a tame hereditary setting with specified weights and exceptional points, and their module and homological structure is closely controlled by the data of a decorated species. Of particular importance is their intrinsic link to tilting theory, derived equivalence with canonical and squid algebras, and the compatibility of their Grothendieck group data with Saito's classification of marked extended affine root systems.

1. Construction: Species, Weights, and Bimodules

The definition of a Coxeter-Dynkin algebra of canonical type begins with a tame $F$–$G$-bimodule ${}_F M_G$ (with center $k$). One also specifies a sequence of integers $p_1,\ldots,p_t$ (each $p_i \geq 2$) corresponding to the "arms" of the associated species, and for each $i$ an exceptional point (regular-simple representation) $\rho_i$ with underlying modules $U_i$, $V_i$ and endomorphism division algebra $D_i$.

A central ingredient is the canonical $F$–$G$-bimodule morphism

$$\theta_0\colon {}_F M_G \longrightarrow \bigoplus_{i=1}^t \mathrm{Hom}_{D_i}(U_i,V_i),\quad m \mapsto (\rho_i(-\otimes m))_i.$$

The cokernel ${}_F W_G = \mathrm{coker}(\theta_0)$ furnishes an additional "edge" block in the resulting algebra structure.

The Coxeter-Dynkin algebra $B$ is then the matrix algebra whose structure is fully encoded by this data. In block form, for arms of lengths $p_i-1$, and with summands for $U_i^\vee$ and $V_i$, it takes the shape: $$B = \begin{pmatrix} F & 0 & \cdots & U_1^\vee & \cdots & U_t^\vee & W\ \vdots & D_1 & \cdots & D_1 & 0 & 0 & V_1\ \vdots & & \ddots & \vdots & \ddots & & \vdots \ 0 & & & D_t & \cdots & D_t & V_t \ G & & & & & & \ \end{pmatrix}$$ Multiplication is inherited from the species structure and the induced maps from $U_i^\vee \to V_i$ into $W$; the precise formalism is given by the decorated species construction and the projection to $\mathrm{coker}(\theta_0)$ (Perniok, 22 Sep 2025).

The algebra $B$ is finite-dimensional and is constructed to have the same derived category as the associated canonical or squid algebra. This block-matrix presentation not only generalizes path algebras but also systematically encodes "glued" extra algebraic structure reflective of the non-simply laced nature of canonical-type representation theory.

2. Tilting Objects and Derived Equivalence

Coxeter-Dynkin algebras of canonical type are derived-equivalent to squid algebras and hence to canonical algebras, with explicit tilting objects providing the connection. Two principal constructions yield tilting modules or complexes whose endomorphism algebra is the Coxeter-Dynkin algebra $B$:

(a) Generalized APR (Auslander-Platzeck-Reiten) Tilting

Given the squid algebra $A$, let $S = \mathrm{top}(e_F A)$ be the simple module at the $F$-vertex. If $S$ satisfies

• non-injectivity,
• $\mathrm{Hom}_A({}_A A, S) = 0$, and
• $Ae_F$ is not a direct summand of the injective hull of $Ae_F/S$,

then the tilting module

$$T = e_G A \oplus \bigoplus_{i=1}^t \bigoplus_{j=1}^{p_i - 1} e_i(j) A \oplus X, \quad X = \tau^{-1}(\mathrm{top}(e_F A))$$

is a classical tilting object in $\mathrm{mod}\text{-}A$. One constructs a short exact sequence

$$0 \longrightarrow e_F A \longrightarrow \bigoplus_{i=1}^t (U_i^\vee \otimes_{D_i} e_i(1)A) \longrightarrow X \longrightarrow 0$$

and, via diagram chasing and homological computations, shows

$\mathrm{End}_A(T) \cong B.$

Thus, the derived functor $\mathrm{RHom}_A(T,-)$ gives a triangle equivalence $$D^b(\mathrm{mod}\text{-}A) \cong D^b(\mathrm{mod}\text{-}B)$$ (Perniok, 22 Sep 2025).

(b) Tilting via One-Point Extensions and Reflection Functors

Alternatively, expressing $A$ as a one-point extension of a hereditary algebra $A_0$, and identifying a hereditary subalgebra $B_0$ inside $B$, the theory of reflection functors applies. Under suitable compatibility between the extension bimodule $N$ and the derived equivalence $$D^b(\mathrm{mod}\text{-}A_0) \to D^b(\mathrm{mod}\text{-}B_0)$$, a tilting complex over $A$ is formed as

$$T' = (A e_G)[0] \oplus \left(\bigoplus_{i,j} A e_i(j)[0]\right) \oplus (\mathrm{top}(e_F A))[1]$$

with

$$\mathrm{End}_{D^b(\mathrm{mod}\text{-}A)}(T') \cong B.$$

This construction uses adjunctions and extends derived equivalences between hereditary subalgebras to the full setting, encapsulating the effect of the additional arms and the one-point extension (Perniok, 22 Sep 2025).

3. Block Structure and Connection to Species

The algebra $B$ has a block-matrix structure that encodes the division algebras $D_i$ along the arms, the exceptional points $\rho_i$, the dual modules $U_i^\vee$, $V_i$, and the crucial cokernel $W = \mathrm{coker}(\theta_0)$. This structure generalizes the representation of a canonical algebra as a path algebra modulo relations, incorporating explicitly the effect of "folding," arm lengths, and the extra module-theoretic data from the regular simples and their endomorphism algebras. The choice and arrangement of blocks, together with the connecting maps, mirror the data of marked diagrams appearing in Saito's classification of extended affine root systems (see Section 5 below).

4. Compatibility with Saito's Classification of Marked Extended Affine Root Systems

A profound aspect is the link between the Grothendieck group $K_0(\mathrm{mod}\text{-}B)$ with its Euler bilinear form and canonical lattices associated with Saito's marked extended affine root systems. Specifically, for $B$ arising from parameters $(p_1,\ldots,p_t)$ and corresponding division algebra data $(d_1,\ldots,d_t)$, one defines a canonical lattice $V$ with basis

$$a,\, w,\, s_i^{(j)} \quad (i = 1,\ldots,t;\, 0 \leq j \leq p_i-2)$$

and bilinear form specified by

$$\langle a,a\rangle = \kappa, \;\; \langle a,s_i^{(0)}\rangle = \kappa\epsilon f_i,\; \langle a,w\rangle = \kappa\epsilon, \ldots$$

where the symbol $$\sigma = (p_1p_2\cdots p_t|d_1\cdots d_t, \epsilon, f_1\cdots f_t)$$ fully encodes the combinatorics. The established isomorphism between this lattice and $(K_0(\mathrm{mod}\text{-}B), q_B)$ (the Euler form) shows that the canonical structure of $B$ reproduces the invariants used by Saito to classify extended affine root systems. This correspondence also clarifies the nature of the Weyl group actions and the representation-theoretic moduli arising from these algebras (Perniok, 22 Sep 2025).

5. Canonical Type, Derived Equivalence, and Classification

Coxeter-Dynkin algebras of canonical type form a derived equivalence class together with squid algebras and canonical algebras for the same data $(p_1, \ldots, p_t)$ and module-theoretic information. The explicit tilting constructions above provide the mechanism for this equivalence. This class plays a fundamental role in the paper of:

• fractional Calabi–Yau properties,
• periodicity phenomena,
• and connections to weighted projective lines, as well as their moduli spaces and links to singularity theory.

These algebras encapsulate and generalize the earlier concepts of canonical algebras, incorporating additional symmetry and rigidity via their species-inspired block structure, and making transparent their relationship to the geometry of marked Dynkin and extended affine diagrams. Their construction realizes, in algebraic terms, the combinatorial and geometric features emphasized in Saito’s extended affine root system theory.

6. Significance for Representation Theory and Singularity Theory

The construction and properties of Coxeter-Dynkin algebras of canonical type provide a bridge between representation-theoretic concepts (such as tilting theory, derived equivalence, and module categories of canonical/squid algebras) and the infinite-dimensional Lie theoretic and singularity-theoretic structures classified by marked Dynkin diagrams. The connection to Saito's work offers robust tools for understanding the moduli of representations, automorphism groups, and the spectrum of the Coxeter (Auslander–Reiten) transformation in the derived context. The canonical, functorially defined nature of these algebras makes them central objects in modern approaches to the classification and paper of derived categories associated with weighted projective lines and surface singularities (Perniok, 22 Sep 2025).