Completing the classification of Erdmann's algebras of dihedral, semi-dihedral and quaternion type up to stable equivalence of Morita type: the surviving scalar-parameter problems

Erdmann's algebras of dihedral, semi-dihedral and quaternion type contain, up to Morita equivalence, all blocks of finite group algebras of tame representation type. Zhou and Zimmermann (2011) classified these algebras up to stable equivalence of Morita type up to explicit scalar-parameter identification problems, recorded as open in their paper. Later literature resolved most of them: Taillefer (2019) proved that the local dihedral algebras $D(1\mathcal{A})^k_2(0)$ and $D(1\mathcal{A})^k_2(1)$ are not stably equivalent of Morita type and settled the semi-dihedral scalar question in characteristic 2 for all parameters, while Zimmermann (2018) showed that over algebraically closed fields of characteristic $\ne2$ the quaternion two-simple scalar parameters collapse up to isomorphism, except at the corner $(k,s)=(1,3)$. What survives is precise: in characteristic 2 it is open whether $Q(2\mathcal{B})^{k,s}_1(a,1)$ and $Q(2\mathcal{B})^{k,s}_1(a',1)$ with $a\neq a'$ are stably equivalent of Morita type — open even at the derived-equivalence level — together with the corner $(k,s)=(1,3)$ with $c=0$ (open in every characteristic) and the within-family questions for the local algebras $SD(1\mathcal{A})^k_2(c,d)$ and $Q(1\mathcal{A})^k_2(c,d)$. The problem is to settle these remaining identifications, in either direction, thereby completing the classification of Zhou and Zimmermann without scalar caveats.

Background

Let $K$ be an algebraically closed field. In the modular representation theory of a finite group $G$ over a field of characteristic $p>0$, the group algebra decomposes as a direct product of indecomposable two-sided ideals called blocks. Each block has a defect group, a $p$-subgroup of $G$ measuring its homological size, and a block is of tame representation type when its indecomposable modules, though infinite in number, occur in finitely many one-parameter families. The tame blocks arise only in characteristic 2, with dihedral, semi-dihedral or generalised quaternion defect groups, and Erdmann classified them by giving an explicit list, up to Morita equivalence (equivalence of module categories), of 'algebras of dihedral, semi-dihedral and quaternion type' defined by quiver and relations in her Lecture Notes volume; the list is infinite only through integer parameters and a few scalar parameters.

All algebras in the list are self-injective (projective modules coincide with injective ones) and symmetric. For self-injective algebras a natural coarsening of Morita equivalence is an equivalence of stable module categories: the stable category is the quotient of the module category by the ideal of morphisms that factor through a projective module. Broué isolated the well-behaved subclass of stable equivalences of Morita type: algebras $A$ and $B$ are stably equivalent of Morita type if there is an $A$–$B$-bimodule $M$ and a $B$–$A$-bimodule $N$, both projective on either side, such that $M\otimes_B N\cong A\oplus P$ and $N\otimes_A M\cong B\oplus Q$ for projective bimodules $P$ and $Q$. A derived equivalence (an equivalence of unbounded derived categories, which for these algebras is generated by a tilting complex in Rickard's sense) between self-injective algebras induces a stable equivalence of Morita type, so the stable classification is a coarsening of the derived one; Holm determined the derived-equivalence classification of Erdmann's list, and Zhou and Zimmermann (2011) determined the stable-equivalence-of-Morita-type classification, using invariants preserved by such equivalences: the stable centre (the centre modulo the projective centre, i.e. modulo the bimodule endomorphisms factoring through a projective bimodule); the absolute value of the determinant of the Cartan matrix (the matrix recording the composition-factor multiplicities of the indecomposable projective modules); the stable Grothendieck group; the Külshammer ideals — a descending sequence of ideals $T_n(A)^\perp$ of the centre of a symmetric algebra defined from its Frobenius form (for example $T_1(A)$ consists of the elements $x$ with $x^p$ a sum of commutators, and $T_1(A)^\perp$ generalises the classical Reynolds ideal); and the Hochschild cohomology $HH^\ast(A)=\mathrm{Ext}^\ast_{A^e}(A,A)$, whose degree-one part carries a Lie bracket that is preserved, for symmetric algebras, under stable equivalences of Morita type by results of König–Liu–Zhou. Their classification coincided with the derived classification except for explicitly recorded scalar-parameter identification problems (Remarks 3.4, 3.6, 4.2 and 7.2 of that paper).

The algebras entering the surviving questions are concrete. Writing $1\mathcal{A}$ for the one-vertex quiver with two loops $x,y$ and $2\mathcal{B}$ for the two-vertex quiver with loops $\alpha,\eta$ and a $2$-cycle $\beta,\gamma$ between the vertices, the two-simple quaternion algebra $Q(2\mathcal{B})^{k,s}_1(a,c)$ with $k\ge1$, $s\ge3$, $a\in K^\times$ (with $a\neq1$ when $k+s=4$), $c\in K$ is the quotient of the path algebra by the relations $\gamma\beta=\eta^{s-1}$, $\beta\eta=(\alpha\beta\gamma)^{k-1}\alpha\beta$, $\eta\gamma=(\gamma\alpha\beta)^{k-1}\gamma\alpha$, $\alpha^2=a(\beta\gamma\alpha)^{k-1}\beta\gamma+c(\beta\gamma\alpha)^k$, $\alpha^2\beta=0$, $\gamma\alpha^2=0$. The local (one-simple) quaternion algebra $Q(1\mathcal{A})^k_2(c,d)$ ($k\ge2$, $(c,d)\ne(0,0)$) has relations $x^2-(yx)^{k-1}y-c(xy)^k$, $y^2-(xy)^{k-1}x-d(xy)^k$, $(xy)^k-(yx)^k$, $(xy)^kx$, $(yx)^ky$, and the local semi-dihedral algebra $SD(1\mathcal{A})^k_2(c,d)$ ($k\ge2$, $(c,d)\ne(0,0)$) has relations $x^2-(yx)^{k-1}y+c(xy)^k$, $y^2-d(xy)^k$, $(xy)^k-(yx)^k$, $(xy)^kx$.

The two-simple semi-dihedral algebras, with $k\ge2$, $s\ge1$, scalar $c\in\{0,1\}$ (and $k+s\ge4$ in the second family), are $SD(2\mathcal{B})^{k,s}_1(c)$, with relations $\gamma\beta=0$, $\eta\gamma=0$, $\beta\eta=0$, $\alpha^2=(\beta\gamma\alpha)^{k-1}\beta\gamma+c(\alpha\beta\gamma)^k$, $\eta^s=(\gamma\alpha\beta)^k$, $(\alpha\beta\gamma)^k=(\beta\gamma\alpha)^k$, and $SD(2\mathcal{B})^{k,s}_2(c)$, with relations $\beta\eta=(\alpha\beta\gamma)^{k-1}\alpha\beta$, $\eta\gamma=(\gamma\alpha\beta)^{k-1}\gamma\alpha$, $\gamma\beta=\eta^{s-1}$, $\alpha^2=c(\alpha\beta\gamma)^k$, $\beta\eta^2=0$, $\eta^2\gamma=0$. For blocks with generalised quaternion defect groups of order $2^n$ the relevant instances of the two-simple quaternion family are $(k,s)=(2,2^{n-2})$.

Holm and Zimmermann (2008) used their generalised Reynolds ideals to separate the two semi-dihedral scalars in certain parity cases. Zimmermann (2018) computed the Külshammer-ideal quotient $Z(A)/T_1(A)^\perp$ for all $A=Q(2\mathcal{B})^{k,s}_1(a,c)$: over algebraically closed fields of characteristic $\ne2$ all these algebras with $(k,s)\ne(1,3)$ are isomorphic to $Q(2\mathcal{B})^{k,s}_1(1,0)$, so no scalar problem survives there except at the corner $(k,s)=(1,3)$; in characteristic 2 the quotient, which is invariant under stable equivalences of Morita type between symmetric algebras, distinguishes exactly whether $c=0$, and rescaling reduces the family to the normal forms $(1,0)$ and $(a,1)$ with $a\in K^\times$ (the reduction of $Q(2\mathcal{B})^{k,s}_1(a,0)$ to $(1,0)$ again holding only for $(k,s)\ne(1,3)$). Taillefer (2019) used dimensions and Lie algebra structures of $HH^1$ to resolve the dihedral question and the whole characteristic-2 semi-dihedral scalar question, and to separate $Q(2\mathcal{B})^{k,s}_1(1,c)$ from $Q(2\mathcal{B})^{k,s}_1(1,0)$ for $k+s>4$. What no invariant developed so far detects is the parameter $a$ itself in characteristic 2: all computed invariants depend only on whether $c$ vanishes. Closing this gap, together with the corner and local-family analogues, is the surviving core of the classification problem. The number-of-simples invariance question for stable equivalences of Morita type (which Zhou–Zimmermann prove for these classes, following their formulation of the Auslander–Reiten conjecture) is settled for the three type-families; the conjecture in general is a separate famous problem and is not part of this question.

Problem Statement

Complete the classification of Erdmann's algebras of dihedral, semi-dihedral and quaternion type — hence of all tame blocks of finite group algebras — up to stable equivalence of Morita type, by settling the scalar-parameter identification questions that remain open after Zhou–Zimmermann (2011), Holm–Zimmermann (2008), Zimmermann (2018) and Taillefer (2019). Concretely, over an algebraically closed field $K$:

(1) (Quaternion two-simple family, characteristic 2.) Let $k\ge1$, $s\ge3$ (with $a,a'\ne1$ when $k+s=4$). Determine for which $a,a'\in K^\times$ the algebras $Q(2\mathcal{B})^{k,s}_1(a,1)$ and $Q(2\mathcal{B})^{k,s}_1(a',1)$ are stably equivalent of Morita type; equivalently, decide whether the scalar $a$ is an invariant of the stable equivalence class of Morita type. This is open even at the derived-equivalence level. The block instances are $(k,s)=(2,2^{n-2})$.

(2) (Quaternion two-simple family, corner $(k,s)=(1,3)$, any characteristic.) Decide whether $Q(2\mathcal{B})^{1,3}_1(a,0)$ and $Q(2\mathcal{B})^{1,3}_1(a',0)$, with $a\neq a'$ in $K\setminus\{0,1\}$, are stably equivalent of Morita type: the isomorphism reduction of $Q(2\mathcal{B})^{k,s}_1(a,0)$ to the normal form $(1,0)$ fails at this corner both in characteristic $\ne2$ and in characteristic 2 (Zimmermann 2018).

(3) (Local families.) (a) In characteristic 2, decide whether $SD(1\mathcal{A})^k_2(c,1)$ and $SD(1\mathcal{A})^k_2(c',1)$ with $k\ge2$ and distinct nonzero $c,c'$ are stably equivalent of Morita type. (b) In characteristic 2, decide whether $Q(1\mathcal{A})^k_2(0,d)$ and $Q(1\mathcal{A})^k_2(0,d')$ with $d\neq d'$ nonzero, and whether $Q(1\mathcal{A})^k_2(c,d)$ and $Q(1\mathcal{A})^k_2(c',d')$ with $cd\ne0\ne c'd'$ and $\{c,d\}\ne\{c',d'\}$, are stably equivalent of Morita type. (c) In characteristic 2 with $ks$ odd and $c=c'=0$, decide whether the cross-family pair $SD(2\mathcal{B})^{k,s}_1(0)$ and $SD(2\mathcal{B})^{k,s}_2(0)$ is stably equivalent of Morita type.

Settling these questions, together with the published results cited above, would complete the classification of the three type-families up to stable equivalence of Morita type and remove the scalar caveats from Theorem 7.1 of Zhou–Zimmermann.

References

1. Guodong Zhou and Alexander Zimmermann, "Classifying tame blocks and related algebras up to stable equivalences of Morita type", Journal of Pure and Applied Algebra 215 (2011), no. 12, 2969–2986. DOI: 10.1016/j.jpaa.2011.04.017; arXiv:1005.0222. https://doi.org/10.1016/j.jpaa.2011.04.017
2. Karin Erdmann, "Blocks of Tame Representation Type and Related Algebras", Lecture Notes in Mathematics 1428, Springer-Verlag, Berlin, 1990. DOI: 10.1007/BFb0084003; ISBN 9783540527091. https://doi.org/10.1007/BFb0084003
3. Thorsten Holm and Alexander Zimmermann, "Generalized Reynolds ideals and derived equivalences for algebras of dihedral and semidihedral type", Journal of Algebra 320 (2008), no. 9, 3425–3437. DOI: 10.1016/j.jalgebra.2008.07.026. https://doi.org/10.1016/j.jalgebra.2008.07.026
4. Alexander Zimmermann, "Külshammer ideals of algebras of quaternion type", Journal of Algebra and Its Applications 17 (2018), no. 8, 1850157. DOI: 10.1142/S0219498818501578; arXiv:1605.07757. https://arxiv.org/abs/1605.07757
5. Rachel Taillefer, "First Hochschild cohomology group and stable equivalence classification of Morita type of some tame symmetric algebras", Homology, Homotopy and Applications 21 (2019), no. 1, 19–48. DOI: 10.4310/hha.2019.v21.n1.a2; arXiv:1706.10044. https://arxiv.org/abs/1706.10044
6. Alex Dugas and Bill Trok, "The stable Picard group of a local dihedral algebra", Communications in Algebra 46 (2018), no. 6, 2428–2439. DOI: 10.1080/00927872.2017.1383998. https://doi.org/10.1080/00927872.2017.1383998
7. Benjamin Briggs and Lleonard Rubio y Degrassi, "Stable invariance of the restricted Lie algebra structure of Hochschild cohomology", Pacific Journal of Mathematics 321 (2022), no. 1, 45–71. DOI: 10.2140/pjm.2022.321.45; arXiv:2006.13871. https://arxiv.org/abs/2006.13871
8. Lleonard Rubio y Degrassi, Sibylle Schroll and Andrea Solotar, "The first Hochschild cohomology as a Lie algebra", Quaestiones Mathematicae 46 (2023), no. 9, 1955–1980 (online 2022). DOI: 10.2989/16073606.2022.2115424; arXiv:1903.12145. https://doi.org/10.2989/16073606.2022.2115424
9. Steffen König, Yuming Liu and Guodong Zhou, "Transfer maps in Hochschild (co)homology and applications to stable and derived invariants and to the Auslander–Reiten conjecture", Transactions of the American Mathematical Society 364 (2012), no. 1, 195–232. DOI: 10.1090/s0002-9947-2011-05358-4. https://doi.org/10.1090/s0002-9947-2011-05358-4