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Path Subgroupoids of Weyl groupoids,
Rainbow Boomerang Graphs
and Verma modules for
Nichols algebras of diagonal type

Shunsuke Hirota
Abstract

We extend the exchange property of Weyl groupoids in the sense of Heckenberger and Yamane to a newly introduced class called path subgroupoids of Weyl groupoids. This extension allows us, in particular, to interpret the fact that odd reflections of basic Lie superalgebras satisfy the exchange property as a consequence of the combinatorics of the Weyl groupoid.

We also establish an analogous statement within the same framework for Nichols algebras of diagonal type, generalizing our previous results on homomorphisms between Verma modules sharing same characters but associated with different Borel subalgebras in the case of basic Lie superalgebras.

1 Introduction

1.1 Background and motivation

The Weyl groupoids in the sense of Heckenberger and Yamane [20, 19, 11], introduced with the classification of Nichols algebras of diagonal type [18, 1] in mind, are recognized as a good generalization of Conway-Coxeter’s frieze patterns [19, 12, 10] (Example 2.11) and root systems of basic Lie superalgebras [6, 19, 5] (Remark 3.12).

Importantly, they inherit the favorable properties of Weyl groups. For instance, they retain properties such as the exchange property, the existence and uniqueness of the longest element, and the existence of Hamiltonian cycles in their Cayley graphs [22, 33].

On the other hand, as observed in [15, Proposition 3.5.3], even when considering odd reflections of basic Lie superalgebras [30], they still satisfy a similar exchange property. In [21], the exchange property was utilized to study homomorphisms between Verma modules.

The goal of this work is to unify the exchange property of Weyl groupoids with that of odd reflections. Furthermore, through this approach, we explain analogous results of [21] for Verma modules of Nichols algebras of diagonal type, showing that they can be understood in the same framework as in the case of basic Lie superalgebras. This unified perspective will also be useful for future studies of other algebraic structures that can be understood within the Weyl groupoid framework [4].

The root systems of basic Lie superalgebras are known to serve as fundamental concrete examples of Weyl groupoids, particularly through the classification of Weyl proupoids [13]. Technically, the Weyl groupoid of a basic Lie superalgebra is better described by a more informative object, namely a non-simply connected Weyl groupoid [19]. For example, the simply connected Weyl groupoid of 𝔤​𝔩​(m|n)\mathfrak{gl}(m|n) depends only on m+nm+n. However, the formulation given in [19] posed certain difficulties when applied to actual studies of basic Lie superalgebras. Extending the formulation of [19], we introduce supplementary concepts to address these issues.

In [21], the exchange property was formulated in terms of edge-colored graphs by introducing the class of rainbow boomerang graphs (Proposition 2.26). We utilize this framework to achieve the aforementioned unification.

This paper is structured as follows.

In Section 2.1, following Heckenberger and Yamane, we discuss the foundations of Weyl groupoids, paying particular attention to simply connectedness due to its relevance to our setting.

In Section 2.2, we introduce the class of path subgroupoids and explain, using the rainbow boomerang graph, that the exchange property holds within this class.

In Section 3, we explain that the groupoid generated by the odd reflections of basic Lie superalgebras forms a path subgroupoid by referencing the convenient formulation of [6].

In Section 4, we study the composition of homomorphisms between Verma modules with the same character for Nichols algebras of diagonal type. We demonstrate that, in this setting, the structure closely resembles the case of basic Lie superalgebras [21].

1.2 Acknowledgements

I would like to express my heartfelt gratitude to my supervisor, Syu Kato, for his patient and extensive guidance, as well as for his helpful suggestions and constructive feedback. The author is also grateful to Istvan Heckenberger for his valuable discussions, comments and helpful advices. I would also like to sincerely thank Yoshiyuki Koga and Hiroyuki Yamane for engaging in insightful discussions on related topics. The author would like to thank the Kumano Dormitory community at Kyoto University for their generous financial and living assistance.

2 Weyl groupoids

2.1 Basics and examples

See [19, Section 9,10] for basic material about Weyl groupoids.

Definition 2.1.

[19] An edge-colored graph GG with vertex set VV is called a semi Cartan graph (also known as a Cartan scheme) if it is equipped with:

  • •

    a non-empty finite set II of colors,

  • •

    and a label set {Ax}x∈V\{A^{x}\}_{x\in V}, where each AxA^{x} is a generalized Cartan matrix of size #​I×#​I\#I\times\#I (in the sense of [24]),

satisfying the following conditions:

  1. (CG1)

    GG is properly colored (i.e., edges emanating from the same vertex have distinct colors) and #​I\#I-regular (i.e., each vertex is incident to exactly #​I\#I edges).

  2. (CG2)

    If two vertices xx and yy are connected by an edge of color ii, then the ii-th row of AxA^{x} equals the ii-th row of AyA^{y}.

The underlying edge-colored graph of a semi Cartan graph GG is called the exchange graph and is denoted by E​(G)E(G). When illustrating GG, we omit loops for simplicity, thanks to (CG1).

The size of II is called the rank of GG.

For x∈Vx\in V, define ri​x∈Vr_{i}x\in V as the vertex connected to xx by an edge of color ii. Then, rir_{i} is an involution on VV.

For each x∈Vx\in V, consider a copy (ℤI)x(\mathbb{Z}^{I})^{x} of ℤI\mathbb{Z}^{I} associated with xx. The standard basis of (ℤI)x(\mathbb{Z}^{I})^{x} is denoted by {αix}i∈I\{\alpha_{i}^{x}\}_{i\in I}.

The standard basis of ℤI\mathbb{Z}^{I} is also denoted by {αi}i∈I\{\alpha_{i}\}_{i\in I}. We define a standard isomorphism φx:ℤI→ℤI\varphi^{x}:\mathbb{Z}^{I}\to\mathbb{Z}^{I} for each xx, which maps αix\alpha_{i}^{x} to αi\alpha_{i} for i∈Ii\in I.

For each i∈Ii\in I and x∈Vx\in V, define six∈Homℤ⁡((ℤI)x,(ℤI)ri​x)s_{i}^{x}\in\operatorname{Hom}_{\mathbb{Z}}((\mathbb{Z}^{I})^{x},(\mathbb{Z}^{I})^{r_{i}x}) by the mapping:

αjx↦αjri​x−ai​jx​αiri​x,for ​j∈I.\alpha_{j}^{x}\mapsto\alpha_{j}^{r_{i}x}-a_{ij}^{x}\alpha_{i}^{r_{i}x},\quad\text{for }j\in I.

When the context is clear, the subscript xx in sixs_{i}^{x} may be omitted. Additionally, it is sometimes expressed as a composition with the identity map idx\operatorname{id}_{x} at a vertex xx to emphasize the starting or ending points of the mapping.

Remark 2.2.

Our (CG1) is equivalent to (CG1) in [19].

Definition 2.3 (Semi Weyl Groupoid).

The semi Weyl groupoid W​(G)W(G) of GG is the category with objects VV, where the morphisms from xx to rit​⋯​ri1​xr_{i_{t}}\cdots r_{i_{1}}x are elements of Homℤ⁡((ℤI)x,(ℤI)rit​⋯​ri1​x)\operatorname{Hom}_{\mathbb{Z}}((\mathbb{Z}^{I})^{x},(\mathbb{Z}^{I})^{r_{i_{t}}\cdots r_{i_{1}}x}) of the form

sitrit−1​⋯​ri1​x​⋯​si2ri1​x​si1x.s_{i_{t}}^{r_{i_{t-1}}\cdots r_{i_{1}}x}\cdots s_{i_{2}}^{r_{i_{1}}x}s_{i_{1}}^{x}.

We denote the set of such morphisms as HomW​(G)⁡(x,rit​⋯​ri1​x)\operatorname{Hom}_{W(G)}(x,r_{i_{t}}\cdots r_{i_{1}}x). The composition of morphisms is defined by the natural composition of these maps.

By the above construction, the semi Weyl groupoid indeed becomes a groupoid due to (CG2). For a general connected groupoid WW, note that the group structure of AutW⁡(x)=HomW⁡(x,x)\operatorname{Aut}_{W}(x)=\operatorname{Hom}_{W}(x,x) does not depend on the choice of xx. An element of HomW​(G)⁡(x,y)\operatorname{Hom}_{W(G)}(x,y) can be regarded as an element of Autℤ⁡(ℤI)\operatorname{Aut}_{\mathbb{Z}}(\mathbb{Z}^{I}) via φx\varphi^{x} and φy\varphi^{y}.

Definition 2.4 (Real Roots).

[19] For each x∈Vx\in V, define the set of real roots RxR^{x} as subsets of (ℤI)x(\mathbb{Z}^{I})^{x} of the form:

Rx:={w​αiy∣w∈HomW​(G)⁡(y,x),y∈V,i∈I}.R^{x}:=\{w\alpha_{i}^{y}\mid w\in\operatorname{Hom}_{W(G)}(y,x),\,y\in V,\,i\in I\}.

Let the set of positive real roots be defined as:

Rx+:=(Rx∩(∑i∈Iℤ≥0​αix)).R^{x+}:=\bigl{(}R^{x}\cap\bigl{(}\sum_{i\in I}\mathbb{Z}_{\geq 0}\alpha_{i}^{x}\bigr{)}\bigr{)}.

A semi Cartan graph is said to be finite when #​Rx<∞\#R^{x}<\infty.

Definition 2.5.

[19] A semi Cartan graph GG is called a Cartan graph if it satisfies the following conditions:

  1. (CG3)

    For all x∈Vx\in V, Rx=Rx+∪(−Rx+)R^{x}=R^{x+}\cup(-R^{x+}).

  2. (CG4)

    If w∈HomW⁡(G)​(x,y)w\in\operatorname{Hom}_{W}(G)(x,y) and w​αiy∈Rx+w\alpha_{i}^{y}\in R^{x+} for all i∈Ii\in I, then w=idxw=\operatorname{id}_{x}. In particular, we have x=yx=y.

A semi Weyl groupoid arising from a (finite) Cartan graph is called a (finite) Weyl groupoid.

Remark 2.6.

Our (CG4) is equivalent to (CG4) of [19, Remark 1.6] by [5] , [[19], Corollary 9.3.8] and Lemma 2.8.

Remark 2.7.

In existing literature, such as [19], groupoids arising from semi Cartan graphs are also referred to as Weyl groupoids. On the other hand, there is a convention of using the term Weyl groupoid where generalized root system would be more appropriate. Indeed, as in the case of classical BC types, groupoids associated with distinct Cartan graphs can be isomorphic. While adhering to this convention, we distinguish groupoids associated with semi Cartan graphs, which are not Cartan graphs, by calling them semi Weyl groupoids to avoid confusion.

Lemma 2.8 ([19], Lemma 9.1.19).

Let GG be a semi Cartan graph satisfying (CG3). Then sixs_{i}^{x} provides a bijection between the sets

(Rx∖{−αix})and(Rri​x∖{−αiri​x}).\big{(}R^{x}\setminus\{-\alpha_{i}^{x}\}\big{)}\quad\text{and}\quad\big{(}R^{r_{i}x}\setminus\{-\alpha_{i}^{r_{i}x}\}\big{)}.
Theorem 2.9 ([19] Theorem 9.3.5).

Let GG be a Cartan graph and w∈HomW​(G)⁡(x,y)w\in\operatorname{Hom}_{W(G)}(x,y). Define

l​(w):=min⁡{n∣idx⁡sin​…​si1=w}l(w):=\min\{n\mid\operatorname{id}_{x}s_{i_{n}}\dots s_{i_{1}}=w\}

and

N​(w):=#​{α∈Ry+∣w​α∈−Rx+}.N(w):=\#\{\alpha\in R^{y+}\mid w\alpha\in-R^{x+}\}.

Then, l​(w)=N​(w)l(w)=N(w).

Remark 2.10.

[19] A semi Cartan graph is called standard if AxA^{x} is independent of x∈Vx\in V.

For a standard Cartan graph GG:

G​ is finite⇔Ax​ is of finite type.G\text{ is finite}\iff A^{x}\text{ is of finite type}.

This result and the term ”real root” are from Kac [24] and are consistent with the definitions provided therein.

In particular, the Weyl groupoid arising from a finite Cartan graph with a single vertex can be identified with the Weyl group of type AxA^{x}.

Example 2.11.

[19, 12, 10]

[2−2−22]\begin{bmatrix}2&-2\\ -2&2\end{bmatrix}[2−2−12]\begin{bmatrix}2&-2\\ -1&2\end{bmatrix}[2−3−12]\begin{bmatrix}2&-3\\ -1&2\end{bmatrix}[2−3−12]\begin{bmatrix}2&-3\\ -1&2\end{bmatrix}[2−2−12]\begin{bmatrix}2&-2\\ -1&2\end{bmatrix}[2−2−22]\begin{bmatrix}2&-2\\ -2&2\end{bmatrix}[2−1−22]\begin{bmatrix}2&-1\\ -2&2\end{bmatrix}[2−1−32]\begin{bmatrix}2&-1\\ -3&2\end{bmatrix}[2−1−32]\begin{bmatrix}2&-1\\ -3&2\end{bmatrix}[2−1−22]\begin{bmatrix}2&-1\\ -2&2\end{bmatrix}1212121212

From the semi Cartan graph of rank two above, considering (CG2), the sequence (2,1,3,1,2,2,1,3,1,2)(2,1,3,1,2,2,1,3,1,2) naturally corresponds to it. Determining the real root system of this semi Cartan graph can be confirmed to be equivalent to considering a frieze with this sequence as the quiddity sequence. In this case, the frieze is as follows, confirming that it is a finite Cartan graph.

For example, when the top-left vertex of the graph above is denoted as xx, the set

Rx+={α1x,2​α1x+α2x,α1x+α2x,α1x+2​α2x,α2x},R^{x+}=\{\alpha_{1}^{x},2\alpha_{1}^{x}+\alpha_{2}^{x},\alpha_{1}^{x}+\alpha_{2}^{x},\alpha_{1}^{x}+2\alpha_{2}^{x},\alpha_{2}^{x}\},

corresponds to the bold column in the following frieze. Similarly, it can be confirmed that the real root system of the adjacent vertex corresponds to the sequence shifted by one position. Furthermore, the frieze extended to negative entries can also be interpreted in terms of negative roots.

000000000000000111111111111112213122131221313122131221312211111111111111100000000000000

By a similar argument, it can be seen that a connected (simply connected) finite Cartan graphs of rank two is equivalent to the concept of frieze patterns. In particular, according to the classification results of Conway and Coxeter [9], the isomorphism classes are parametrized by the triangulations of regular polygons. In particular, the current example corresponds to a triangulation of regular pentagon.

Definition 2.12.

A morphism of vertex-labeled edge-colored graphs is a graph morphism that preserves both the labels of the vertices and the colors of the edges.

Below, let the semi Cartan graph be connected. Consider a vertex-labeled edge-colored graph morphism G~→G\widetilde{G}\to G between semi Cartan graphs with the same color set II. We call (G~,G,π)(\widetilde{G},G,\pi) a covering.

Proposition 2.13 ([19], Proposition 10.1.5).

Let (G~,G,π)(\widetilde{G},G,\pi) be a covering. Then there exists a natural functor on the semi-Weyl groupoid:

Fπ:W​(G~)→W​(G),F_{\pi}:W(\widetilde{G})\to W(G),

which induces an injective homomorphism

AutW​(G~)⁡(y)→AutW​(G)⁡(π​(y))\operatorname{Aut}_{W(\widetilde{G})}(y)\rightarrow\operatorname{Aut}_{W(G)}(\pi(y))

for each vertex y∈G~y\in\widetilde{G}.

Definition 2.14.

[19] A semi-Cartan graph GG is called simply connected if the map π\pi is an isomorphism for every covering (G~,G,π)(\widetilde{G},G,\pi).

Equivalently, GG is simply connected if

#​HomW​(G)⁡(x,y)≤1for all ​x,y∈V.\#\operatorname{Hom}_{W(G)}(x,y)\leq 1\quad\text{for all }x,y\in V.
Proposition 2.15 ([19], Proposition 10.1.6).

Let GG be a Cartan graph. For x∈V​(G)x\in V(G) and a subgroup U⊆AutW​(G)⁡(x)U\subseteq\operatorname{Aut}_{W(G)}(x), there exists a covering (G~,G,π)(\widetilde{G},G,\pi) and a vertex x~∈V​(G~)\widetilde{x}\in V(\widetilde{G}) such that:

π​(x~)=xandFπ​(AutW​(G~)⁡(x~))=U.\pi(\widetilde{x})=x\quad\text{and}\quad F_{\pi}(\operatorname{Aut}_{W(\widetilde{G})}(\widetilde{x}))=U.

Moreover, such a covering is unique up to isomorphism, and

#π−1(x)=[AutW​(G)(x):U].\#\pi^{-1}(x)=[\operatorname{Aut}_{W(G)}(x):U].

In particular, a simply connected covering SC⁡(G)\operatorname{SC}(G) of GG, as a Cartan graph, always exists and is unique up to isomorphism.

Example 2.16.

By (CG4), the vertex set VV of a connected simply connected Cartan graph can be identified with a set {w​idx∣w∈HomW​(G)⁡(x,y),y∈V}\{w\operatorname{id}_{x}\mid w\in\operatorname{Hom}_{W(G)}(x,y),\,y\in V\}, where x∈Vx\in V is fixed. Clearly, a connected Cartan graph is loopless if and only if it is simply connected. If GG is standard, then S​C​(G)SC(G), as a graph, is the same as the Cayley graph of the Weyl group. By [22, 33], a simply connected Cartan graph is Hamiltonian (i.e. there exist a path that visits every vertex of a graph exactly once and returns to the starting vertex).

Example 2.17.

The isomorphism classes of connected standard Cartan graphs of type A2A_{2} correspond to the conjuate classes of subgroups of S3S_{3} via the following Galois correspondence:

S3S_{3}ℤ/3​ℤ\mathbb{Z}/3\mathbb{Z}ℤ/2​ℤ\mathbb{Z}/2\mathbb{Z}e

In more detail, the graph:

is represented as:

[2−1−12]\begin{bmatrix}2&-1\\ -1&2\end{bmatrix}[2−1−12]\begin{bmatrix}2&-1\\ -1&2\end{bmatrix}[2−1−12]\begin{bmatrix}2&-1\\ -1&2\end{bmatrix}21

which is the Cartan graph of 𝔤​𝔩​(2|1)\mathfrak{gl}(2|1) in the sense of Theorem 3.8. The corresponding Weyl group is isomorphic to ℤ/2​ℤ\mathbb{Z}/2\mathbb{Z}.

Additionally, the graph:

is represented as:

[2−1−12]\begin{bmatrix}2&-1\\ -1&2\end{bmatrix}

which is the Cartan graph of 𝔰​𝔩3\mathfrak{sl}_{3} in the sense of Theorem 3.8. The corresponding Weyl group is isomorphic to S3S_{3}.

2.2 Path subgroupoids and rainbow boomerang graph

Below, let the semi Cartan graph be connected.

Definition 2.18.

The path subgroupoid P​(G)P(G) of a semi Cartan graph GG is defined as the subgroupoid of the semi Weyl groupoid W​(G)W(G) generated by morphisms of the form:

{sit​⋯​si1​idx|ris+1​⋯​ri1​x≠ris​⋯​ri1​x​ for ​1≤s≤t−1},\left\{s_{i_{t}}\dotsb s_{i_{1}}\operatorname{id}_{x}\,\middle|\,r_{i_{s+1}}\dotsb r_{i_{1}}x\neq r_{i_{s}}\dotsb r_{i_{1}}x\text{ for }1\leq s\leq t-1\right\},

where x∈Vx\in V. For x,y∈Vx,y\in V, the set of morphisms between xx and yy in this subgroupoid is denoted by HomP​(G)⁡(x,y)\operatorname{Hom}_{P(G)}(x,y).

For α∈Rx\alpha\in R^{x}, we define:

orb⁡(α):={w​α|w∈HomP​(G)⁡(x,y)}⊆⨆y∈VRy,\operatorname{orb}(\alpha):=\left\{w\alpha\,\middle|\,w\in\operatorname{Hom}_{P(G)}(x,y)\right\}\subseteq\bigsqcup_{y\in V}R^{y},

and

Δ:={orb⁡(α)|α∈Rx}.\Delta:=\left\{\operatorname{orb}(\alpha)\,\middle|\,\alpha\in R^{x}\right\}.

This definition does not depend on the choice of xx.

A semi Cartan graph GG is said to be path simply connected if

#​HomP​(G)⁡(x,y)=1for any ​x,y.\#\operatorname{Hom}_{P(G)}(x,y)=1\quad\text{for any }x,y.

Moreover, if GG satisfies (CG3), this condition is equivalent to the following: For a fixed point xx and any O∈ΔO\in\Delta, #​(O∩Rx)=1\#(O\cap R^{x})=1 holds.

Furthermore, if GG is finite, this condition is also equivalent to #​Δ=#​Rx\#\Delta=\#R^{x}.

Lemma 2.19.

path simply connected semi Cartan graph is multiedge free

Proof. If there were two edges with the labels ii and jj between two nodes xx and yy, then we would have:

sj​si⋅αix=sj​(−αiy)=−αix−ai​jx​αjx≠αix=si​si⋅αix.s_{j}s_{i}\cdot\alpha_{i}^{x}=s_{j}(-\alpha_{i}^{y})=-\alpha_{i}^{x}-a_{ij}^{x}\alpha_{j}^{x}\neq\alpha_{i}^{x}=s_{i}s_{i}\cdot\alpha_{i}^{x}.

Thus, we have : #​HomP​(G)⁡(x,y)>1\#\operatorname{Hom}_{P(G)}(x,y)>1. □\square

Example 2.20.

The following finite Cartan graph is multiedge-free but not path-simply connected.
[2−1−22]\begin{bmatrix}2&-1\\ -2&2\end{bmatrix}[2−1−22]\begin{bmatrix}2&-1\\ -2&2\end{bmatrix}[2−1−22]\begin{bmatrix}2&-1\\ -2&2\end{bmatrix}[2−1−22]\begin{bmatrix}2&-1\\ -2&2\end{bmatrix}\pgfmathresultpt1\pgfmathresultpt2\pgfmathresultpt1\pgfmathresultpt2

Example 2.21.

The path subgroupoid of a simply connected Weyl groupoid is the Weyl groupoid itself. Hence, by the definition of simply connectedness, it is path simply connected.

Example 2.22.

semi Cartan trees are trivially path simply connected.

Definition 2.23.

When GG is path simply connected, for O∈ΔO\in\Delta, let Ox∈RxO_{x}\in R^{x} be the unique element in O∩RxO\cap R^{x}. Define Δx+\Delta^{x+} as

Δx+={O∈Δ∣Ox∈Rx+},\Delta^{x+}=\{O\in\Delta\mid O_{x}\in R^{x+}\},

and Δpure+\Delta^{\text{pure}+} as

Δpure+=⋂x∈VΔx+.\Delta^{\text{pure}+}=\bigcap_{x\in V}\Delta^{x+}.

For instance, if GG is simply connected, then Δpure+=∅\Delta^{\text{pure}+}=\emptyset.

Definition 2.24.

For a path simply connected Cartan graph GG, we define the edge-colored graph R​B​(G)RB(G) as follows:

  • •

    Underlying graph: The underlying graph of GG, with loops removed.

  • •

    Color set CC: For a fixed x∈Vx\in V,

    C=Δx+−Δpure+C=\Delta^{x+}-\Delta^{\text{pure+}}
  • •

    Coloring: Replace each edge between zz and yy colored ii with an edge colored by a unique O∈C{O}\in C such that Oz∈{±αiz}{O}_{z}\in\{\pm\alpha_{i}^{z}\} (see Lemma 2.8).

Below, following [21], we recall the definition of a rainbow boomerang.

Definition 2.25.

[21, Definition 2.3]([Rainbow Boomerang Graph] A edge-colored graph GG which is properly colored (i.e. for each vertex the insident edges have distinct colors) is called a rainbow boomerang graph when a walk is shortest if and only if it is rainbow.

The following is the property of our interest.

Proposition 2.26 (Exchange property: Proposition 2.10 in [21]).

Let GG be a rainbow boomerang graph. Let kk be a positive integer. If there exists a rainbow walk v0​c0​v1​c1​…​ck​vk+1v_{0}c_{0}v_{1}c_{1}\dots c_{k}v_{k+1} and an edge vk+1​c0​vk+2v_{k+1}c_{0}v_{k+2}, then there exists a rainbow walk vk+2​d1​vk+3​d2​…​v2​k+1​dk​v0v_{k+2}d_{1}v_{k+3}d_{2}\dots v_{2k+1}d_{k}v_{0} such that {c1,c2,…,ck}={d1,d2,…,dk}\{c_{1},c_{2},\dots,c_{k}\}=\{d_{1},d_{2},\dots,d_{k}\}.

v1v_{1}v2v_{2}v3v_{3}vkv_{k}vk+1v_{k+1}v0v_{0}v2​k+1v_{2k+1}v2​kv_{2k}vk+3v_{k+3}vk+2v_{k+2}c1c_{1}c2c_{2}ckc_{k}dkd_{k}dk−1d_{k-1}d1d_{1}c0c_{0}c0c_{0}
Corollary 2.27.

[21, Corollary 2.13] For a connected rainbow boomerang graph GG and a color cc of GG, the edge-colored graph obtained by removing all edges of color cc from GG consists of two connected components, each of which is a rainbow boomerang graph.

Theorem 2.28.

R​B​(G)RB(G) of a path simply connected Cartan graph GG is a rainbow boomerang graph.

Proof. In the simply connected case, this follows immediately from Theorem 2.9.

In general, if GG is connected and path-simply connected, then under the natural identification of S​C​(G)SC(G) with the root system of GG, the edge-colored graph obtained from R​B​(S​C​(G))RB(SC(G)) by removing edges with colors belonging to Δpure+\Delta^{\text{pure+}} is a disjoint union of copies of R​B​(G)RB(G), with the number of components equal to the order of the group of automorphisms of an object of W​(G)W(G) by Proposition 2.15. Consequently, R​B​(G)RB(G) is a rainbow booerang graph by Corollary 2.27.

□\square

Example 2.29.

By appropriately removing edges from the Cayley graph of the symmetric group on 4 elements with respect to its simple reflections, we obtain a disjoint union of four finite Young lattices L​(2,2)L(2,2). This is consistent with the fact that the order of the Weyl group of 𝔤​𝔩​(2|2)\mathfrak{gl}(2|2) is 4.

Example 2.30.

Let GG be a finite Cartan graph of rank 2. Then, GG is multiedge-free (if #​V≠2\#V\neq 2, this is the case) if and only if GG is path simply connected. In this case, R​B​(G)RB(G) is one of the following:

  • •

    a line segment ;

  • •

    a cycle graph C2​nC_{2n} of length 2​n2n (n>0n>0) .

3 Non simply connected Weyl groupoids of basic Lie superalgebras

From now on, our 𝔤\mathfrak{g} will be a direct sum of one of the finite-dimensional basic Lie superalgebras from the following list:

𝔰​𝔩​(m|n),m≠n,𝔤​𝔩​(m|n),𝔬​𝔰​𝔭​(m|2​n),D​(2,1;α),G​(3),F​(4).\mathfrak{sl}(m|n),\,m\neq n,\,\mathfrak{gl}(m|n),\,\mathfrak{osp}(m|2n),\,D(2,1;\alpha),\,G(3),\,F(4).

For concrete definitions, we refer to [29][Chapters 1-4].

We denote the even and odd parts of 𝔤\mathfrak{g} as 𝔤0¯\mathfrak{g}_{\overline{0}} and 𝔤1¯\mathfrak{g}_{\overline{1}}, respectively.

Definition 3.1 ([29, 7]).

A Cartan subalgebra and the Weyl group of the reductive Lie algebra 𝔤0¯\mathfrak{g}_{\overline{0}} are denoted by 𝔥\mathfrak{h} and WW, respectively.

A basic Lie superalgebra 𝔤\mathfrak{g} has a supersymmetric, superinvariant, even bilinear form ⟨,⟩\langle\,,\,\rangle, which induces a WW-invariant bilinear form (,)(\,,\,) on 𝔥∗\mathfrak{h}^{*} via duality.

The root space 𝔤α\mathfrak{g}_{\alpha} associated with α∈𝔥∗\alpha\in\mathfrak{h}^{*} is defined as 𝔤α:={x∈𝔤∣[h,x]=α​(h)​x​for all ​h∈𝔥}.\mathfrak{g}_{\alpha}:=\{x\in\mathfrak{g}\mid[h,x]=\alpha(h)x\,\text{for all }h\in\mathfrak{h}\}.

The set of roots Δ\Delta is defined as Δ:={α∈𝔥∗∣𝔤α≠0}∖{0}.\Delta:=\{\alpha\in\mathfrak{h}^{*}\mid\mathfrak{g}_{\alpha}\neq 0\}\setminus\{0\}. Each 𝔤α\mathfrak{g}_{\alpha} is either purely even or purely odd and is one-dimensional (our list does not include 𝔰​𝔩​(n|n)\mathfrak{sl}(n|n) and 𝔭​𝔰​𝔩​(n|n)\mathfrak{psl}(n|n)). Therefore, the notions of even roots and odd roots are well defined. An odd root α\alpha is said to be isotropic if (α,α)=0(\alpha,\alpha)=0. The sets of all even roots, even positive roots, odd roots and odd isotropic roots are denoted by Δ0¯\Delta_{\overline{0}}, Δ0¯+\Delta_{\overline{0}}^{+}), Δ1¯\Delta_{\overline{1}} and Δ⊗\Delta_{\otimes}, respectively.

Definition 3.2 ([29, 7]).

We fix a Borel subalgebra 𝔟0¯\mathfrak{b}_{\overline{0}} of 𝔤0¯\mathfrak{g}_{\overline{0}}. The set of all Borel subalgebras 𝔟\mathfrak{b} of 𝔤\mathfrak{g} that contain 𝔟0¯\mathfrak{b}_{\overline{0}} is denoted by 𝔅​(𝔤)\mathfrak{B(g)}.

For a Borel subalgebra 𝔟∈𝔅​(𝔤)\mathfrak{b}\in\mathfrak{B(g)}, we express the triangular decomposition of 𝔤\mathfrak{g} as

𝔤=𝔫𝔟−⊕𝔥⊕𝔫𝔟+,\mathfrak{g}=\mathfrak{n}^{\mathfrak{b}-}\oplus\mathfrak{h}\oplus\mathfrak{n}^{\mathfrak{b}+},

where 𝔟=𝔥⊕𝔫𝔟+\mathfrak{b}=\mathfrak{h}\oplus\mathfrak{n}^{\mathfrak{b}+}.

The sets of positive roots, odd positive roots, and odd isotropic positive roots corresponding to 𝔟\mathfrak{b} are denoted by Δ𝔟+\Delta^{\mathfrak{b}+}, Δ1¯𝔟+\Delta_{\overline{1}}^{\mathfrak{b}+}, and Δ⊗𝔟+\Delta_{\otimes}^{\mathfrak{b}+}, respectively. The set of simple roots (basis) corresponding to Δ𝔟+\Delta^{\mathfrak{b}+} is denoted by Π𝔟\Pi^{\mathfrak{b}}. We define Π⊗𝔟:=Π𝔟∩Δ⊗\Pi_{\otimes}^{\mathfrak{b}}:=\Pi^{\mathfrak{b}}\cap\Delta_{\otimes}. We define

Δpure+:=⋂𝔟∈𝔅​(𝔤)Δ𝔟+,\Delta^{\operatorname{pure+}}:=\bigcap_{\mathfrak{b}\in\mathfrak{B(g)}}\Delta^{\mathfrak{b}+},
Δ⊗pure+:=⋂𝔟∈𝔅​(𝔤)Δ⊗𝔟+=Δpure+∩Δ⊗.\Delta_{\otimes}^{\operatorname{pure+}}:=\bigcap_{\mathfrak{b}\in\mathfrak{B(g)}}\Delta_{\otimes}^{\mathfrak{b}+}=\Delta^{\operatorname{pure+}}\cap\Delta_{\otimes}.
Theorem 3.3 (Odd reflection [29] 3.5).

For α∈Π⊗𝔟\alpha\in\Pi_{\otimes}^{\mathfrak{b}}, define rα𝔟∈Map⁡(Π𝔟,Δ)r^{\mathfrak{b}}_{\alpha}\in\operatorname{Map}(\Pi^{\mathfrak{b}},\Delta) by

rα𝔟​(β)={−α(β=α),α+β(α+β∈Δ),β(otherwise).r^{\mathfrak{b}}_{\alpha}(\beta)=\begin{cases}-\alpha&(\beta=\alpha),\\ \alpha+\beta&(\alpha+\beta\in\Delta),\\ \beta&(\text{otherwise}).\end{cases}

for β∈Π𝔟\beta\in\Pi^{\mathfrak{b}}. (When there is no risk of confusion, rα𝔟r_{\alpha}^{\mathfrak{b}} is abbreviated as rαr_{\alpha}.) A Borel subalgebra rα​𝔟∈𝔅​(𝔤)r_{\alpha}\mathfrak{b}\in\mathfrak{B(g)} exists, with the corresponding basis given by

Πrα​𝔟:={rα𝔟​(β)}β∈Π𝔟.\Pi^{r_{\alpha}\mathfrak{b}}:=\{r^{\mathfrak{b}}_{\alpha}(\beta)\}_{\beta\in\Pi^{\mathfrak{b}}}.

The linear transformation of 𝔥∗\mathfrak{h}^{*} induced by an odd reflection does not necessarily map a Borel subalgebra to another Borel subalgebra.

The following is well-known:

Proposition 3.4 ([29, 7]).

Each pair of elements 𝔟,𝔟′∈𝔅​(𝔤)\mathfrak{b},\mathfrak{b^{\prime}}\in\mathfrak{B(g)} due to transferred to each other by a finite number of odd reflections.

Definition 3.5.

The edge-colored graph R​B​(𝔤)RB(\mathfrak{g}) is defined as follows:

  • •

    Vertex set: 𝔅​(𝔤)\mathfrak{B(g)}.

  • •

    Color set: For a fixed 𝔟∈𝔅​(𝔤)\mathfrak{b}\in\mathfrak{B(g)}, the set Δ𝔟+∖Δpure+\Delta^{\mathfrak{b}+}\setminus\Delta^{\text{pure+}}.

  • •

    Edges and colors: An edge is drawn between two vertices if they are related by an odd reflection. The edge is assigned a color corresponding to the unique α∈Δ𝔟+∖Δpure+\alpha\in\Delta^{\mathfrak{b}+}\setminus\Delta^{\text{pure+}} such that α\alpha belongs to the positive root system of one vertex but not the other.

Since the positive root systems associated with different Borel subalgebras are in one-to-one correspondence, the structure of the edge-colored graph does not depend on the choice of 𝔟\mathfrak{b}.

Definition 3.6 ([6, 30]).

Let 𝔟∈𝔅​(𝔤)\mathfrak{b}\in\mathfrak{B(g)}, and consider a total ordering ≤\leq on Π𝔟\Pi^{\mathfrak{b}}. We call the pair (𝔟,≤)(\mathfrak{b},\leq) an ordered root basis. This ordering is denoted by

Π(𝔟,≤)={α1(𝔟,≤),…,αθ(𝔟,≤)}.\Pi^{(\mathfrak{b},\leq)}=\{\alpha_{1}^{(\mathfrak{b},\leq)},\dots,\alpha_{\theta}^{(\mathfrak{b},\leq)}\}.

For a composition of odd reflections rβt​…​rβ1r_{\beta_{t}}\dots r_{\beta_{1}}, we define the ordered root basis

rβt​…​rβ1​((𝔟,≤))r_{\beta_{t}}\dots r_{\beta_{1}}((\mathfrak{b},\leq))

by

αjrβt​…​rβ1​(𝔟,≤):=rβt​…​rβ1​(αj(𝔟,≤)).\alpha_{j}^{r_{\beta_{t}}\dots r_{\beta_{1}}(\mathfrak{b},\leq)}:=r_{\beta_{t}}\dots r_{\beta_{1}}(\alpha_{j}^{(\mathfrak{b},\leq)}).

In this way, the ordered root bases are mapped to each other under odd reflections.

Definition 3.7.

Recall Definition 3.6. Given a fixed ordered root basis (𝔟¯,≤¯)(\overline{\mathfrak{b}},\overline{\leq}), we define E​(𝔤)E(\mathfrak{g}) as an edge-colored graph with the following structure:

  • •

    Vertex set VV: Each vertex (𝔟,≤)(\mathfrak{b},\leq) represents an ordered root basis obtained from (𝔟¯,≤¯)(\overline{\mathfrak{b}},\overline{\leq}) through a finite sequence of odd reflections.

  • •

    Color set : The total orbdered set II as Definition 3.6.

  • •

    Edges: Draw an edge of color ii between vertices that are related by an odd reflection corresponding to the ii-th simple root. Additionally, assign a loop of color ii at a vertex if the ii-th simple root is non-isotropic for that vertex.

We rely on the following result (see [6, Definition 2.10], [19, Corollary 2.14], or [5]).

Theorem 3.8.

Under the above settings, for each (𝔟,≤)∈V(\mathfrak{b},\leq)\in V, there exists a unique family of generalized Cartan matrices {A(𝔟,≤)}\{A^{(\mathfrak{b},\leq)}\}, such that the vertex labeling by this family of matrices makes E​(𝔤)E(\mathfrak{g}) a finite connected Cartan graph, and for each (𝔟,≤)∈V(\mathfrak{b},\leq)\in V, there is an additive bijection

R(𝔟,≤)+≃Δ𝔟+∖2​Δ𝔟+R^{(\mathfrak{b},\leq)+}\simeq\Delta^{\mathfrak{b}+}\setminus 2\Delta^{\mathfrak{b}+}

given by mapping αi(𝔟,≤)↦αi𝔟\alpha_{i}^{(\mathfrak{b},\leq)}\mapsto\alpha_{i}^{\mathfrak{b}}.

We denote the Cartan graph constructed above by G​(𝔤)G(\mathfrak{g}).

Corollary 3.9 ([6] Remark 2.18).

If (𝔟,≤),(𝔟,≤′)∈V(\mathfrak{b},\leq),(\mathfrak{b},\leq^{\prime})\in V, then ≤⁣=⁣≤′\leq=\leq^{\prime}. In particular, VV can be identified with 𝔅​(𝔤)\mathfrak{B}(\mathfrak{g}).

Proof. This directly follows from (CG4) and Theorem 3.8. □\square

Theorem 3.10.

G​(𝔤)G(\mathfrak{g}) is path simply connected. Furthermore, Δ\Delta in the sense of Definition 3.1 can be identified with the root system Δ\Delta in the sense of Definition 2.18.

As edge-colored graphs, R​B​(𝔤)RB(\mathfrak{g}) in the sense of Definition 3.5 is isomorphic to R​B​(G​(𝔤))RB(G(\mathfrak{g})) in the sense of Definition 2.24.

In particular, R​B​(𝔤)RB(\mathfrak{g}) is a connected rainbow boomerang graph.

Proof. This directly follows from Theorem 3.8. □\square

Remark 3.11.

Here are a few remarks about the above facts:

  1. 1.

    By this construction, E​(𝔤)E(\mathfrak{g}) is indeed the exchange graph of G​(𝔤)G(\mathfrak{g}).

  2. 2.

    The set R(𝔟,≤)R^{(\mathfrak{b},\leq)} is a subset of (ℤI)(𝔟,≤)(\mathbb{Z}^{I})^{(\mathfrak{b},\leq)}, and Δ\Delta is a subset of 𝔥∗\mathfrak{h}^{*}. We strictly distinguish between these two.

  3. 3.

    The map si(𝔟,≤)s_{i}^{(\mathfrak{b},\leq)} is a linear transformation from (ℤI)(𝔟,≤)(\mathbb{Z}^{I})^{(\mathfrak{b},\leq)} to (ℤI)ri​(𝔟,≤)(\mathbb{Z}^{I})^{r_{i}{(\mathfrak{b},\leq)}}, while the odd reflection ri𝔟r_{i}^{\mathfrak{b}} is a map from Π𝔟\Pi^{\mathfrak{b}} to Δ\Delta.

  4. 4.

    The equality Δ∖2​Δ=Δ\Delta\setminus 2\Delta=\Delta holds unless 𝔤=𝔬​𝔰​𝔭​(2​m+1|2​n)\mathfrak{g}=\mathfrak{osp}(2m+1|2n) or 𝔤=G​(3)\mathfrak{g}=G(3).

  5. 5.

    By the above, G​(𝔤)G(\mathfrak{g}) does not depend on the choice of (𝔟,≤)(\mathfrak{b},\leq) and is uniquely determined by 𝔤\mathfrak{g}.

  6. 6.

    For a vertex xx in G​(𝔤)G(\mathfrak{g}), the automorphism group Aut⁡(x)\operatorname{Aut}(x) can be identified with the Weyl group WW ([6, Proposition 2.15]).

  7. 7.

    As noted in [6], similar considerations make sense in the broader setting of regular symmetrizable contragredient Lie superalgebras, which are not necessarily finite-dimensional.

Remark 3.12.

While it may not be explicitly stated in the literature, it seems reasonable to expect that the classification of basic Lie superalgebras by Kac [23] could, conceptually (albeit highly nontrivially), be rederived as a consequence of Heckenberger’s results [18] (see also [34], Introduction). Indeed, as noted in ([5] Lemma5.1), when the inner product of simple roots is a rational number, one can canonically establish an isomorphism with the Cartan graph of a Nichols algebra of diagonal type with generic parameters. From the list in [1], only two exceptions exist beyond those associated with basic Lie superalgebras. This allows for a contemporary explanation of all cases except D​(2,1;α)D(2,1;\alpha).

Example 3.13.

Let V=V0¯⊕V1¯V=V_{\overline{0}}\oplus V_{\overline{1}} be a ℤ/2​ℤ\mathbb{Z}/2\mathbb{Z}-graded vector space, where V0¯V_{\overline{0}} (the even part) is spanned by v1,…,vmv_{1},\dots,v_{m} and V1¯V_{\overline{1}} (the odd part) is spanned by vm+1,…,vm+nv_{m+1},\dots,v_{m+n}.

The space End⁡(V)\operatorname{End}(V) is spanned by basis elements Ei​jE_{ij}, defined by:

Ei​j⋅vk=δj​k​vi.E_{ij}\cdot v_{k}=\delta_{jk}v_{i}.

The general linear Lie superalgebra 𝔤​𝔩​(m|n)\mathfrak{gl}(m|n) is defined as the Lie superalgebra spanned by all Ei​jE_{ij} with 1≤i,j≤m+n1\leq i,j\leq m+n, under the supercommutator:

[Ei​j,Ek​l]=Ei​j​Ek​l−(−1)|Ei​j|​|Ek​l|​Ek​l​Ei​j,[E_{ij},E_{kl}]=E_{ij}E_{kl}-(-1)^{|E_{ij}||E_{kl}|}E_{kl}E_{ij},

where |Ei​j|=0¯|E_{ij}|=\overline{0} if Ei​jE_{ij} acts within V0¯V_{\overline{0}} or V1¯V_{\overline{1}} (even), and |Ei​j|=1¯|E_{ij}|=\overline{1} if it maps between V0¯V_{\overline{0}} and V1¯V_{\overline{1}} (odd).

The Cartan subalgebra 𝔥\mathfrak{h} is given by 𝔥=⨁k​Ei​i\mathfrak{h}=\bigoplus kE_{ii}.

Let Ei​iE_{ii} be associated with dual basis elements εi\varepsilon_{i} for 1≤i≤m+n1\leq i\leq m+n. Then we have 𝔤εi−εj=k​Ei​j\mathfrak{g}_{\varepsilon_{i}-\varepsilon_{j}}=kE_{ij}.

The bilinear form (,)(\,,\,) is computed as follows:

(εi,εj)={1if ​i=j≤m,−1if ​i=j≥m+1,0if ​i≠j.(\varepsilon_{i},\varepsilon_{j})=\begin{cases}1&\text{if }i=j\leq m,\\ -1&\text{if }i=j\geq m+1,\\ 0&\text{if }i\neq j.\end{cases}

Define δi=εm+i\delta_{i}=\varepsilon_{m+i} for 1≤i≤n1\leq i\leq n. The sets of roots are as follows:

Δ0¯={εi−εj,δi−δj∣i≠j},\Delta_{\overline{0}}=\{\varepsilon_{i}-\varepsilon_{j},\delta_{i}-\delta_{j}\mid i\neq j\},
Δ1¯={εi−δj∣1≤i≤m, 1≤j≤n}.\Delta_{\overline{1}}=\{\varepsilon_{i}-\delta_{j}\mid 1\leq i\leq m,\,1\leq j\leq n\}.

For the even part 𝔤0¯=𝔤​𝔩​(m)⊕𝔤​𝔩​(n)\mathfrak{g}_{\overline{0}}=\mathfrak{gl}(m)\oplus\mathfrak{gl}(n), we fix the standard Borel subalgebra 𝔟0¯\mathfrak{b}_{\overline{0}} as:

𝔟0¯=⨁1≤i≤j≤mk​Ei​j⊕⨁m+1≤i≤j≤nk​Ei​j.\mathfrak{b}_{\overline{0}}=\bigoplus_{1\leq i\leq j\leq m}kE_{ij}\oplus\bigoplus_{m+1\leq i\leq j\leq n}kE_{ij}.

We assume that the Borel subalgebras we consider all contain 𝔟0¯\mathfrak{b}_{\overline{0}}.

When 𝔤=𝔤​𝔩​(m|n)\mathfrak{g}=\mathfrak{gl}(m|n), we can identify εi−εi+1\varepsilon_{i}-\varepsilon_{i+1} with orb⁡(αi𝔟)\operatorname{orb}(\alpha_{i}^{\mathfrak{b}}).

According to [6], fixing the total order determined by

αi𝔟:=εi−εi+1,1≤i≤m+n−1,\alpha_{i}^{\mathfrak{b}}:=\varepsilon_{i}-\varepsilon_{i+1},\quad 1\leq i\leq m+n-1,

E​(𝔤​𝔩​(m|n))E(\mathfrak{gl}(m|n)) is defined as an edge-colored graph with the following structure [6]:

  • •

    Vertex set : V=𝔅​(𝔤)=Pm×nV=\mathfrak{B}(\mathfrak{g})=P_{m\times n} (Young diagrams fitting in a m×n rectangle.)

  • •

    Color set : I={1,2,…,m+n−1}I=\{1,2,\dots,m+n-1\};

  • •

    Edges: There is an edge of color ii between vertices 𝔟1\mathfrak{b}_{1} and 𝔟2\mathfrak{b}_{2} if and only if 𝔟1\mathfrak{b}_{1} and 𝔟2\mathfrak{b}_{2} are related by adding or subtracting a box at coordinates (x,y)(x,y) in French notation, with x−y+m=ix-y+m=i.

Furthermore, the graph G​(𝔤​𝔩​(m|n))G(\mathfrak{gl}(m|n)) is the labeled graph obtained by labeling each vertex 𝔟\mathfrak{b} of E​(𝔤​𝔩​(m|n))E(\mathfrak{gl}(m|n)) with A𝔟=Am+n−1A^{\mathfrak{b}}=A_{m+n-1}.

Example 3.14.

In 𝔤​𝔩​(2|1)\mathfrak{gl}(2|1), we have the following identifications:

ε1−ε2↔orb⁡(α1∅)={α1∅,α1(1)+α2(1),α2(12)}.\varepsilon_{1}-\varepsilon_{2}\leftrightarrow\operatorname{orb}(\alpha^{\emptyset}_{1})=\bigl{\{}\alpha^{\emptyset}_{1},\alpha^{(1)}_{1}+\alpha^{(1)}_{2},\alpha^{(1^{2})}_{2}\bigr{\}}.
ε1−δ1↔orb⁡(α1∅+α2∅)={α1∅+α2∅,α1(1),−α1(12)}.\varepsilon_{1}-\delta_{1}\leftrightarrow\operatorname{orb}(\alpha^{\emptyset}_{1}+\alpha^{\emptyset}_{2})=\bigl{\{}\alpha^{\emptyset}_{1}+\alpha^{\emptyset}_{2},\alpha^{(1)}_{1},-\alpha^{(1^{2})}_{1}\bigr{\}}.
ε2−δ1↔orb⁡(α2∅)={α2∅,−α2(1),−α1(12)−α2(12)}.\varepsilon_{2}-\delta_{1}\leftrightarrow\operatorname{orb}(\alpha^{\emptyset}_{2})=\bigl{\{}\alpha^{\emptyset}_{2},-\alpha^{(1)}_{2},-\alpha^{(1^{2})}_{1}-\alpha^{(1^{2})}_{2}\bigr{\}}.
Example 3.15.

The edge-colored graph E​(𝔤​𝔩​(3|2))E(\mathfrak{gl}(3|2)) (excluding loops) is as follows.

∅\emptyset{ytableau}\ytableau~{}{ytableau}​&\ytableau~{}&~{}\\ {ytableau}​&\ytableau~{}\\ ~{}&~{}{ytableau}​&\ytableau~{}&~{}\\ ~{}~{}{ytableau}​&\ytableau~{}\\ ~{}&~{}\\ ~{}~{}{ytableau}​&\ytableau~{}&~{}\\ ~{}~{}\\ ~{}~{}{ytableau}\ytableau~{}\\ ~{}{ytableau}\ytableau~{}\\ ~{}\\ ~{}{ytableau}​&\ytableau~{}\\ ~{}\\ ~{}&~{}342312241314

.

Example 3.16.

Let 𝔤=D​(2,1;α)\mathfrak{g}=D(2,1;\alpha). See [8] for more information on this type of Lie superalgebra.

The vector space 𝔥∗\mathfrak{h}^{*} has an orthogonal basis {δ,ε1,ε2}\{\delta,\varepsilon_{1},\varepsilon_{2}\} with respect to the inner product (⋅,⋅)(\cdot,\cdot), where

(δ,δ)=−(1+α),(ε1,ε1)=1,(ε2,ε2)=α.(\delta,\delta)=-(1+\alpha),\quad(\varepsilon_{1},\varepsilon_{1})=1,\quad(\varepsilon_{2},\varepsilon_{2})=\alpha.

The sets of roots are as follows:

Δ0¯={±2​δ,±2​ε1,±2​ε2}\Delta_{\overline{0}}=\{\pm 2\delta,\pm 2\varepsilon_{1},\pm 2\varepsilon_{2}\}
Δ1¯=Δ⊗={±(δ−ε1−ε2),±(δ+ε1−ε2),±(δ−ε1+ε2),±(δ+ε1+ε2)}\Delta_{\overline{1}}=\Delta_{\otimes}=\{\pm(\delta-\varepsilon_{1}-\varepsilon_{2}),\pm(\delta+\varepsilon_{1}-\varepsilon_{2}),\pm(\delta-\varepsilon_{1}+\varepsilon_{2}),\pm(\delta+\varepsilon_{1}+\varepsilon_{2})\}

The exchange graph E​(D​(2,1;α))E(D(2,1;\alpha)) is described as follows.

𝔟1\mathfrak{b}_{1}𝔟2\mathfrak{b}_{2}𝔟3\mathfrak{b}_{3}𝔟4\mathfrak{b}_{4}213

The Cartan graph G​(D​(2,1;α))G(D(2,1;\alpha)) is defined as follows.

A𝔟1=(2−10−12−10−12),A𝔟2=(2−1−1−120−102),A𝔟3=(2−1−1−12−1−1−12),A𝔟4=(20−102−1−1−12).\begin{array}[]{cc}A^{\mathfrak{b}_{1}}=\begin{pmatrix}2&-1&0\\ -1&2&-1\\ 0&-1&2\end{pmatrix},&A^{\mathfrak{b}_{2}}=\begin{pmatrix}2&-1&-1\\ -1&2&0\\ -1&0&2\end{pmatrix},\\[10.0pt] A^{\mathfrak{b}_{3}}=\begin{pmatrix}2&-1&-1\\ -1&2&-1\\ -1&-1&2\end{pmatrix},&A^{\mathfrak{b}_{4}}=\begin{pmatrix}2&0&-1\\ 0&2&-1\\ -1&-1&2\end{pmatrix}.\end{array}

The corresponding positive root systems for each vertex are:

R𝔟1+\displaystyle R^{\mathfrak{b}_{1}+} ={α1𝔟1,α1𝔟1+α2𝔟1,α1𝔟1+α2𝔟1+α3𝔟1,α1𝔟1+2​α2𝔟1+α3𝔟1,α2𝔟1,α2𝔟1+α3𝔟1,α3𝔟1}\displaystyle=\{\alpha_{1}^{\mathfrak{b}_{1}},\alpha_{1}^{\mathfrak{b}_{1}}+\alpha_{2}^{\mathfrak{b}_{1}},\alpha_{1}^{\mathfrak{b}_{1}}+\alpha_{2}^{\mathfrak{b}_{1}}+\alpha_{3}^{\mathfrak{b}_{1}},\alpha_{1}^{\mathfrak{b}_{1}}+2\alpha_{2}^{\mathfrak{b}_{1}}+\alpha_{3}^{\mathfrak{b}_{1}},\alpha_{2}^{\mathfrak{b}_{1}},\alpha_{2}^{\mathfrak{b}_{1}}+\alpha_{3}^{\mathfrak{b}_{1}},\alpha_{3}^{\mathfrak{b}_{1}}\}
R𝔟2+\displaystyle R^{\mathfrak{b}_{2}+} ={α2𝔟2,α2𝔟2+α1𝔟2,α2𝔟2+α1𝔟2+α3𝔟2,α2𝔟2+2​α1𝔟2+α3𝔟2,α1𝔟2,α1𝔟2+α3𝔟2,α3𝔟2}\displaystyle=\{\alpha_{2}^{\mathfrak{b}_{2}},\alpha_{2}^{\mathfrak{b}_{2}}+\alpha_{1}^{\mathfrak{b}_{2}},\alpha_{2}^{\mathfrak{b}_{2}}+\alpha_{1}^{\mathfrak{b}_{2}}+\alpha_{3}^{\mathfrak{b}_{2}},\alpha_{2}^{\mathfrak{b}_{2}}+2\alpha_{1}^{\mathfrak{b}_{2}}+\alpha_{3}^{\mathfrak{b}_{2}},\alpha_{1}^{\mathfrak{b}_{2}},\alpha_{1}^{\mathfrak{b}_{2}}+\alpha_{3}^{\mathfrak{b}_{2}},\alpha_{3}^{\mathfrak{b}_{2}}\}
R𝔟3+\displaystyle R^{\mathfrak{b}_{3}+} ={α1𝔟3,α1𝔟3+α2𝔟3,α1𝔟3+α3𝔟3,α1𝔟3+α2𝔟3+α3𝔟3,α2𝔟3,α2𝔟3+α3𝔟3,α3𝔟3}\displaystyle=\{\alpha_{1}^{\mathfrak{b}_{3}},\alpha_{1}^{\mathfrak{b}_{3}}+\alpha_{2}^{\mathfrak{b}_{3}},\alpha_{1}^{\mathfrak{b}_{3}}+\alpha_{3}^{\mathfrak{b}_{3}},\alpha_{1}^{\mathfrak{b}_{3}}+\alpha_{2}^{\mathfrak{b}_{3}}+\alpha_{3}^{\mathfrak{b}_{3}},\alpha_{2}^{\mathfrak{b}_{3}},\alpha_{2}^{\mathfrak{b}_{3}}+\alpha_{3}^{\mathfrak{b}_{3}},\alpha_{3}^{\mathfrak{b}_{3}}\}
R𝔟4+\displaystyle R^{\mathfrak{b}_{4}+} ={α1𝔟1,α1𝔟1+α3𝔟1,α1𝔟1+α3𝔟1+α2𝔟1,α1𝔟1+2​α3𝔟1+α2𝔟1,α3𝔟1,α3𝔟1+α2𝔟1,α2𝔟1}\displaystyle=\{\alpha_{1}^{\mathfrak{b}_{1}},\alpha_{1}^{\mathfrak{b}_{1}}+\alpha_{3}^{\mathfrak{b}_{1}},\alpha_{1}^{\mathfrak{b}_{1}}+\alpha_{3}^{\mathfrak{b}_{1}}+\alpha_{2}^{\mathfrak{b}_{1}},\alpha_{1}^{\mathfrak{b}_{1}}+2\alpha_{3}^{\mathfrak{b}_{1}}+\alpha_{2}^{\mathfrak{b}_{1}},\alpha_{3}^{\mathfrak{b}_{1}},\alpha_{3}^{\mathfrak{b}_{1}}+\alpha_{2}^{\mathfrak{b}_{1}},\alpha_{2}^{\mathfrak{b}_{1}}\}

For example, the following correspondences hold:

2​ε1↔orb⁡(α1𝔟1),δ−ε1−ε2↔orb⁡(α2𝔟1),2​ε2↔orb⁡(α3𝔟1),2\varepsilon_{1}\leftrightarrow\operatorname{orb}(\alpha_{1}^{\mathfrak{b}_{1}}),\quad\delta-\varepsilon_{1}-\varepsilon_{2}\leftrightarrow\operatorname{orb}(\alpha_{2}^{\mathfrak{b}_{1}}),\quad 2\varepsilon_{2}\leftrightarrow\operatorname{orb}(\alpha_{3}^{\mathfrak{b}_{1}}),
δ+ε1+ε2↔orb⁡(α1𝔟1+α2𝔟1+α3𝔟1),2​δ↔orb⁡(α1𝔟1+2​α2𝔟1+α3𝔟1).\delta+\varepsilon_{1}+\varepsilon_{2}\leftrightarrow\operatorname{orb}(\alpha_{1}^{\mathfrak{b}_{1}}+\alpha_{2}^{\mathfrak{b}_{1}}+\alpha_{3}^{\mathfrak{b}_{1}}),\quad 2\delta\leftrightarrow\operatorname{orb}(\alpha_{1}^{\mathfrak{b}_{1}}+2\alpha_{2}^{\mathfrak{b}_{1}}+\alpha_{3}^{\mathfrak{b}_{1}}).

we also note that

Δpure+={2​δ,2​ε1,2​ε2,δ+ε1+ε2},Δ⊗pure+={δ+ε1+ε2}.\Delta^{\text{pure}+}=\{2\delta,2\varepsilon_{1},2\varepsilon_{2},\delta+\varepsilon_{1}+\varepsilon_{2}\},\quad\Delta_{\otimes}^{\text{pure}+}=\{\delta+\varepsilon_{1}+\varepsilon_{2}\}.

For other types of G​(𝔤)G(\mathfrak{g}), see [29, 6, 1].

4 Simply connected Weyl groupoids of Nichols algebras of diagonal type

We retain the setting of the previous section.

4.1 Nichols algebras of diagonal type

The braided monoidal category 𝒴GG​𝒟{}^{G}_{G}\mathcal{YD} (resp. 𝒴GG​𝒟fd{}^{G}_{G}\mathcal{YD}^{\mathrm{fd}}) of Yetter-Drinfeld modules (resp. finite-dimensional Yetter-Drinfeld modules) over a group GG, as well as the Nichols algebra over them, are discussed in detail in [14, 19].

Example 4.1.
  1. 1.

    Let ee be the trivial group. In this case,

    𝒴ee​𝒟fd≅Vecfd{}_{e}^{e}\mathcal{YD}^{\mathrm{fd}}\cong\text{Vec}^{\mathrm{fd}}

    as braided tensor categories, where Vecfd\text{Vec}^{\mathrm{fd}} denotes the symmetic tensor category of finite-dimensional vector spaces.

  2. 2.

    Let sVecfd\text{sVec}^{\mathrm{fd}} be a full subcategory of 𝒴ℤ/2​ℤℤ/2​ℤ​𝒟fd{}_{\mathbb{Z}/2\mathbb{Z}}^{\mathbb{Z}/2\mathbb{Z}}\mathcal{YD}^{\mathrm{fd}} consisting of ℤ/2​ℤ\mathbb{Z}/2\mathbb{Z}-graded vector spaces M=M0¯⊕M1¯M=M_{\overline{0}}\oplus M_{\overline{1}} such that M0¯M_{\overline{0}} (resp. M1¯M_{\overline{1}}) is a direct sum of the trivial (resp. sign) representations of ℤ/2​ℤ\mathbb{Z}/2\mathbb{Z}.

    Then, sVecfd\text{sVec}^{\mathrm{fd}} is a symmetric tensor category with braiding given by:

    cM,N​(m⊗n)=(−1)|m|​|n|​n⊗m,c_{M,N}(m\otimes n)=(-1)^{|m||n|}n\otimes m,

    where |m||m| and |n||n| denote the degrees (parities) of m∈Mm\in M and n∈Nn\in N.

Definition 4.2.

G​-grVecfdG\text{-grVec}^{\mathrm{fd}} denotes the tensor category of finite-dimensional GG-graded vector spaces.

Consider the forgetful functor as underlying tensor categories (but not as braided tensor categories!):

F:𝒴GG​𝒟fd≅G​-grVecfd→Vecfd.F:{}_{G}^{G}\mathcal{YD}^{\mathrm{fd}}\cong G\text{-grVec}^{\mathrm{fd}}\to\text{Vec}^{\mathrm{fd}}.

Define

dimV:=dimkF​(V).\dim V:=\dim_{k}F(V).

Let θ∈ℕ\theta\in\mathbb{N}, and set I={1,2,…,θ}I=\{1,2,\ldots,\theta\}. We denote {α1,…,αθ}\{\alpha_{1},\ldots,\alpha_{\theta}\} as the canonical ℤ\mathbb{Z}-basis of ℤθ\mathbb{Z}^{\theta}.

For a bicharacter 𝔮​(−,−):ℤθ×ℤθ→k×\mathfrak{q}(-,-):\mathbb{Z}^{\theta}\times\mathbb{Z}^{\theta}\to k^{\times}, there exists a direct sum

V=k​x1⊕⋯⊕k​xθ∈𝒴ℤθℤθ​𝒟V=kx_{1}\oplus\dots\oplus kx_{\theta}\in{{}^{\mathbb{Z}^{\theta}}_{\mathbb{Z}^{\theta}}\mathcal{YD}}

of θ\theta one-dimensional Yetter-Drinfeld modules such that the following holds:

cV,V​(xi⊗xj)=𝔮​(αi,αj)​xj⊗xii,j∈Ic_{V,V}(x_{i}\otimes x_{j})=\mathfrak{q}(\alpha_{i},\alpha_{j})x_{j}\otimes x_{i}\quad i,j\in I

We write qi​j=𝔮​(αi,αj)q_{ij}=\mathfrak{q}(\alpha_{i},\alpha_{j}).

This VV’s Nichols algebra is denoted by B𝔮B_{\mathfrak{q}}. Such Nichols algebras are called of diagonal type. The algebra and coalgebra structures of B𝔮B_{\mathfrak{q}} are fully determined by the braiding matrix (qi​j)(q_{ij}). The number θ\theta is called the rank of B𝔮B_{\mathfrak{q}}.

B𝔮B_{\mathfrak{q}} inherits a natural ℤ≥0θ\mathbb{Z}_{\geq 0}^{\theta}-grading deg⁡xi:=αi\deg x_{i}:=\alpha_{i}. This grading is compatible with both the algebra and coalgebra structures of B𝔮B_{\mathfrak{q}}, by the construction of B𝔮B_{\mathfrak{q}} as a Nichols algebra. By these definitions, B𝔮B_{\mathfrak{q}} is a bimonoid object in ℤ≥0θ​-gr​𝒴ℤθℤθ​𝒟\mathbb{Z}^{\theta}_{\geq 0}\text{-gr}{}_{\mathbb{Z}^{\theta}}^{\mathbb{Z}^{\theta}}\mathcal{YD}.

Example 4.3 (Classification of rank 1 Nichols algebras [19] Example 1.10.1).

When θ=1\theta=1, a bicharacter 𝔮\mathfrak{q} can be identified with an element q∈k×q\in k^{\times}. The graded algebra B𝔮B_{\mathfrak{q}} is classified as follows:

B𝔮≃{k​[x]/(xord⁡q)1<ord⁡q<∞,k​[x]q=1​ or ​ord⁡q=∞.B_{\mathfrak{q}}\simeq\begin{cases}k[x]/(x^{\operatorname{ord}q})&1<\operatorname{ord}q<\infty,\\ k[x]&q=1\text{ or }\operatorname{ord}q=\infty.\end{cases}
Example 4.4 ([19] Theorem 16.2.5).

Let A=(di​ai​j)A=(d_{i}a_{ij}) be a symmetrized generalized Cartan matrix. If qi​j=qdi​ai​jq_{ij}=q^{d_{i}a_{ij}} and qq is not a root of unity, then we have B𝔮≃Uq+​(𝔤A)B_{\mathfrak{q}}\simeq U_{q}^{+}(\mathfrak{g}_{A}).

Here, Uq+​(𝔤A)U_{q}^{+}(\mathfrak{g}_{A}) represents the positive part of the quantum group associated with Kac-Moody Lie algebra 𝔤A\mathfrak{g}_{A}. This result was first proven by Lusztig in [28] in the case of finite type.

Using the theory of Lyndon words, a PBW-type basis for a Nichols algebra can be constructed.

Theorem 4.5 ([1] 2.6, [25]).

For a bicharacter 𝔮\mathfrak{q}, there exists a totally ordered set (S,≤)(S,\leq) such that for each s∈Ss\in S, there exists a homogeneous element Xs∈B𝔮X_{s}\in B_{\mathfrak{q}} satisfying:

{Xl1m1​⋯​Xlkmk∣k≥0,l1<⋯<lk∈S, 0≤mi<ord⁡𝔮​(deg⁡Xli,deg⁡Xli)}\left\{X_{l_{1}}^{m_{1}}\cdots X_{l_{k}}^{m_{k}}\mid k\geq 0,\,l_{1}<\cdots<l_{k}\in S,\,0\leq m_{i}<\operatorname{ord}\mathfrak{q}(\deg X_{l_{i}},\deg X_{l_{i}})\right\}

is a basis for B𝔮B_{\mathfrak{q}}.

Proposition 4.6 ([2] Lemma 2.18).

If #​R𝔮+<∞\#R_{\mathfrak{q}}^{+}<\infty, we define R𝔮+={deg⁡(Xs)∣s∈S}⊂ℤ≥0θR_{\mathfrak{q}}^{+}=\{\deg(X_{s})\mid s\in S\}\subset\mathbb{Z}_{\geq 0}^{\theta}. Then R𝔮+R_{\mathfrak{q}}^{+} does not depend on the choice of the ordered set SS.

Corollary 4.7.

If #​R𝔮+<∞\#R_{\mathfrak{q}}^{+}<\infty, then there is a ℤ≥0θ\mathbb{Z}_{\geq 0}^{\theta}-graded Yetter-Drinfeld module isomorphism

B𝔮≃⨂α∈R𝔮+B​(k​Xα).B_{\mathfrak{q}}\simeq\bigotimes_{\alpha\in R_{\mathfrak{q}}^{+}}B(kX_{\alpha}).

Proof. This follows from Example 4.3 and Theorem 4.5. □\square

Corollary 4.8.

dimB𝔮<∞\dim B_{\mathfrak{q}}<\infty if and only if #​R𝔮+<∞\#R_{\mathfrak{q}}^{+}<\infty and 1<ord⁡𝔮​(α,α)<∞1<\operatorname{ord}\mathfrak{q}(\alpha,\alpha)<\infty for all α∈R𝔮+\alpha\in R_{\mathfrak{q}}^{+}.

4.2 Lusztig autmorphisms of small quantum groups

In this subsection, we introduce the algebras in which we are interested in. We will follow [32].

Definition 4.9.

We denote U~𝔮\tilde{U}_{\mathfrak{q}} as the Hopf algebra generated by the symbols Ki,Ki−1,Li,Li−1,EiK_{i},K_{i}^{-1},L_{i},L_{i}^{-1},E_{i}, and FiF_{i}, with i∈Ii\in I, subject to the relations:

Ki​Ej=qi​j​Ej​Ki,Li​Ej=qj​i−1​Ej​Li,K_{i}E_{j}=q_{ij}E_{j}K_{i},\quad L_{i}E_{j}=q_{ji}^{-1}E_{j}L_{i},
Ki​Fj=qi​j−1​Fj​Ki,Li​Fj=qj​i​Fj​Li,K_{i}F_{j}=q_{ij}^{-1}F_{j}K_{i},\quad L_{i}F_{j}=q_{ji}F_{j}L_{i},
Ei​Fj−Fj​Ei=δi,j​(Ki−Li),E_{i}F_{j}-F_{j}E_{i}=\delta_{i,j}(K_{i}-L_{i}),
X​Y=Y​X,Ki​Ki−1=Li​Li−1=1,XY=YX,\quad K_{i}K_{i}^{-1}=L_{i}L_{i}^{-1}=1,

for all i,j∈Ii,j\in I and X,Y∈{Ki±1,Li±1∣i∈I}X,Y\in\{K_{i}^{\pm 1},L_{i}^{\pm 1}\mid i\in I\}.

The counit ε:U𝔮→k\varepsilon:U_{\mathfrak{q}}\to k is defined as

ε​(Ki±1)=ε​(Li±1)=ε​(Ei)=ε​(Fi)=0for all ​i∈I.\varepsilon(K_{i}^{\pm 1})=\varepsilon(L_{i}^{\pm 1})=\varepsilon(E_{i})=\varepsilon(F_{i})=0\quad\text{for all }i\in I.

Let τ\tau be the algebra antiautomorphism of U~q\tilde{U}_{q} defined by

τ​(Ki)=Ki,τ​(Li)=Li,τ​(Ei)=Fi,andτ​(Fi)=Ei\tau(K_{i})=K_{i},\quad\tau(L_{i})=L_{i},\quad\tau(E_{i})=F_{i},\quad\text{and}\quad\tau(F_{i})=E_{i}

for all i∈𝕀i\in\mathbb{I}.

Let J𝔮J_{\mathfrak{q}} be the defining relation of the Nichols algebra of diagonal type, generated by EiE_{i}, which is determined by the braiding matrix (qi​j)(q_{ij}).

Let U𝔮{U}_{\mathfrak{q}} be the Hopf algebra obtained by quotienting U~q\tilde{U}_{q} by J𝔮J_{\mathfrak{q}} and τ​(J𝔮)\tau(J_{\mathfrak{q}}).

We have that U𝔮=⨁μ∈ℤθ(U𝔮)μU_{\mathfrak{q}}=\bigoplus_{\mu\in\mathbb{Z}^{\theta}}(U_{\mathfrak{q}})_{\mu} is a ℤθ\mathbb{Z}^{\theta}-graded Hopf algebra with

deg⁡Ei=−deg⁡Fi=αianddeg⁡Ki±1=deg⁡Li±1=0∀i∈I.\deg E_{i}=-\deg F_{i}=\alpha_{i}\quad\text{and}\quad\deg K_{i}^{\pm 1}=\deg L_{i}^{\pm 1}=0\quad\forall i\in I.

The multiplication of U𝔮U_{\mathfrak{q}} induces a linear isomorphism

U𝔮−⊗U𝔮0⊗U𝔮+≅U𝔮,U_{\mathfrak{q}}^{-}\otimes U_{\mathfrak{q}}^{0}\otimes U_{\mathfrak{q}}^{+}\cong U_{\mathfrak{q}},

where

U𝔮+=k​⟨Ei∣i∈I⟩≅𝔅𝔮,U𝔮0=k​⟨Ki±1,Li±1∣i∈I⟩,U𝔮−=k​⟨Fi∣i∈I⟩.U_{\mathfrak{q}}^{+}=k\langle E_{i}\mid i\in I\rangle\cong\mathfrak{B}_{\mathfrak{q}},\quad U_{\mathfrak{q}}^{0}=k\langle K_{i}^{\pm 1},L_{i}^{\pm 1}\mid i\in I\rangle,\quad U_{\mathfrak{q}}^{-}=k\langle F_{i}\mid i\in I\rangle.

are ℤθ\mathbb{Z}^{\theta}-graded subalgebras of U𝔮U_{\mathfrak{q}}. We remark that U𝔮0≅k​(ℤθ×ℤθ)U_{\mathfrak{q}}^{0}\cong k(\mathbb{Z}^{\theta}\times\mathbb{Z}^{\theta}).

Remark 4.10.

We do not require an explicit presentation of the defining relations of ℬ𝔮\mathcal{B}_{\mathfrak{q}} or the coproduct and antipode structures. All we need is the following remarkable Lusztig automorphism, which creates distinctions from the highest weight theory of more general Hopf algebras with trianglar decomposition [31].

There are variations of what is called ”small quantum groups”. However, as shown in [32, Corollary 8.17] , results on our algebra U𝔮U_{\mathfrak{q}} can be applied to a broad class of small quantum groups, such as those discussed in [27] or [26].

Theorem 4.11 ([17, 16, 5]).

Let 𝔮¯\bar{\mathfrak{q}} be a bicharacter such that B𝔮¯B_{\bar{\mathfrak{q}}} is finite-dimensional. Then, there exists a simply connected finite Cartan graph G​[𝔮¯]G[\bar{\mathfrak{q}}] with a vertex set V​(G​[𝔮¯])V(G[\bar{\mathfrak{q}}]) consisting of bicharacters with finite-dimensional Nichols algebras. For each vertex 𝔮\mathfrak{q}, there is an additive bijection between R𝔮+R_{\mathfrak{q}}^{+} (in the sense of Proposition 4.6) and R𝔮+R^{\mathfrak{q}+} (in the sense of Definition 2.4).

Moreover, for w∈HomW​(G​(𝔮¯))⁡(𝔮1,𝔮2)w\in\operatorname{Hom}_{W(G(\bar{\mathfrak{q}}))}(\mathfrak{q}_{1},\mathfrak{q}_{2}), there exists an algebra isomorphism

Tw:U𝔮1→U𝔮2T_{w}:U_{\mathfrak{q}_{1}}\to U_{\mathfrak{q}_{2}}

satisfying

Tw​((U𝔮1)α)=(U𝔮2)w​α,for any ​α∈ℤθ.T_{w}((U_{\mathfrak{q}_{1}})_{\alpha})=(U_{\mathfrak{q}_{2}})_{w\alpha},\quad\text{for any }\alpha\in\mathbb{Z}^{\theta}.
Example 4.12.

The representation theory of a small quantum group corresponding to the rank 2 Nichols algebra B𝔮B_{\mathfrak{q}} of type ufo​(7)\text{ufo}(7) is described in detail ([3]). For this 𝔮\mathfrak{q}, G​[𝔮]G[\mathfrak{q}] is the Cartan graph given in Example 2.11, and it is known that such objects do not arise from (modular) contragredient Lie (super) algebras. The ℤ2\mathbb{Z}^{2}-degree of the PBW basis of B𝔮B_{\mathfrak{q}} can be easily read from the frieze pattern in Example 2.11.

Remark 4.13.

G​[𝔮]G[\mathfrak{q}] is the simply connected cover of the small Cartan graph of 𝔮\mathfrak{q} in the sense of [19]. For the Nichols algebra 𝔅𝔮\mathfrak{B}_{\mathfrak{q}} of super type with the same Weyl groupoid as the basic Lie superalgebra 𝔤\mathfrak{g}, we have G​[𝔮]=S​C​(G​(𝔤))G[\mathfrak{q}]=SC(G(\mathfrak{g})).

Definition 4.14.

For a bicharacter 𝔮\mathfrak{q} with finite-dimensional Nichols algebra, we define the rainbow boomerang graph R​B​[𝔮]:=R​B​(G​[𝔮])RB[\mathfrak{q}]:=RB(G[\mathfrak{q}]) (Definition 2.24). Note that G​[𝔮]G[\mathfrak{q}] is simply connected, so it is trivially path simply connected.

4.3 Homomorphisms between Verma modules

For α=n1​α1+⋯+nθ​αθ∈ℤθ\alpha=n_{1}\alpha_{1}+\cdots+n_{\theta}\alpha_{\theta}\in\mathbb{Z}^{\theta}, we set

Kα=K1n1​⋯​KθnθandLα=L1n1​⋯​Lθnθ.K_{\alpha}=K_{1}^{n_{1}}\cdots K_{\theta}^{n_{\theta}}\quad\text{and}\quad L_{\alpha}=L_{1}^{n_{1}}\cdots L_{\theta}^{n_{\theta}}.

In particular, Kαi=KiK_{\alpha_{i}}=K_{i} for i∈Ii\in I.

Definition 4.15.

Fix a bicharacter 𝔮¯:ℤθ×ℤθ→k×.\bar{\mathfrak{q}}:\mathbb{Z}^{\theta}\times\mathbb{Z}^{\theta}\to k^{\times}. If w∈HomW​(G​[𝔮¯])⁡(𝔮,𝔮¯)w\in\operatorname{Hom}_{W(G[\bar{\mathfrak{q}}])}(\mathfrak{q},\bar{\mathfrak{q}}), then the triangular decomposition of U𝔮U_{\mathfrak{q}} induces a new triangular decomposition on U𝔮¯U_{\bar{\mathfrak{q}}}. Explicitly,

Tw​(U𝔮−)⊗U𝔮¯0⊗Tw​(U𝔮+)≅U𝔮¯,T_{w}(U_{\mathfrak{q}}^{-})\otimes U_{\bar{\mathfrak{q}}}^{0}\otimes T_{w}(U_{\mathfrak{q}}^{+})\cong U_{\bar{\mathfrak{q}}},

since Tw​(U𝔮0)=U𝔮¯0T_{w}(U_{\mathfrak{q}}^{0})=U_{\bar{\mathfrak{q}}}^{0}.

Given λ∈ℤθ\lambda\in\mathbb{Z}^{\theta}, we consider kλ𝔮=k​vλ𝔮k_{\lambda}^{\mathfrak{q}}=kv_{\lambda}^{\mathfrak{q}} as a ℤθ\mathbb{Z}^{\theta}-graded U𝔮¯0​Tw​(U𝔮+)U_{\bar{\mathfrak{q}}}^{0}T_{w}(U_{\mathfrak{q}}^{+})-module concentrated in degree λ\lambda with the action

Kα​Lβ​u​vλ𝔮=ε​(u)​𝔮¯​(α,λ)𝔮¯​(λ,β)​vλ𝔮,∀Kα​Lβ∈U𝔮¯0,u∈Tw​(U𝔮+).K_{\alpha}L_{\beta}uv_{\lambda}^{\mathfrak{q}}=\varepsilon(u)\frac{\mathfrak{\bar{q}}(\alpha,\lambda)}{\mathfrak{\bar{q}}(\lambda,\beta)}v_{\lambda}^{\mathfrak{q}},\quad\forall K_{\alpha}L_{\beta}\in U_{\bar{\mathfrak{q}}}^{0},\,u\in T_{w}(U_{\mathfrak{q}}^{+}).

Osing this, we introduce the ℤθ\mathbb{Z}^{\theta}-graded U𝔮¯U_{\bar{\mathfrak{q}}}-module

M𝔮¯​(λ)=U𝔮¯⊗U𝔮¯0​Tw​(U𝔮+)kλ𝔮.M^{\mathfrak{\bar{q}}}(\lambda)=U_{\bar{\mathfrak{q}}}\otimes_{U_{\bar{\mathfrak{q}}}^{0}T_{w}(U_{\mathfrak{q}}^{+})}k_{\lambda}^{\mathfrak{q}}.

Note that for all v∈(U𝔮¯)λv\in(U_{\mathfrak{\bar{q}}})_{\lambda}, we can compute:

Kα​Lβ​v=𝔮¯​(α,λ)𝔮¯​(λ,β)​v​Kα​Lβ,K_{\alpha}L_{\beta}v=\frac{\mathfrak{\bar{q}}(\alpha,\lambda)}{\mathfrak{\bar{q}}(\lambda,\beta)}vK_{\alpha}L_{\beta},

(see [32][((4.3)).

We also define an analog of the Weyl vector. (It differs from the one for [32] by a factor of −1-1.) Specifically, we define:

ρ𝔮:=−12​∑β∈R𝔮+(ord⁡𝔮​(β,β)−1)​β.\rho^{\mathfrak{q}}:=-\frac{1}{2}\sum_{\beta\in R_{\mathfrak{q}}^{+}}(\operatorname{ord}\mathfrak{q}(\beta,\beta)-1)\beta.
Remark 4.16.

Instead of our special kλ𝔮k_{\lambda}^{\mathfrak{q}}, we could consider a more general situation. However, by [32, Proposition 5.5], all blocks are equivalent to the block containing our Verma module (the principal block). Thus, for simplicity, we restrict our discussion to this case.

Our Verma module corresponds to so called a type I representation when U𝔮¯U_{\bar{\mathfrak{q}}} is of the classical Drinfeld-Jimbo type.

We consider the category ℤθ​-gr​(U𝔮¯​-Mod)\mathbb{Z}^{\theta}\text{-gr}(U_{\bar{\mathfrak{q}}}\text{-Mod}), where morphisms respects this ℤθ\mathbb{Z}^{\theta}-grading. (This is the module category of a monoid object in the category of ℤθ​-graded vector spaces\mathbb{Z}^{\theta}\text{-graded vector spaces} in the sense of [14].)

Let M=⨁ν∈ℤθMνM=\bigoplus_{\nu\in\mathbb{Z}^{\theta}}M_{\nu} be a ℤθ\mathbb{Z}^{\theta}-graded vector space. The formal character of MM is defined as:

ch​M=∑μ∈ℤθdim​Mμ​eμ.\mathrm{ch}\,M=\sum_{\mu\in\mathbb{Z}^{\theta}}\mathrm{dim}M_{\mu}e^{\mu}.

The following can be shown in the same way as in [21]. Note that

ch​U𝔮−=∏β∈R𝔮+1−e−ord⁡𝔮¯​(β,β)​β1−e−β=∏β∈R𝔮+(1+e−β+⋯+e(1−ord⁡𝔮¯​(β,β))​β).\mathrm{ch}\,U_{\mathfrak{q}}^{-}=\prod_{\beta\in R_{\mathfrak{q}}^{+}}\frac{1-e^{-\operatorname{ord}\mathfrak{\bar{q}}(\beta,\beta)\beta}}{1-e^{-\beta}}=\prod_{\beta\in R_{\mathfrak{q}}^{+}}\left(1+e^{-\beta}+\cdots+e^{(1-\operatorname{ord}\mathfrak{\bar{q}}(\beta,\beta))\beta}\right).
Proposition 4.17.

[32, Lemma 6.1] Let w∈HomW​(G​[𝔮¯])⁡(𝔮,𝔮¯)w\in\operatorname{Hom}_{W(G[\bar{\mathfrak{q}}])}(\mathfrak{q},\bar{\mathfrak{q}}) and w′∈HomW​(G​[𝔮¯])⁡(𝔮′,𝔮¯)w^{\prime}\in\operatorname{Hom}_{W(G[\bar{\mathfrak{q}}])}(\mathfrak{q}^{\prime},\bar{\mathfrak{q}}). For a pair of vertices 𝔮,𝔮′\mathfrak{q},\mathfrak{q}^{\prime} in G​[𝔮¯]G[\bar{\mathfrak{q}}] and λ∈𝔥∗\lambda\in\mathfrak{h}^{*}, the following statements hold:

  1. 1.

    ch⁡M𝔮​(λ−w​ρ𝔮)=ch⁡M𝔮′​(λ−w′​ρ𝔮′)\operatorname{ch}M^{\mathfrak{q}}(\lambda-w\rho^{\mathfrak{q}})=\operatorname{ch}M^{\mathfrak{q}^{\prime}}(\lambda-w^{\prime}\rho^{\mathfrak{q}^{\prime}}).

  2. 2.

    dimM𝔮′​(λ−w′​ρ𝔮′)λ−w​ρ𝔮=1\dim M^{\mathfrak{q}^{\prime}}(\lambda-w^{\prime}\rho^{\mathfrak{q}^{\prime}})_{\lambda-w\rho^{\mathfrak{q}}}=1.

  3. 3.

    dimHom⁡(M𝔮​(λ−w​ρ𝔮),M𝔮′​(λ−w′​ρ𝔮′))=1\dim\operatorname{Hom}(M^{\mathfrak{q}}(\lambda-w\rho^{\mathfrak{q}}),M^{\mathfrak{q}^{\prime}}(\lambda-w^{\prime}\rho^{\mathfrak{q}^{\prime}}))=1.

Definition 4.18.

For λ∈ℤθ\lambda\in\mathbb{Z}^{\theta}, we denote a nonzero homomorphism from M𝔮​(λ−w​ρ𝔮)M^{\mathfrak{q}}(\lambda-w\rho^{\mathfrak{q}}) to M𝔮′​(λ−w′​ρ𝔮′)M^{\mathfrak{q}^{\prime}}(\lambda-w^{\prime}\rho^{\mathfrak{q}^{\prime}}) by ψλ𝔮​𝔮′\psi_{\lambda}^{\mathfrak{q}\mathfrak{q}^{\prime}}. Let the highest weight vector of M𝔮​(λ)M^{\mathfrak{q}}(\lambda) be vλ𝔮v_{\lambda}^{\mathfrak{q}}.

Definition 4.19.

For q∈k×q\in k^{\times} and n∈ℕn\in\mathbb{N}, we recall the quantum numbers

(n)q=∑j=0n−1qj(n)_{q}=\sum_{j=0}^{n-1}q^{j}

.

Proposition 4.20.

[32, Section 7.1] For λ∈ℤθ\lambda\in\mathbb{Z}^{\theta}, exactly one of the following holds:

  1. 1.

    ψλri​𝔮,𝔮\psi_{\lambda}^{r_{i}\mathfrak{q},\mathfrak{q}} and ψλ𝔮,ri​𝔮\psi_{\lambda}^{\mathfrak{q},r_{i}\mathfrak{q}} are isomorphisms.

  2. 2.
    ψλri​𝔮,𝔮∘ψλ𝔮,ri​𝔮=ψλ𝔮,ri​𝔮∘ψλri​𝔮,𝔮=0.\psi_{\lambda}^{r_{i}\mathfrak{q},\mathfrak{q}}\circ\psi_{\lambda}^{\mathfrak{q},r_{i}\mathfrak{q}}=\psi_{\lambda}^{\mathfrak{q},r_{i}\mathfrak{q}}\circ\psi_{\lambda}^{r_{i}\mathfrak{q},\mathfrak{q}}=0.

Proof. By the Lusztig automorphism, the case of positive roots can be reduced to that of simple roots, so we may assume 𝔮=𝔮¯\mathfrak{q}=\overline{\mathfrak{q}}. (This is formalized in [32, Section 7.3][Section 7.3] by constructing a suitable category equivalence.)

From the defining relations, we can compute:

Ei​Fin=Fin​Ei+Fin−1​((n)qi​i−1​Ki−(n)qi​i​Li),Fi​Ein=Ein​Fi+Ein−1​((n)qi​i−1​Li−(n)qi​i​Ki),\begin{split}E_{i}F_{i}^{n}&=F_{i}^{n}E_{i}+F_{i}^{n-1}\left((n)_{q_{ii}^{-1}}K_{i}-(n)_{q_{ii}}L_{i}\right),\\ F_{i}E_{i}^{n}&=E_{i}^{n}F_{i}+E_{i}^{n-1}\left((n)_{q_{ii}^{-1}}L_{i}-(n)_{q_{ii}}K_{i}\right),\end{split} (1)

, we calculate as follows:

ψλri​𝔮,𝔮∘ψλ𝔮,ri​𝔮​(vλ−ρ𝔮𝔮)\displaystyle\psi_{\lambda}^{r_{i}\mathfrak{q},\mathfrak{q}}\circ\psi_{\lambda}^{\mathfrak{q},r_{i}\mathfrak{q}}\left(v_{\lambda-\rho^{\mathfrak{q}}}^{\mathfrak{q}}\right) =ψλri​𝔮,𝔮​(Eiord⁡qi​i−1​vλ−si​ρri​𝔮ri​𝔮)\displaystyle=\psi_{\lambda}^{r_{i}\mathfrak{q},\mathfrak{q}}\left(E_{i}^{\operatorname{ord}q_{ii}-1}v_{\lambda-s_{i}\rho^{r_{i}\mathfrak{q}}}^{r_{i}\mathfrak{q}}\right)
=Eiord⁡qi​i−1​Fiord⁡qi​i−1​(vλ−ρ𝔮𝔮)\displaystyle=E_{i}^{\operatorname{ord}q_{ii}-1}F_{i}^{\operatorname{ord}q_{ii}-1}\left(v_{\lambda-\rho^{\mathfrak{q}}}^{\mathfrak{q}}\right)
=∏n=1ord⁡qi​i−1((n)qi​i−1​𝔮¯​(αi,λ)−(n)qi​i​𝔮¯​(λ,αi)−1)​vλ−ρ𝔮𝔮.\displaystyle=\prod_{n=1}^{\operatorname{ord}q_{ii}-1}\left((n)_{q_{ii}^{-1}}\bar{\mathfrak{q}}(\alpha_{i},\lambda)-(n)_{q_{ii}}\bar{\mathfrak{q}}(\lambda,\alpha_{i})^{-1}\right)v_{\lambda-\rho^{\mathfrak{q}}}^{\mathfrak{q}}.

Similarly, we have

ψλ𝔮,ri​𝔮∘ψλri​𝔮,𝔮​(vλ−si​ρri​𝔮ri​𝔮)=∏n=1ord⁡qi​i−1((n)qi​i−1​𝔮¯​(λ,αi)−1−(n)qi​i​𝔮¯​(αi,λ))​vλ−si​ρri​𝔮ri​𝔮.\psi_{\lambda}^{\mathfrak{q},r_{i}\mathfrak{q}}\circ\psi_{\lambda}^{r_{i}\mathfrak{q},\mathfrak{q}}\left(v_{\lambda-s_{i}\rho^{r_{i}\mathfrak{q}}}^{r_{i}\mathfrak{q}}\right)=\prod_{n=1}^{\operatorname{ord}q_{ii}-1}\left((n)_{q_{ii}^{-1}}\bar{\mathfrak{q}}(\lambda,\alpha_{i})^{-1}-(n)_{q_{ii}}\bar{\mathfrak{q}}(\alpha_{i},\lambda)\right)v_{\lambda-s_{i}\rho^{r_{i}\mathfrak{q}}}^{r_{i}\mathfrak{q}}.

Finally, we observe:

((n)qi​i−1​𝔮¯​(αi,λ)−(n)qi​i​𝔮¯​(λ,αi)−1)=0⇔((n)qi​i−1​𝔮¯​(λ,αi)−1−(n)qi​i​𝔮¯​(αi,λ))=0,\left((n)_{q_{ii}^{-1}}\bar{\mathfrak{q}}(\alpha_{i},\lambda)-(n)_{q_{ii}}\bar{\mathfrak{q}}(\lambda,\alpha_{i})^{-1}\right)=0\iff\left((n)_{q_{ii}^{-1}}\bar{\mathfrak{q}}(\lambda,\alpha_{i})^{-1}-(n)_{q_{ii}}\bar{\mathfrak{q}}(\alpha_{i},\lambda)\right)=0,

by the identity (n)q=qn−1​(n)q−1(n)_{q}=q^{n-1}(n)_{q^{-1}}.

□\square

Remark 4.21.

The ability to provide a more detailed description of the Verma module associated with the bicharacter corresponding to the longest element is a phenomenon unique to the finite-dimensional case, which led to stronger results ([32, Lemma 6.4])

Here, following [21], we define the quotient of the rainbow boomerang graph.

Definition 4.22.

[21, Definition 2.14] Let GG be an edge-colored graph with color set CC. Let D⊆CD\subseteq C. We define an equivalence relation ∼D\sim_{D} on VV as follows: For x,y∈Vx,y\in V, we say x∼Dyx\sim_{D}y if there exists a walk from xx to yy consisting only of edges with colors in DD. We denote the equivalence class of xx by [x][x].

We define the edge-colored graph G/DG/D as follows:

  • •

    The vertex set is V/∼DV/\sim_{D}, the set of equivalence classes under ∼D\sim_{D}.

  • •

    The color set is C∖DC\setminus D.

  • •

    There is an edge of color c∈C∖Dc\in C\setminus D between [x][x] and [y][y] in G/DG/D if and only if there exist u∈[x]u\in[x] and v∈[y]v\in[y] such that there is an edge of color cc between uu and vv in GG.

Given a walk WW in GG:

v0​c0​v1​c1​…​ck−1​vk,v_{0}c_{0}v_{1}c_{1}\dotsc c_{k-1}v_{k},

we define the induced walk W¯\overline{W} in G/DG/D as:

[v0]​[c0]​[v1]​[c1]​…​[ck−1]​[vk],[v_{0}][c_{0}][v_{1}][c_{1}]\dotsc[c_{k-1}][v_{k}],

where [ci]=ci[c_{i}]=c_{i} if [vi]≠[vi+1][v_{i}]\neq[v_{i+1}], and [ci][c_{i}] represents an empty walk if [vi]=[vi+1][v_{i}]=[v_{i+1}].

Proposition 4.23.

[21, Propositlon 2.15] The graph G/DG/D is a rainbow boomerang graph.

Definition 4.24.

We identify the color set of R​B​[𝔮¯]RB[\bar{\mathfrak{q}}] with R𝔮¯+R_{\bar{\mathfrak{q}}}^{+}.

For λ∈ℤθ\lambda\in\mathbb{Z}^{\theta}, let DλD_{\lambda} denote the collection of roots α\alpha in R𝔮¯+R_{\bar{\mathfrak{q}}}^{+} such that

∏n=1ord⁡q​(α,α)−1((n)q​(α,α)−1​𝔮¯​(α,λ)−(n)q​(α,α)​𝔮¯​(λ,α)−1)≠0.\prod_{n=1}^{\operatorname{ord}q(\alpha,\alpha)-1}\left((n)_{q(\alpha,\alpha)^{-1}}\bar{\mathfrak{q}}(\alpha,\lambda)-(n)_{q(\alpha,\alpha)}\bar{\mathfrak{q}}(\lambda,\alpha)^{-1}\right)\neq 0.

We set R​B​[𝔮¯,λ]:=R​B​[𝔮¯]/DλRB[\bar{\mathfrak{q}},\lambda]:=RB[\bar{\mathfrak{q}}]/D_{\lambda}.

The following is exactly the same as in [21, Corollary 3.27].

Proposition 4.25.

The vertex set of R​B​[𝔮¯,λ]RB[\bar{\mathfrak{q}},\lambda] can be identified with the isomorphism classes of {M𝔮​(λ−ρ𝔮)}𝔮∈V​(G​[𝔮¯])\{M^{\mathfrak{q}}(\lambda-\rho^{\mathfrak{q}})\}_{\mathfrak{q}\in V(G[\bar{\mathfrak{q}}])}.

Definition 4.26.

Let w=𝔮0​c1​𝔮1​…​ct​𝔮tw=\mathfrak{q}_{0}c_{1}\mathfrak{q}_{1}\dots c_{t}\mathfrak{q}_{t} be a walk in R​B​[𝔮¯,λ]RB[\bar{\mathfrak{q}},\lambda]. Take wi∈HomW​(G​[𝔮¯])⁡(𝔮i,𝔮¯)w_{i}\in\operatorname{Hom}_{W(G[\bar{\mathfrak{q}}])}(\mathfrak{q}_{i},\bar{\mathfrak{q}}) for i=0,1,…,ti=0,1,\dots,t.

The corresponding composition of nonzero homomorphisms

M𝔮0​(λ−w0​ρ𝔮0)→ψλ𝔮0,𝔮1M𝔮1​(λ−w1​ρ𝔮1)→ψλ𝔮1,𝔮2⋯M^{\mathfrak{q}_{0}}(\lambda-w_{0}\rho^{\mathfrak{q}_{0}})\xrightarrow{\psi_{\lambda}^{\mathfrak{q}_{0},\mathfrak{q}_{1}}}M^{\mathfrak{q}_{1}}(\lambda-w_{1}\rho^{\mathfrak{q}_{1}})\xrightarrow{\psi_{\lambda}^{\mathfrak{q}_{1},\mathfrak{q}_{2}}}\cdots
⋯→ψλ𝔮t−2,𝔮t−1M𝔮t−1​(λ−wt−1​ρ𝔮t−1)→ψλ𝔮t−1,𝔮tM𝔮t​(λ−wt​ρ𝔮t).\cdots\xrightarrow{\psi_{\lambda}^{\mathfrak{q}_{t-2},\mathfrak{q}_{t-1}}}M^{\mathfrak{q}_{t-1}}(\lambda-w_{t-1}\rho^{\mathfrak{q}_{t-1}})\xrightarrow{\psi_{\lambda}^{\mathfrak{q}_{t-1},\mathfrak{q}_{t}}}M^{\mathfrak{q}_{t}}(\lambda-w_{t}\rho^{\mathfrak{q}_{t}}).

is denoted by ψλw\psi_{\lambda}^{w}.

The following theorem is an analogue of [21, Theorem 4.9], which we have been aiming for.

Theorem 4.27.

Let λ∈ℤθ\lambda\in\mathbb{Z}^{\theta}. For a walk ww in R​B​[𝔮¯,λ]RB[\bar{\mathfrak{q}},\lambda], the following are equivalent:

  1. 1.

    ψλw≠0\psi_{\lambda}^{w}\neq 0.

  2. 2.

    ww is rainbow.

  3. 3.

    ww is shortest.

Proof. Using the discussion in this subsection, the argument proceeds exactly as in Section 4 in [21]. □\square

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Shunsuke Hirota
Department of Mathematics, Kyoto University
Kitashirakawa Oiwake-cho, Sakyo-ku, 606-8502, Kyoto
E-mail address: [email protected]