Root Systems in Lie Theory: From the Classic Definition to Nowadays

1. Root Systems: The Origin

The purpose of this article is to discuss the role played by root systems in the theory of Lie algebras and related objects in representation theory, with focus on the combinatorial description and properties.

1.1. Semisimple Lie algebras

The study of Lie algebras began toward the end of the 19th century. They emerged as the algebraic counterpart of a purely geometric object: Lie groups, which we can briefly define as groups that admit a differentiable structure such that multiplication and the function that computes inverses are differentiable. Lie algebras appeared as some algebraic structure attached to the tangent space of the unit of this group.

Initially Lie algebras were only considered over complex or real numbers, but the abstraction of the definition led to Lie algebras over arbitrary fields.

Definition 1.1.

Let $\mathbb{k}$ be a field. A Lie algebra is a pair $(\mathfrak{g}, [,])$, where $\mathfrak{g}$ is a $\mathbb{k}$-vector space and $[,]:\mathfrak{g}\times \mathfrak{g}\to \mathfrak{g}$ is a bilinear map (called the bracket) such that the following equalities hold for all $x,y,z\in \mathfrak{g}$:

$$\begin{align*} [x,y]&= -[y,x], & &\text{antisymmetry}, \\ [x,[y,z]] &= [[x,y],z]+[y,[x,z]], & &\text{Jacobi identity}. \end{align*}$$

There is a subtle difference when the field is of characteristic two: the antisymmetry is replaced by $[x,x]=0$ for all $x\in \mathfrak{g}$ (which implies the former one). From now on all Lie algebras considered here are assumed to be finite-dimensional.

An easy example is to pick a vector space $\mathfrak{g}$ together with trivial bracket $[x,y]=0$ for all $x,y\in \mathfrak{g}$; these Lie algebras are called abelian.

There is a general way to move from an associative algebra $A$ to a Lie algebra: take $\mathfrak{g}=A$ as vector space and set $[a,b]\coloneq ab-ba$ for each pair $a,b\in A$. A prominent example of this construction is the general linear algebra $\mathfrak{gl}(V)$, which is the set of linear endomorphisms of a finite-dimensional vector space $V$. Other classical examples appear as Lie subalgebras (that is, subspaces closed under the bracket) of $\mathfrak{gl}(V)$:

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$\mathfrak{sl}(V)$, those endomorphisms whose trace is zero; if $V=\mathbb{k}^n$, then we simply denote $\mathfrak{sl}(V)$ by $\mathfrak{sl}(n,\mathbb{k})$, or $\mathfrak{sl}(n)$ when the field $\mathbb{k}$ is clear from the context.

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The orthogonal and symplectic Lie subalgebras $\mathfrak{so}(V,b)$, respectively $\mathfrak{sp}(V,b)$, of those endomorphisms $T$ such that$$\begin{align*} b\left(T(v),w\right)+b\left(v,T(w)\right)&=0 &\text{for all }&v,w\in V, \end{align*}$$

where $b$ is a symmetric, respectively antisymmetric, nondegenerate bilinear form on $V$.

Analogously, we may start with $A=\mathbb{k}^{n\times n}$, the algebra of $n\times n$ matrices, and take some subalgebras, as the subspaces of upper triangular matrices, those of trace 0, the orthogonal matrices, between others.

Once we have a notion of algebra, it is natural to ask for ideals: in the case of Lie algebras, these are subspaces $\mathfrak{I}\subseteq \mathfrak{g}$ such that $[\mathfrak{I},\mathfrak{g}]\subseteq \mathfrak{I}$. This leads to consider simple Lie algebras, those Lie algebras $\mathfrak{g}$ such that $\dim \mathfrak{g}>1$ and the unique ideals are the trivial ones: $0$ and $\mathfrak{g}$. In addition, we say that a Lie algebra $\mathfrak{g}$ is semisimple if $\mathfrak{g}$ is isomorphic to the direct sum of simple Lie algebras.

For the rest of this section we fix $\mathbb{k}=\mathbb{C}$. We know that a Lie algebra $\mathfrak{g}$ is simple if and only if $\mathfrak{g}$ is (isomorphic to) $\mathfrak{sl}(V)$, $\mathfrak{so}(V,b)$, $\mathfrak{sp}(V,b)$, and a few exceptional examples $E_k$, $k=6,7,8$, $F_4$, $G_2$. That is, up to 5 exceptions, all the complex simple Lie algebras are subalgebras of matrices. Thus one may wonder if some properties of the algebras of matrices still hold for simple Lie algebras. We will recall some of them by the end of this section, following Hum78.

As for associative algebras, we can study modules over Lie algebras. A $\mathfrak{g}$-module is a pair $(V,\cdot )$, where $V$ is a $\mathbb{k}$-vector space and $\cdot :\mathfrak{g}\otimes V\to V$ is a linear map such that

$$\begin{align*} [x,y]\cdot v &= x\cdot (y\cdot v)-y\cdot (x\cdot v), &&x,y\in \mathfrak{g}, v\in V. \end{align*}$$

For example, the bracket gives an action of $\mathfrak{g}$ over itself, called the adjoint action.

For each $x\in \mathfrak{g}$ we look at the inner derivation

$$\begin{align*} \operatorname {ad}x:&\mathfrak{g}\to \mathfrak{g}, & \operatorname {ad}x(y)&=[x,y], & &y\in \mathfrak{g}, \end{align*}$$

associated to the adjoint action. These endomorphisms induce a symmetric bilinear form on $\mathfrak{g}$, called the Killing form:

$$\begin{align*} \kappa & :\mathfrak{g}\times \mathfrak{g}\to \mathbb{k}, & \kappa (x,y) &\coloneq \operatorname {Tr}(\operatorname {ad}x \operatorname {ad}y), & & x,y\in \mathfrak{g}. \end{align*}$$

The Killing form and the $\mathfrak{g}$-modules give other characterizations of semisimplicity: $\mathfrak{g}$ is semisimple if and only if $\kappa$ is nondegenerate if and only if every module is semisimple, i.e., every $\mathfrak{g}$-submodule admits a complement which is a $\mathfrak{g}$-submodule.

When $\mathfrak{g}$ is one of the Lie algebras of matrices above, the action of diagonal matrices is, in fact, diagonalizable. Mimicking this fact we look for subalgebras such that the action of their elements is diagonalizable, called toral subalgebras.

From now on assume that $\mathfrak{g}$ is also semisimple. It can be shown that toral algebras are abelian, and we pick a maximal one ${\mathfrak{h}}$. Thus $\mathfrak{g}$ decomposes as the direct sum of the ${\mathfrak{h}}$-eigenspaces:

$$\begin{align*} \mathfrak{g}&=\oplus _{\alpha \in {\mathfrak{h}}^*}\mathfrak{g}_{\alpha }, &&\text{where }\mathfrak{g}_{\alpha }\coloneq \{x\in \mathfrak{g}| [h,x]=\alpha (h)x\}. \end{align*}$$

As ${\mathfrak{h}}$ is abelian, we have that ${\mathfrak{h}}\subseteq \mathfrak{g}_0$: one can show that we have an equality, ${\mathfrak{h}}=\mathfrak{g}_0$. Thus, if we set $\Delta \coloneq \{\alpha \in {\mathfrak{h}}^*| \alpha \ne 0, \mathfrak{g}_{\alpha } \ne 0 \}$, then $\Delta$, a finite set called the root system of $\mathfrak{g}$, gives a decomposition of $\mathfrak{g}$ into ${\mathfrak{h}}$-eigenspaces as follows:

$$\begin{equation*} \mathfrak{g}={\mathfrak{h}}\oplus \left( \oplus _{\alpha \in \Delta } \mathfrak{g}_{\alpha } \right). \end{equation*}$$

This decomposition is compatible with the bracket,

$$\begin{align*} [\mathfrak{g}_{\alpha },\mathfrak{g}_{\beta }]&\subseteq \mathfrak{g}_{\alpha +\beta } && \text{for all }\alpha ,\beta \in {\mathfrak{h}}^*, \end{align*}$$

and the Killing form

$$\begin{align*} \kappa _{\mathfrak{g}_{\alpha }\times \mathfrak{g}_{\beta }}&=0 &\text{if }&\alpha +\beta \ne 0. \end{align*}$$

We can derive that $\kappa _{{\mathfrak{h}}\times {\mathfrak{h}}}$ is nondegenerate, thus it induces a symmetric nondegenerate bilinear form $(\cdot ,\cdot ):{\mathfrak{h}}^*\times {\mathfrak{h}}^*\to \mathbb{C}$.

Example 1.2.

If $\mathfrak{g}=\mathfrak{sl}(4)$, the Lie algebra of $4\times 4$ matrices with trace 0, then ${\mathfrak{h}}$ is the subspace of diagonal matrices, with basis $h_i\coloneq E_{ii}-E_{i+1,i+1}$, $i=1,2,3$. Here, $E_{ij}$ is the matrix with 1 in the $(i,j)$-entry and 0 otherwise. Let $A\coloneq \left[\begin{smallmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{smallmatrix}\right]$, and set $\alpha _j\in {\mathfrak{h}}^*$ as the element such that $\alpha _j(h_i)=a_{ij}$. Then

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$e_i\coloneq E_{i,i+1}\in \mathfrak{g}_{\alpha _i}$, $i=1,2,3$;

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$E_{13}\in \mathfrak{g}_{\alpha _1+\alpha _2}$, $E_{24}\in \mathfrak{g}_{\alpha _2+\alpha _3}$, $E_{14}\in \mathfrak{g}_{\alpha _1+\alpha _2+\alpha _3}$;

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for all $i<j$, if $E_{ij}\in \mathfrak{g}_{\alpha }$, then $E_{ji}\in \mathfrak{g}_{-\alpha }$. In particular, $f_i\coloneq E_{i+1,i}\in \mathfrak{g}_{-\alpha _i}$, $i=1,2,3$.

Thus, if we set $\alpha _{ij}\coloneq \sum _{k=i}^j\alpha _k$, $i\le j$, then

$$\begin{equation*} \Delta =\{ \pm \alpha _{ij} | 1\le i\le j\le 3 \}. \end{equation*}$$

This example has a straightforward generalization to $\mathfrak{sl}(n)$ for any $n\ge 2$.

1.2. Root systems for Lie algebras

We may derive strong properties of the root system $\Delta$ using the representation theory of $\mathfrak{sl}(2)$, we refer to Bou02Hum78 for more details.

(i)

${\mathfrak{h}}^*$ is spanned by $\Delta$.

(ii)

If $\alpha \in \Delta$, then $-\alpha \in \Delta$. Moreover, for each $\alpha \in \Delta$, $\Delta \cap \mathbb{C}\alpha =\{\pm \alpha \}$.

(iii)

For each $\alpha \in \Delta$, the eigenspace $\mathfrak{g}_{\alpha }$ is one-dimensional. Moreover, $S_{\alpha }\coloneq \mathfrak{g}_{\alpha }\oplus \mathfrak{g}_{-\alpha }\oplus [\mathfrak{g}_{\alpha },\mathfrak{g}_{-\alpha }]$ is a subalgebra isomorphic to $\mathfrak{sl}(2)$. Notice that $[\mathfrak{g}_{\alpha },\mathfrak{g}_{-\alpha }]\subset {\mathfrak{h}}$.

(iv)

If $\alpha ,\beta ,\alpha +\beta \in \Delta$, then $[\mathfrak{g}_{\alpha },\mathfrak{g}_{\beta }]=\mathfrak{g}_{\alpha +\beta }$.

(v)

Let $\alpha ,\beta \in \Delta$ be such that $\alpha \ne \pm \beta$. Then there exist $q,r\in \mathbb{N}_0$ such that$$\begin{align*} \{i\in \mathbb{Z}|\beta +i\alpha \in \Delta \}&=\{-r\le i\le q\}. \end{align*}$$

Moreover, $r-q=\tfrac{2(\beta ,\alpha )}{(\alpha ,\alpha )}$. That is, the root string over $\beta$ in the direction of $\alpha$ has no holes.

By (i) there exists a basis $B$ of ${\mathfrak{h}}^*$ contained in $\Delta$. We can check that all the coefficients of any $\beta \in \Delta$, written in terms of $B$, are rational numbers, so we may consider the $\mathbb{Q}$-linear subspace ${\mathfrak{h}}^*_{\mathbb{Q}}$ generated by $\Delta$ and take the extension to $\mathbb{R}$: we get a finite-dimensional $\mathbb{R}$-vector space $V$ which contains all the information and the geometry of $\Delta$.

Remark 1.3.

$V$ becomes an Euclidean vector space with the scalar product induced by the Killing form.

For each $\alpha \in \Delta$ set $s_{\alpha }:V\to V$,

$$\begin{align*} s_{\alpha }(\beta )&=\beta -\tfrac{2(\beta ,\alpha )}{(\alpha ,\alpha )}\alpha , & &\beta \in V. \end{align*}$$

Then $s_{\alpha }$ is a linear automorphism of the Euclidean space $V$ such that $s_{\alpha }^2=\operatorname {id}$, and by (v), $s_{\alpha }(\Delta )=\Delta$.

2. Classical Root Systems

From the information above one may wonder if there exists an abstract notion of root system. The answer is yes, and we will recall it following Bou02, see also Hum78. We can classify all finite root systems in terms of so-called finite Cartan matrices. We will also recall a way to come back from (abstract) root systems to complex Lie algebras.

2.1. Abstract definition

Definition 2.1 (Bou02).

Let $V$ be a finite-dimensional $\mathbb{R}$-vector space. A finite subset $\varDelta \subset V$ is a root system in $V$ if

(RS1)

$0\notin \varDelta$ and $V$ is spanned by $\varDelta$.

(RS2)

For each $\alpha \in \varDelta$, there exists $\alpha ^{\vee }\in V^*$ such that $\alpha ^{\vee }(\alpha )=2$ and the reflection$$\begin{align*} s_{\alpha } &:V\to V, & s_{\alpha }(\beta )&=\beta -\alpha ^{\vee }(\beta )\alpha , & &\beta \in V, \end{align*}$$

satisfies that $s_{\alpha }(\varDelta )=\varDelta$.

(RS3)

For all $\alpha ,\beta \in \varDelta$, $\alpha ^{\vee }(\beta )\in \mathbb{Z}$.

The elements of $\varDelta$ are called roots, and $\dim V$ is the rank of $\varDelta$. The (finite) subgroup ${\mathcal{W}}$ of $\operatorname {Aut}(V)$ generated by $s_{\alpha }$, $\alpha \in \varDelta$, is the Weyl group of $\varDelta$.

In Hum78 one also requires that for each $\alpha \in \varDelta$, $\varDelta \cap \mathbb{R}\alpha =\{\pm \alpha \}$. In other references, root systems with this extra property are called reduced.

The reflections $s_{\alpha }$, $\alpha \in \varDelta$ are univocally determined and there exists a symmetric invariant nondegenerate bilinear form $(\cdot |\cdot ):V\times V\to \mathbb{R}$, which is moreover invariant by ${\mathcal{W}}$ and positive definite. Now, the elements $\alpha ^{\vee }$ are recovered using this form:

$$\begin{equation*} \alpha ^{\vee }(\beta )= \tfrac{2(\beta ,\alpha )}{(\alpha ,\alpha )}\text{ for all }\alpha ,\beta \in \varDelta . \end{equation*}$$

Also, the set $\varDelta ^{\vee }\coloneq \{\alpha ^{\vee }: \alpha \in \varDelta \}$ is a root system of $V^*$, with $(\alpha ^{\vee })^{\vee }=\alpha$. There are four examples of reduced root systems in rank 2: $A_1\times A_1$, $A_2$, $B_2$ and $G_2$, with $2$, $6$, $8$, and $12$ roots, respectively. The third one is depicted in Figure 1.

Figure 1.

Root system of type $B_2$.

Let $\alpha ,\beta \in \varDelta$ be such that $\alpha \ne \pm \beta$. One may check that $\alpha +\beta \in \varDelta$ (respectively, $\alpha -\beta \in \varDelta$) if $(\alpha ,\beta )<0$ (respectively, $(\alpha ,\beta )>0$). This is the starting point, together with (RS2) and (RS3), to check that an analogue of (v) holds for (abstract) root systems.

Another key point is the existence of a base of a root system. It means a subset $B\subset \varDelta$ such that $B$ is a basis of $V$ (as a vector space), and every $\beta \in \varDelta$ is written, in terms of $B$, as a linear combination whose coefficients are all nonnegative integers, or all nonpositive integers.

The proof of existence of bases gives the geometric flavor behind root systems. We take a vector $\gamma$ such that the orthogonal hyperplane $P$ to $\gamma$ does not contain any root. Indeed $\gamma$ belongs to $V-\cup _{\alpha \in \varDelta } H_{\alpha }$, where $H_{\alpha }$ is the kernel of $\alpha ^{\vee }$, i.e., the hyperplane orthogonal to $\alpha$: the connected components of $V-\cup _{\alpha \in \varDelta } H_{\alpha }$ are called the Weyl chambers. Thus $\varDelta =\varDelta ^+(\gamma )\cup \varDelta ^-(\gamma )$, where

$$\begin{align*} \varDelta ^{\pm }(\gamma )&=\{\beta \in \varDelta | \pm (\beta ,\gamma )>0\}. \end{align*}$$

A base is made by those indecomposable roots in $\varDelta ^+(\gamma )$: those $\beta \in \varDelta ^+(\gamma )$ which cannot be written as a sum $\beta =\beta _1+\beta _2$, with $\beta _i\in \varDelta ^+(\gamma )$. Moreover every base can be constructed in this way.

For example, in Figure 1 we take the green hyperplane: the positive roots are the red ones, the negative are the blue ones, and $B=\{\alpha _1,\alpha _2\}$ is a base.

The Weyl group ${\mathcal{W}}$ permutes bases (and Weyl chambers as well), and the action is simply transitive. We check then that any root $\alpha \in \varDelta$ belongs to a base, and for each base $B$, ${\mathcal{W}}$ is generated by $s_{\alpha }$, $\alpha \in B$ (we reduce the number of generators of ${\mathcal{W}}$ to the rank of the root system). This leads to the study of groups generated by reflections and Coxeter groups considered in Bou02, which became an important subject of research on its own, and remains active until now.

2.2. The classification

As for algebraic objects, we may ask for irreducible root systems: those which cannot split into two orthogonal subsets (otherwise each subset is itself a root system). Every root system $\varDelta$ of $V$ decomposes uniquely as a union of irreducible root systems $\varDelta _i$ corresponding to the subspaces $V_i$ of $V$ spanned by $\varDelta _i$. Thus, in order to classify root systems, we can restrict to the irreducible ones.

Assume now that $\varDelta$ is an irreducible root system of rank $\theta$. Set $A^{\varDelta }\in \mathbb{Z}^{\theta \times \theta }$ as the matrix with entries

$$\begin{align*} a_{ij} &\coloneq \alpha _i^{\vee }(\alpha _j)=\tfrac{2(\alpha _j,\alpha _i)}{(\alpha _i,\alpha _i)}, && 1\le i,j\le \theta , \end{align*}$$

where $B=\{\alpha _i\}_{1\le i\le \theta }$ is a base. One can check that $A^{\varDelta }$ is well-defined; i.e., it does not depend on the chosen base. In addition, $A$ is indecomposable: for all $i<j$ there exist $i_k\in \{1,\cdots ,\theta \}$ such that $a_{ii_1}a_{i_1i_2}\cdots a_{i_tj}\ne 0$. Moreover,

(GCM1)

$a_{ii}=2$ for all $1\le i\le \theta$,

(GCM2)

$a_{ij}=0$ if and only if $a_{ji}=0$,

(GCM3)

for all $i\ne j$, $a_{ij}\le 0$.

Any $A\in \mathbb{Z}^{\theta \times \theta }$ satisfying (GCM1)–(GCM3) is called a generalized Cartan matrix (GCM) Kac90. The information of GCM is encoded in a graph called the Dynkin diagram: it has $\theta$ vertices, labelled with $1,2,\ldots ,\theta$, and for each pair $1\le i< j\le \theta$,

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if $a_{ij}a_{ji}\le 4$, then we add $\max \{|a_{ij}|,|a_{ji}|\}$ edges between vertices $i$ and $j$, with an arrow from $j$ to $i$ (respectively $i$ to $j$) if $|a_{ij}|>1$ (respectively, $|a_{ji}|>1$); in particular, if $a_{ij}=0$ (so $a_{ji}=0$ as well) then we draw no edges between $i$ and $j$, and if $a_{ij}=a_{ji}=-1$, then we draw just a line;

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if $a_{ij}a_{ji}>4$, then we draw a thick line between $i$ and $j$ labelled with $(|a_{ij}|,|a_{ji}|)$ .

For example, the Dynkin diagrams of $\left[\begin{smallmatrix} 2 & -2 \\ -2 & 2\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix} 2 & -3 \\ -2 & 2\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{smallmatrix}\right]$ are, respectively

$$\begin{align*} & \vcenter{\img[][39pt][7pt][{$\xymatrix{ \circ\ar@2{<->}[r] & \circ}$}]{Images/img9ff6b7b05ac243f7d3bb75b35bb4a506.svg}} && \vcenter{\img[][39pt][13pt][{$\xymatrix{ \circ\ar@{-}@<0.2mm>[r]^{(3,2)} & \circ}$}]{Images/img80425a5eb9fd9e03bb0a0f92cee4d7ae.svg}} && & \vcenter{\img[][73pt][4pt][{$\xymatrix{ \circ\ar@{-}[r] & \circ\ar@{-}[r] & \circ}$}]{Images/imgd4cb2567529fb68a0b7b1306c169a179.svg}} . \end{align*}$$

One reason to differentiate between $a_{ij}a_{ji}\le 4$ and $a_{ij}a_{ji}>4$ is all finite and affine Dynkin diagrams satisfy the first condition, and these are probably the most studied cases. We refer to Bou02Hum78 for the definition of affine Dynkin diagrams while finite ones are depicted in Figure 2, in connection with finite-dimensional complex Lie algebras.

One may define the Weyl group ${\mathcal{W}}^A$ of a GCM $A$ as the subgroup of $\operatorname {Aut}(V)$ generated by reflections $s_i:V\to V$, $s_i(\alpha _j)=\alpha _j-a_{ij}\alpha _i$, where $(\alpha _i)_{1\le i\le \theta }$ is the canonical basis of $V=\mathbb{R}^{\theta }$: if $A$ is the Cartan matrix of a Lie algebra as above, then the Weyl group of $\varDelta$ is generated by these $s_i$’s. Analogously, we can define

$$\begin{equation*} \varDelta ^A=\{w(\alpha _i)| w\in {\mathcal{W}}^A, 1\le i\le \theta \}. \end{equation*}$$

Then one can prove that ${\mathcal{W}}^A$ is finite if and only if $\varDelta ^A$ is finite, which is equivalent to the notion of finite GCM. Finite GCM are parametrized by finite Dynkin diagrams, i.e., those in Figure 2.

Figure 2.

Finite connected Dynkin diagrams.

$A_{m}$	$\vcenter{\img[][122pt][4pt][{$\xymatrix@C-5pt{ \circ\ar@{-}[r] & \circ\ar@{.}[r]& \circ\ar@{-}[r] & \circ\ar@{-}[r] & \circ}$}]{Images/imged4c23b6c41150737f9af3eeffa20c0d.svg}}$
$B_{m}$	$\vcenter{\img[][122pt][7pt][{$\xymatrix@C-5pt{ \circ\ar@{-}[r] & \circ\ar@{.}[r]& \circ\ar@{-}[r] & \circ\ar@{=>}[r] & \circ}$}]{Images/img73c37c5ee48f99e154cce4ef4a33e562.svg}}$
$C_{m}$	$\vcenter{\img[][122pt][7pt][{$\xymatrix@C-5pt{ \circ\ar@{-}[r] & \circ\ar@{.}[r]& \circ\ar@{-}[r] & \circ\ar@{<=}[r] & \circ}$}]{Images/img75347e66ca9ba02b37384d52d0abf5ed.svg}}$
$D_{m}$	$\vcenter{\img[][122pt][23pt][{$\xymatrix@C-5pt@R-15pt{ & & & \circ\ar@{-}[d] & \\ \circ\ar@{-}[r] & \circ\ar@{.}[r]& \circ\ar@{-}[r] & \circ\ar@{-}[r] & \circ}$}]{Images/imgf8bd0607fac17205859c40659dd95be9.svg}}$
$E_{m}$, $m=6,7,8$	$\vcenter{\img[][122pt][23pt][{$\xymatrix@C-5pt@R-15pt{ & & \circ\ar@{-}[d] & &\\ \circ\ar@{-}[r] & \circ\ar@{-}[r]& \circ\ar@{.}[r] & \circ\ar@{-}[r] & \circ}$}]{Images/img33b3c49924aa5023a7915d2672695e8d.svg}}$
$F_{4}$	$\vcenter{\img[][92pt][7pt][{$\xymatrix@C-5pt{ \circ\ar@{-}[r] & \circ\ar@{<=}[r] & \circ\ar@{-}[r] & \circ}$}]{Images/img2b9ba07699452750ed1b25411eb5bc04.svg}}$
$G_2$	$\vcenter{\img[][39pt][7pt][{$\xymatrix{ \circ\ar@3{<-}[r] & \circ}$}]{Images/img34e3aec3154387b5fe19a137a20ce710.svg}}$

Theorem 2.2.

Reduced irreducible root systems are parametrized by Dynkin diagrams in Figure 2.

Up to now we deal with three notions:

(i)

Simple Lie algebras over $\mathbb{C}$,

(ii)

Irreducible root systems,

(iii)

Finite Cartan matrices, or the corresponding Dynkin diagrams.

We moved first from (i) to (ii), and then state a correspondence (ii)$\longleftrightarrow$ (iii). Now we need to come back to (i). We can check that $\mathfrak{sl}(n+1)$ has Cartan matrix of type $A_n$ (see Example 1.2), while matrices of types $B_n$, $C_n$ and $D_n$ appear for orthogonal and symplectic Lie algebras. For each one of the exceptional finite Cartan matrices $A$ in Figure 2 we can construct by hand a simple Lie algebra with Cartan matrix $A$. The natural question is if there exists a systematic way to build these Lie algebras. We will recall it in the next subsection, i.e., a correspondence (iii)$\longrightarrow$(i).

2.3. Back to Lie algebras: Kac-Moody construction

Looking at Example 1.2, the Cartan matrix of $\mathfrak{sl}(4)$ can be recovered from the action of the Cartan subalgebra ${\mathfrak{h}}$ on eigenvectors of a base of the root system $\varDelta$. In addition the decomposition $\varDelta =\varDelta ^+ \cup \varDelta ^-$ into positive and negative roots for the chosen base corresponds in this case to the upper and lower triangular matrices $\mathfrak{n}_{\pm }$ of $\mathfrak{sl}(4)$ (recall that ${\mathfrak{h}}$ is spanned by the set of all the diagonal matrices in $\mathfrak{sl}(4)$).

As for associative algebras, we have a notion of a Lie algebra presented by generators and relations as the appropiate quotient of a free Lie algebra. We will attach a Lie algebra $\mathfrak{g}\coloneq \mathfrak{g}(A)$ to each matrix $A\in \mathbb{C}^{\theta \times \theta }$; these algebras were introduced by Serre in 1966 for finite matrices $A$, and by Kac and Moody in two independent and simultaneous works in the late sixties, see Kac90 and the references therein. For the sake of simplicity of the exposition we assume that $\det A\ne 0$.

Let $\widetilde{\mathfrak{g}}\coloneq \widetilde{\mathfrak{g}}(A)$ be the Lie algebra presented by generators $e_i$, $h_i$, $f_i$, $1\le i\le \theta$, and relations

$$\begin{align} \begin{aligned} &&[h_i,h_j] &=0, & [h_i,e_j] &=a_{ij}e_j, \\ &&[e_i,f_j] &=\delta _{ij}h_i, & [h_i,f_j] &=-a_{ij}f_j. \end{aligned} {\tag{1}} \end{align}$$

Let ${\mathfrak{h}}$ be the subspace spanned by $(h_i)_{1\le i\le \theta }$, $\widetilde{\mathfrak{n}}_{\pm }$ the subalgebra generated by $(e_i)_{1\le i\le \theta }$, respectively $(f_i)_{1\le i\le \theta }$. We have the following facts:

(a)

$\widetilde{\mathfrak{n}}_{\pm }$ is a free Lie algebra in $\theta$ generators.

(b)

As a vector space, $\widetilde{\mathfrak{g}}=\widetilde{\mathfrak{n}}_+\oplus {\mathfrak{h}}\oplus \widetilde{\mathfrak{n}}_-$.

(c)

The adjoint action of ${\mathfrak{h}}$ on $\widetilde{\mathfrak{n}}_{\pm }$ is diagonalizable.

(d)

Among all the ideals of $\widetilde{\mathfrak{g}}$ intersecting trivially ${\mathfrak{h}}$, there exists a maximal one $\mathfrak{r}$, which satisfies$$\begin{equation*} \mathfrak{r}=(\mathfrak{r}\cap \widetilde{\mathfrak{n}}_{+})\oplus (\mathfrak{r}\cap \widetilde{\mathfrak{n}}_{-}). \end{equation*}$$

Definition 2.3.

The contragredient Lie algebra $\mathfrak{g}(A)$ associated to $A$ (sometimes called the Kac-Moody algebra) is the quotient $\mathfrak{g}(A)\coloneq \widetilde{\mathfrak{g}}(A)/\mathfrak{r}$.

Because of the definition of $\mathfrak{r}$, $\mathfrak{g}(A)$ is generated by $e_i$, $f_i$, $h_i$, $1\le i\le \theta$, has a triangular decomposition

$$\begin{equation*} \mathfrak{g}(A)=\mathfrak{n}_+\oplus {\mathfrak{h}}\oplus \mathfrak{n}_-, \end{equation*}$$

where $\mathfrak{n}_{\pm }$ is the image of $\widetilde{\mathfrak{n}}_{\pm }$ under the projection $\pi :\widetilde{\mathfrak{g}}(A)\twoheadrightarrow \mathfrak{g}(A)$, i.e., the subalgebra generated by (the image of) $(e_i)_{1\le i\le \theta }$, respectively $(f_i)_{1\le i\le \theta }$, and any other Lie algebra with a triangular decomposition as above, generated by the same set of generators satisfying 1, projects onto $\mathfrak{g}(A)$.

Theorem 2.4.

(A)

Let $A$ be a finite Cartan matrix. Then $\mathfrak{g}(A)$ is a finite-dimensional simple Lie algebra, with Cartan matrix $A$.

(B)

The list of Dynkin diagrams in Figure 2 provides a classification of all finite-dimensional simple Lie algebras over $\mathbb{C}$.

When the generalized Cartan matrix $A$ is not of finite type, the associated Kac-Moody Lie algebra $\mathfrak{g}(A)$ is infinite-dimensional. Although for the purposes of this exposition we are interested in the finite-dimensional examples, the infinite-dimensional Lie algebras $\mathfrak{g}(A)$ (or at least some of them, mainly the affine ones) are quite important since they have appeared in connection either with other areas of mathematics, especially representation theory, or theoretical physics, for example in conformal field theory.

3. Root Systems for Other Kinds of Lie Algebras

Next we deal with contragredient Lie algebras over fields of positive characteristic and later with Lie superalgebras over any field. We will recall the main differences with the picture of Lie algebras over $\mathbb{C}$ which leads to a more general notion of root system. This root system still captures the combinatorics of these Lie theoretic objects.

3.1. Lie algebras over fields of positive characteristic

Let $\mathbb{k}$ be an algebraically closed field of characteristic $p>0$. The study of simple Lie algebras becomes more and more complicated as far as $p$ is smaller, see, e.g., Str04. A main difference with the case of complex numbers is that not all simple Lie algebras have a triangular decomposition as above, and the Cartan subalgebra plays a weaker role in the structure of the whole Lie algebra.

On the other hand, Definition 2.3 still holds over $\mathbb{k}$, so we may ask about the classification of finite-dimensional contragredient Lie algebras. A subtle difference is that we restrict to $\mathbb{Z}$-homogeneous ideals intersecting trivially ${\mathfrak{h}}$, where each $e_i$ has degree 1, each $f_i$ has degree -1 and each $h_i$ has degree 0. Thus, there exists a finer grading of the Lie algebra $\mathfrak{g}(A)$ by $\mathbb{Z}^{\theta }$, where $\deg e_i=\alpha _i$ (the $i$-th element of the canonical basis), $\deg f_i=-\alpha _i$ and $\deg h_i=0$ as well. Let $\varDelta ^A\subset \mathbb{Z}^{\theta }$ be the subset of all nonzero degrees whose homogeneous components are nontrivial.

Remark 3.1.

From the triangular decomposition,

$$\begin{equation*} \varDelta ^A \subset \mathbb{N}_0^{\theta }\cup (-\mathbb{N}_0^{\theta }), \end{equation*}$$

that is, the coefficients of each $\alpha \in \varDelta$ are all nonnegative, or else all nonpositive. Also, there exists an involution $\omega$ of $\mathfrak{g}(A)$ (called the Chevalley involution) such that $e_i\mapsto f_i$, $f_i\mapsto e_i$, $h_i\mapsto -h_i$. As $\omega \left(\mathfrak{g}(A)\right)_{\beta }=\mathfrak{g}(A)_{-\beta }$ for all $\beta \in \mathbb{Z}^{\theta }$, we have that

$$\begin{equation*} \varDelta ^A=-\varDelta ^A. \end{equation*}$$

For example we can consider the finite Cartan matrices over $\mathbb{k}$, since the entries of these Cartan matrices are integer numbers, and show that the associated Lie algebras are finite-dimensional. But, even for contragredient Lie algebras, there are significant differences with the case of complex numbers. As shown in VK71, there are examples of finite-dimensional Lie algebras with diagonal entries $a_{ii}=0$, and two different matrices can give place to isomorphic contragredient Lie algebras. The classification shown in VK71 was incomplete: there was a missing example for $p=3$, the 29-dimensional Brown algebra $\mathfrak{br}(3)$, discovered by Brown in the eighties, whose realization as contragredient Lie algebra with two different matrices was shown in Skr93:

Theorem 3.2.

Fix $p=3$. Let

$$\begin{equation*} A=\left[\begin{smallmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & 1 & 0\end{smallmatrix}\right], \quad B=\left[\begin{smallmatrix} 2 & -1 & 0 \\ -2 & 2 & -1 \\ 0 & 1 & 0\end{smallmatrix}\right]. \end{equation*}$$

Then there exists an isomorphism $\Phi :\mathfrak{g}(A)\to \mathfrak{g}(B)$ such that

$$\begin{align*} \Phi (e_1) &= \underline{e}_1, & \Phi (e_2) &= (\operatorname {ad}\underline{e}_3)^2\underline{e}_2, & \Phi (e_3) &= \underline{f}_3, \\ \Phi (f_1) &= \underline{f}_1, & \Phi (f_2) &= (\operatorname {ad}\underline{f}_3)^2\underline{f}_2, & \Phi (f_3) &= \underline{e}_3. \end{align*}$$

The expression of $\Phi$ is close to that for the action of reflections of the Weyl group on complex Lie algebras, but here $\Phi$ relates two “different” contragredient data.

Remark 3.3.

We fix the following GCM

$$\begin{equation*} C^A=\left[\begin{smallmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -2 & 2\end{smallmatrix}\right], \quad C^B=\left[\begin{smallmatrix} 2 & -1 & 0 \\ -2 & 2 & -1 \\ 0 & -2 & 2\end{smallmatrix}\right], \end{equation*}$$

and set $s_i^A,s_i^B:\mathbb{Z}^3\to \mathbb{Z}^3$ as the corresponding reflections defined by $C^A$, $C^B$, respectively, $i=1,2,3$. Notice that $s_3^A=s_3^B$ and $\Phi \left(g(A)_{\beta }\right)=g(B)_{s_3^A(\beta )}$ for all $\beta \in \mathbb{Z}^3$. Thus $\varDelta ^B=s_3^A(\varDelta ^A)$.

In addition, there exist automorphisms

$$\begin{align*} \Phi ^A_i &:\mathfrak{g}(A)\to \mathfrak{g}(A), & \Phi ^B_i &:\mathfrak{g}(B)\to \mathfrak{g}(B), & &i=1,2, \end{align*}$$

such that $\Phi _i^A\left(\mathfrak{g}(A)_{\beta }\right)=\mathfrak{g}(A)_{s_i^A(\beta )}$, $\Phi _i^B\left(\mathfrak{g}(B)_{\beta }\right)=\mathfrak{g}(B)_{s_i^B(\beta )}$ for all $\beta \in \mathbb{Z}^3$. This implies that

$$\begin{align*} \varDelta ^A&=s_i^A(\varDelta ^A), & \varDelta ^B&=s_i^B(\varDelta ^B), & &i=1,2. \end{align*}$$

3.2. Lie superalgebras

Recall that a Lie superalgebra is a $\mathbb{Z}_2$-graded vector space $\mathfrak{g}=\mathfrak{g}_0\oplus \mathfrak{g}_1$ ($\mathfrak{g}_0$ is the even part and $\mathfrak{g}_1$ is the odd part) together with a linear $\mathbb{Z}_2$-graded map $[,]:\mathfrak{g}\otimes \mathfrak{g}\to \mathfrak{g}$ satisfying analogous versions of antisymmetry and Jacobi identity:

$$\begin{align*} [x,y]&= -(-1)^{|x| |y|}[y,x], \\ [x,[y,z]] &= [[x,y],z]+(-1)^{|x| |y|}[y,[x,z]], \end{align*}$$

for all homogeneous elements $x,y,z\in \mathfrak{g}$, see, e.g., Kac77. Here, $|x|\in \{0,1\}$ denotes the degree of $x$. We have examples from associative algebras, analogous to those of Lie algebras: given $\mathfrak{g}=\mathfrak{g}_0\oplus \mathfrak{g}_1$ a $\mathbb{Z}_2$-graded associative algebra, set

$$\begin{equation*} [x,y]=xy-(-1)^{|x| |y|}yx . \end{equation*}$$

In particular we have, for $V=V_0\oplus V_1$, the Lie superalgebra $\mathfrak{gl}(V)=\operatorname {End}(V)$, with

$$\begin{equation*} \mathfrak{gl}(V)_i=\{T\in \mathfrak{gl}(V): T(V_j)\subseteq V_{i+j}\}. \end{equation*}$$

For each $T\in \mathfrak{gl}(V)$ set $\operatorname {str}(T)\coloneq \operatorname {tr} \left(T_{|V_0}\right)-\operatorname {tr} \left(T_{|V_1}\right)$, the super trace of $T$. We can consider the subalgebra

$$\begin{equation*} \mathfrak{sl}(V)=\left\{T\in \mathfrak{gl}(V): \operatorname {str}(T)= 0\right\}. \end{equation*}$$

Here we consider contragredient data $(A,\mathbf{p})$, where $A=(a_{ij})_{1\le i,j\le \theta }\in \mathbb{k}^{\theta \times \theta }$ is still the matrix of scalars determining the action of the generators $h_i$ on the remaining generators, and $\mathbf{p}=(p_i)_{1\le i\le \theta }\in \mathbb{Z}_2^{\theta }$ gives the $\mathbb{Z}_2$-grading: $e_i,f_i\in \mathfrak{g}(A,\mathbf{p})_{p_i}$ for all $i$. Notice that all $h_i$ are necessarily even.

As observed in Kac77 when $\mathbb{k}=\mathbb{C}$, different contragredient data can give isomorphic Lie superalgebras. But, similar to Lie algebras in positive characteristic, we can give isomorphisms between some pairs $\mathfrak{g}(A,\mathbf{p})$, $\mathfrak{g}(A',\mathbf{p}')$ with formulas close to the action of the Weyl group for complex simple Lie algebras. In this direction, Serganova Ser96 introduced the notion of odd reflection relating two different pairs by a kind of reflection but on a simple odd root $e_i$ such that $a_{ii}=0$ (called isotropic). This is consistent with one of the differences with Lie algebras: there exists a symmetric bilinear form on ${\mathfrak{h}}$, but either the bilinear form can have isotropic roots $\alpha$ (i.e., $(\alpha ,\alpha )=0$) or else the matrix $A\in \mathbb{C}^{\theta \times \theta }$ can take nonintegral values. We have to distinguish the matrix $A$ from the GCM $C^A\in \mathbb{Z}^{\theta \times \theta }$ responsible for the odd reflections.

Example 3.4.

Let $V=(\mathbb{C}^2)_0\oplus \mathbb{C}_1$, that is a $\mathbb{Z}_2$-graded vector space with even component of dimension 2, and odd component of dimension 1. Here $\mathfrak{sl}(V)$ is denoted simply by $\mathfrak{sl}(2|1)$: as for Lie algebras we identify $\mathfrak{sl}(2|1)$ with matrices $\mathtt{A}\in \mathbb{C}^{3\times 3}$, here with zero supertrace, i.e., $\mathtt{a}_{11}+\mathtt{a}_{22}-\mathtt{a}_{33}=0$. The Lie superalgebra $\mathfrak{sl}(2|1)$ is $\mathbb{Z}^2$-graded, with ${\mathfrak{h}}$ (the diagonal matrices) in degree 0, and one-dimensional components of degrees $\pm \alpha _1$ (even roots, since the corresponding spaces are $E_{12}$ and $E_{21}$), $\pm \alpha _2$, $\pm (\alpha _1+\alpha _2)$ (these four roots are odd). The contragredient datum is $A=\left[\begin{smallmatrix} 2 & -1 \\ 1 & 0 \end{smallmatrix}\right]$, $\mathbf{p}_2=(0,1)$. The odd reflection in $\alpha _2$ moves to the pair $B=\left[\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\right]$, $\mathbf{p}_{12}=(1,1)$. Thus we may apply the odd reflection in $\alpha _1$ to $(B,\mathbf{p}_{12})$ and obtain $(C,\mathbf{p}_1)$, where $C=\left[\begin{smallmatrix} 0 & 1 \\ -1 & 2 \end{smallmatrix}\right]$, $\mathbf{p}_1=(1,0)$. These are all the possible movements between pairs whose associated Lie superalgebra is isomorphic to $\mathfrak{sl}(2|1)$.

We see that the set of $\mathbb{Z}^2$-degrees of the nontrivial components is the same, the Cartan matrix is $\left[ \begin{smallmatrix} 2 & -1 \\ -1 & 2 \end{smallmatrix} \right]$ for the three pairs, but the parity of the elements is not the same. For example, for $(B,\mathbf{p}_{12})$, $\pm (\alpha _1+\alpha _2)$ are even roots while $\pm \alpha _1$, $\pm \alpha _2$ are odd.

If we study Lie superalgebras over fields of positive characteristic, then we can have more and more exceptional examples. Finite-dimensional contragredient Lie algebras over fields of prime characteristic were classified in BGL09. The picture is the same: several pairs of contragredient data $(A,\mathbf{p})$ give isomorphic Lie superalgebras.

The question is then how to handle uniformly all possible pairs giving isomorphic Lie superalgebras, and their corresponding roots (i.e., the $\mathbb{Z}^{\theta }$-degrees of the nontrivial components). This will be done with a groupoid, i.e., a category where all the morphisms are invertible. As we look for a generalization of the Weyl group, we will consider a groupoid generated by reflections.

3.3. Generalized root systems

There exist different notions of generalized root systems in the literature. They try to capture different situations, as for example the one by Serganova in Ser96 for complex finite-dimensional Lie superalgebras. A nice axiomatic version was given in HY08, see also HS20 for a refined version of these ideas.

Fix $\theta \in \mathbb{N}$, $\mathbb{I}=\{1,\cdots ,\theta \}$. Let ${\mathcal{X}}\ne \emptyset$ a set (which will correspond to the different contragredient data). A semi-Cartan graph $\mathcal{G}(\mathbb{I},{\mathcal{X}},(C^X)_{X\in {\mathcal{X}}},(\rho _i)_{i\in \mathbb{I}})$ ($\mathcal{G}$ for short) of rank $\theta$ over ${\mathcal{X}}$ consist of

•

functions $\rho _i:{\mathcal{X}}\to {\mathcal{X}}$, $i\in \mathbb{I}$, such that $\rho _i^2=\operatorname {id}_{{\mathcal{X}}}$,

•

GCM $C^X=(c_{ij}^X)_{i,j\in \mathbb{I}}\in \mathbb{Z}^{\theta \times \theta }$, $X\in {\mathcal{X}}$,

such that $c_{ij}^X=c_{ij}^{\rho _i(X)}$ for all $X\in {\mathcal{X}}$, $i,j\in \mathbb{I}$.

As for Lie algebras, we set $s_i^X\in \operatorname {GL}(\mathbb{Z}^{\theta })$ as the reflection $s_i^X(\alpha _j)=\alpha _j-c_{ij}^X\alpha _i$.

Let $\mathcal{M}$ be a monoid. There exists a small category $\mathcal{D}({\mathcal{X}},\mathcal{M})$ whose set of objects is ${\mathcal{X}}$ and the set of morphisms between any two objects is $\mathcal{M}$. Given $f\in \mathcal{M}$ and $X,Y\in {\mathcal{X}}$ we write $(Y,f,X)$ for $f$ viewed as an element of $\operatorname {Hom}(X,Y)$, so the composition becomes

$$\begin{align*} (Z,f,Y)\circ (Y,g,X) &= (Z,fg,X), \end{align*}$$

for any $X,Y,Z\in {\mathcal{X}}$, $f,g\in \mathcal{M}$.

We are interested in the case $\mathcal{M}=\operatorname {GL}(\mathbb{Z}^{\theta })$, the group of automorphisms of $\mathbb{Z}^{\theta }$.

Definition 3.5.

The Weyl groupoid of $\mathcal{G}$ is the full subcategory ${\mathcal{W}}(\mathbb{I},{\mathcal{X}},(A_x)_{x\in {\mathcal{X}}},(\rho _i)_{i\in \mathbb{I}})$ of $\mathcal{D}({\mathcal{X}},\operatorname {GL}(\mathbb{Z}^{\theta }))$ generated by

$$\begin{align*} \sigma _i^X & \coloneq (\rho _i(X),s_i^X,X), && i\in \mathbb{I}, X\in {\mathcal{X}}. \end{align*}$$

Notice that ${\mathcal{W}}$ is indeed a groupoid, since

$$\begin{equation*} \sigma _i^{\rho _i(X)}\sigma _i^X=(X,\operatorname {id}_X,X)\text{ for all }i\in \mathbb{I},X\in {\mathcal{X}}. \end{equation*}$$

Fix $X\in {\mathcal{X}}$. Then $\varDelta ^{X,\operatorname {re}}$ is the set of all elements of the form $w(\alpha _i)\in \mathbb{Z}^{\theta }$, where $i\in \mathbb{I}$, $Y\in {\mathcal{X}}$, $(X,w,Y)\in \operatorname {Hom}_{{\mathcal{W}}}(X,Y)$. This is the set of real roots of $\mathcal{G}$. As for roots of Lie (super)algebras, we consider the subsets

$$\begin{align*} \varDelta ^{X,\operatorname {re}}_{+} & \coloneq \varDelta ^{X,\operatorname {re}} \cap \mathbb{N}_0^{\theta }, & \varDelta ^{X,\operatorname {re}}_{-} & \coloneq \varDelta ^{X,\operatorname {re}} \cap (-\mathbb{N}_0^{\theta }), \end{align*}$$

of positive and negative real roots, and set

$$\begin{equation*} m^X_{ij} \coloneq |\varDelta ^{X,\operatorname {re}} \cap (\mathbb{N}_0\alpha _i+\mathbb{N}_0\alpha _j)|\in \mathbb{N}\cup \{\infty \}. \end{equation*}$$

If $\varDelta ^{X,\operatorname {re}}$ is finite for all $X\in {\mathcal{X}}$ (equivalently, for some $X\in {\mathcal{X}}$), then we say that $\mathcal{G}$ is finite.

A semi-Cartan graph $\mathcal{G}$ is a Cartan graph if the following conditions hold for all $X\in {\mathcal{X}}$:

•

$\varDelta ^{X,\operatorname {re}} = \varDelta ^{X,\operatorname {re}}_{+} \cup \varDelta ^{X,\operatorname {re}}_{-}$;

•

for all $i\ne j$ such that $m^X_{ij}<\infty$, $(\rho _i\rho _j)^{m_{ij}^X}(X)=X$.

Mimicking what happens for Lie algebras, see Remarks 3.1 and 3.3, we introduce the following notion:

Definition 3.6.

A root system over $\mathcal{G}$ is a family $\mathcal{R}=(\varDelta ^X)_{X\in {\mathcal{X}}}$ of subsets $\varDelta ^X\subset \mathbb{Z}^{\theta }$ such that

$$\begin{align*} &0 \notin \varDelta ^X, & &\varDelta ^X \subset \mathbb{N}_0^{\theta }\cup (-\mathbb{N}_0^{\theta }), \\ &\alpha _i\in \varDelta ^X, & &s_i^X(\varDelta ^X)=\varDelta ^{\rho _i(X)}, \end{align*}$$

for all $i\in \mathbb{I}$ and all $X\in {\mathcal{X}}$.

We say that $\mathcal{R}$ is reduced if $\mathbb{Z}\alpha \cap \varDelta ^X=\{\pm \alpha \}$ for all $\alpha \in \varDelta ^X$, $X\in {\mathcal{X}}$. $\mathcal{R}$ is finite if every $\varDelta ^X$ is so.

Cuntz and Heckenberger obtained the classification of finite root systems CH15. The proof involves the bijection between root systems of rank $\theta$ and crystallographic arrangements (certain subsets of hyperplanes) in $\mathbb{R}^{\theta }$. About the list of finite root systems, in rank $\theta =2$ there are infinitely many examples, in bijection with triangulations of $n$-gons for any $n\ge 3$. For $\theta \ge 9$ we only have families corresponding to Lie superalgebras and Lie algebras of types $A,B,C,D$, while for $3\le \theta \le 8$ we have members of the families of Lie (super)algebras and several exceptions.

The definition of a root system seems to carry the possibility to have several examples attached to the same Cartan graph $\mathcal{G}$. But this is not the case when $\mathcal{G}$ is finite. Indeed, by HS20, 10.4.7, if $\mathcal{G}$ is a finite Cartan graph, then $\mathcal{R}=(\varDelta ^{X,\operatorname {re}})_{X\in {\mathcal{X}}}$ is the only reduced root system over $\mathcal{G}$.

Remark 3.7.

In HY08 the authors state the existence of a Weyl groupoid for finite-dimensional complex Lie superalgebras, the ones coming from the $\mathbb{Z}^{\theta }$-grading as above. Moreover, Andruskiewitsch and Angiono proved that the same holds for Lie superalgebras over fields of arbitrary characteristic, in a work in progress. In the same work they derived the classification of finite-dimensional Lie superalgebras from the classification of finite root system in CH15.

It should be noted that not all finite root systems come from a Lie superalgebra.

Once we show the existence of a finite root system for a Lie superalgebra, there are many strong properties derived from the combinatorics of the Weyl groupoid. For example:

•

$\dim \mathfrak{g}(A,\mathbf{p})_{\alpha }=1$ for all $\alpha \in \varDelta ^{(A,\mathbf{p})}$.

•

There might exist roots $\alpha \in \varDelta ^{(A,\mathbf{p})}$ such that $2\alpha \in \varDelta ^{(A,\mathbf{p})}$, which are the odd nonisotropic roots. All of them are the image of simple odd nonisotropic roots of some pair $(A',\mathbf{p}')$ obtained up to odd reflections, and $\dim \mathfrak{g}(A,\mathbf{p})_{2\alpha }=1$, as shown by Andruskiewitsch-Angiono.

•

The whole set $\varDelta ^{(A,\mathbf{p})}$ is obtained up to reflections of the simple roots, attaching $2\alpha$ for each odd nonisotropic root.

In the same line we may wonder if there exists a geometric-combinatoric side on these Lie superalgebras (Weyl chambers and so on) coming from the associated crystallographic arrangements.

Example 3.8.

We continue with the study of $\mathfrak{br}(3)$. Here ${\mathcal{X}}=\{A,B\}$, with $\rho _3(A)=B$ and $\rho _1=\rho _2=\operatorname {id}$. The associated GCM are those in Remark 3.3. Thus we get all the roots applying repeatedly the reflections. For example, $s_2^B(\alpha _1)=\alpha _1+2\alpha _2\in \varDelta ^B$, so

$$\begin{equation*} s_3^B(\alpha _1+2\alpha _2) = \alpha _1+2\alpha _2 +4\alpha _3\in \varDelta ^A. \end{equation*}$$

Using the notation $1^a2^b3^c\coloneq a\alpha _1+b\alpha _2 +c\alpha _3$, we can check that

$$\begin{align*} \varDelta _{+}^{A}= & \{ 1, 12, 123, 1^22^33^4, 12^23^2, 12^23^3, 12^23^4, \\ & \quad 12^33^4, 123^2, 2, 23^2, 23, 3\}, \\ \varDelta _{+}^{B}= & \{ 1, 12^2, 12, 123^2, 12^33^2, 1^22^33^2, 12^23^2, \\ & \quad 123, 12^23, 2, 23^2, 23, 3 \}. \end{align*}$$

Thus, $\dim \mathfrak{n}_{\pm }=13$, so $\dim \mathfrak{br}(3)=29$. In addition one can show that $[\mathfrak{g}_{\alpha },\mathfrak{g}_{\beta }]=\mathfrak{g}_{\alpha +\beta }$ for every pair $\alpha ,\beta \in \varDelta _+^A$ such that $\alpha +\beta \in \varDelta _{+}^A$. Thus we can obtain recursively a nonzero element $e_{\alpha }\in \mathfrak{g}_{\alpha }$:

$$\begin{align*} e_{12}&\coloneq [e_1,e_2], & e_{123}&\coloneq [e_{12},e_3], & e_{123^2} &\coloneq [e_{123},e_3], \end{align*}$$

and so on.

4. Other Contexts and Problems

We finish by recalling other algebraic structures where these generalized root systems appear, as Nichols algebras, and posing some related problems where they could play a key role: Lie algebras in a broad sense and their representations. We are not going to introduce all the involved concepts, we refer to the corresponding papers for more information.

4.1. Nichols algebras

Quantized enveloping algebras are certain deformations of enveloping algebras of semisimple Lie algebras introduced in the eighties by Drinfeld and Jimbo, depending on a parameter $q$. Later on, Lusztig considered Hopf algebras obtained by evaluation of $q$ at a root of unity, which lead to some finite-dimensional examples, usually called Frobenius-Lusztig kernels. These examples have a triangular decomposition whose zero part is a group algebra of copies of finite cyclic groups, and the positive (also, the negative) part is a kind of Hopf algebra.

In the denomination currently used, these positive parts are examples of Nichols algebras. Nichols algebras are Hopf algebras in the category of Yetter-Drinfeld modules over a group algebra (or more precisely, over a Hopf algebra), which play a fundamental role in the classification of finite-dimensional Hopf algebras. Following the line of work of Andruskiewitsch and Schneider, joined by Heckenberger, one can define Hopf algebras with triangular decomposition, whose positive part is a Nichols algebra and which have generalised root systems, see the book HS20, and also AHS10.

The list of all finite-dimensional Nichols algebras is known when the group is finite and abelian, thanks to the work of Heckenberger, and almost complete when the group is finite but nonabelian, by Heckenberger-Vendramin. Both works explode the existence of the generalized root system.

We can see that the list of all generalized root systems appearing for some Nichols algebras contains properly the list of all those appearing for Lie superalgebras, but there are some root systems not attached to any Nichols algebras.

4.2. Lie algebras in symmetric tensor categories and representations

One can extend the definition of Lie algebra to symmetric tensor categories. Indeed Lie superalgebras are essentially Lie algebras in the category of super vector spaces. When $\mathbb{k}$ is of characteristic zero, Deligne proved that any symmetric tensor category (under a mild condition) fibers over the category of supervector spaces, so any Lie algebra over these symmetric tensor categories can be considered as a Lie superalgebra. When $\mathbb{k}$ is of characteristic $p>0$, Coulembier-Etingof-Ostrik proved recently CEO23 that any symmetric tensor category (under a mild condition) fibers over the Verlinde category $\operatorname {Ver}_p$. This category is the semisimplification of the category of representations of $\mathbb{Z}_p$ over $\mathbb{k}$ and contains properly the category of super vector spaces. Thus, in this case, the consideration of Lie algebras in symmetric tensor categories essentially reduces to Lie algebras in $\operatorname {Ver}_p$. One may ask about the existence of contragredient Lie algebras in $\operatorname {Ver}_p$, and root systems.

In the classical case (that is, over $\mathbb{C}$), the root system controls the representation theory of simple Lie algebras, or more precisely a quite interesting subcategory called the category $\mathcal{O}$. For example, finite-dimensional modules are parametrized by nonnegative weights associated to the root system, and the Weyl group describes a character formula for these simple modules. The situation is a bit more complicated for Lie superalgebras, and a character formula exists for certain weights. Recently Sergeev and Veselov used what they called a Weyl groupoid (which is not clearly related to the one considered here) to describe strong properties on the representations. Also, Yamane described very recently character formulas for the so-called atypical weights of quantized enveloping Lie superalgebras by means of the Weyl groupoid. So one may wonder if the Weyl groupoid plays a key role in the description of the representations of Lie algebras in a broad sense.