Standard automorphisms of semisimple Lie algebras and their relations

Let $\mathfrak{g}\coloneqq\mathfrak{sl}(n+1,\mathbb{C})$ be the simple Lie algebra of square matrices $(n+1)\times(n+1)$ with zero trace. We denote the standard basis of the space of square matrices $\operatorname{Mat}_{n+1}(\mathbb{C})$ by $\{E_{ij}\}_{i,j=1}^{n+1}$ and the set $\{1,\,2,\cdots,\,n\}$ by $J_{n}$.

As a Lie algebra, $\mathfrak{g}$ is generated by the elements $e_{i}\coloneqq E_{i\,i+1}$ and $f_{i}\coloneqq E_{i+1\,i}$ for $i\in J_{n}$.

Note that with the corresponding simple co-roots $h_{i}\coloneqq E_{i\,i}-E_{i+1\,i+1}$, with $i\in J_{n}$, we have that $[e_{i},f_{i}]=h_{i}$ and $\mathfrak{h}\coloneqq\bigoplus_{i}^{n}\mathbb{C}h_{i}$ is a Cartan subalgebra of $\mathfrak{g}$.

Let $D\coloneqq\bigoplus_{i}^{n+1}\mathbb{C}E_{i,i}$, $\mathfrak{h}$ be a vector subspace of $D$. The dual base of the standard base of $D$ is denoted as $\{\tilde{\varepsilon}_{i}\}_{i\in J_{n}}$, and the corresponding elements of $\mathfrak{h}^{*}$ are denoted as $\varepsilon_{i}=\tilde{\varepsilon}_{i}|_{\mathfrak{h}}$.

With this notation $\Phi\coloneqq\{\varepsilon_{i}-\varepsilon_{j}|1\leq i,j\leq n+1\}\setminus\{0\}\subset\mathfrak{h}^{*}$ is the root system of $\mathfrak{g}$, that is, the $\mathbb{C}E_{i\,j}=\{x\in\mathfrak{g}|[h,x]=(\varepsilon_{i}-\varepsilon_{j})(h)\cdot x\,\,\,\,\forall h\in\mathfrak{h}\}$ are the root spaces of $\mathfrak{g}$ with respect to $\mathfrak{h}$. The Weyl group of $\mathfrak{g}$, denoted by $\mathcal{W}$, is identified with the symmetric group $S_{n+1}$, which acts linearly on $\mathfrak{h}^{*}$ such that for $\sigma\in\mathcal{W}$

$\mathcal{W}$ is generated by the simple transpositions $s_{i}=(i,i+1)$ with $i\in I$, and the following relations form a complete system of relations

Let us denote the adjoint representation of $\mathfrak{g}$ by $\operatorname{ad}\colon\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{g})$, $x\mapsto\operatorname{ad}_{x}\coloneqq[x,-]$, and define

For all $i\in J_{n}$, it is easy to see that $\tau_{i}(\mathbb{C}E_{k\,l})=\mathbb{C}E_{s_{i}(k)\,s_{i}(l)}$ for all $k,\,l\in I$ whit $k\neq l$. In [3, 21.2], we can see that the weight spaces of the adjoint representation are permuted by the $\tau_{i}$ in the same way that the corresponding weights (roots in this case) are permuted by the $s_{i}$, that is $\tau_{i}(\mathfrak{h})=\mathfrak{h}$.

The aim of this work is to verify the following realations by the $\tau_{i}$’s in an elementary way.

Note that the relations (1.1) for $i=j$ imply $s_{i}^{2}=1$. Under this premise, the remaining relations of (1.1) are equivalent to (1.3), also known as braid relations.

It is evident that directly verifying the relations of Theorem 1.1 is a challenging task. To prove our result, we follow the steps of [5, Proposition 2.14], which is a more general result.

We will study the elements

and verify that these elements of the simple connected Lie group $\operatorname{SL}(n+1,\mathbb{C})$ fulfill “the same” relations as the elements $\tau_{i}$ of Theorem 1.1 inspired by the results of Millson and Toledano Laredo [5]. It is important to note that the “braid relations” in this context are classical ([7],[1]), while the other relations are considered “folklore”.

The theorem’s result will follow once we observe that

It is important to note that the automorphisms $\tau_{i}$ can be defined not only for the adjoint representation, but also for any finite-dimensional representation $V$ of $\mathfrak{g}$, see [3, 21.2]. In this context, they are relevant when one wants to see that the character of $V$ is invariant under the action of the Weyl group.

In this section, we will recall the arguments about generalised braid groups presented by Millson and Toledano ([5, Section 2]).

Let $\mathfrak{g}$ be a complex, semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and $\mathcal{W}$ be the Weyl group. Let $\Phi$ be a root system, $\Pi=\{\alpha_{1},\cdots,\alpha_{n}\}$ a basis of $\Phi$ and $V$ be a finite-dimensional $\mathfrak{g}$-module. We denote by $\mathfrak{h}_{reg}\coloneqq\mathfrak{h}\setminus\bigcup_{\alpha\in\Phi}\ker(\alpha)$ the set of regular elements.

Fix a basepoint $t_{0}\in\mathfrak{h}_{reg}$. Let $[t_{0}]$ be its image in $\mathfrak{h}_{reg}/\mathcal{W}$, and let $P_{\mathfrak{g}}=\pi_{1}(\mathfrak{h}_{reg};t_{0})$, $B_{\mathfrak{g}}=\pi_{1}(\mathfrak{h}_{reg}/\mathcal{W};[t_{0}])$ be the generalized pure and full braid groups of type $\mathfrak{g}$. The fibration $\mathfrak{h}_{reg}\longrightarrow\mathfrak{h}_{reg}/\mathcal{W}$ gives rise to the exact sequence

with $B_{\mathfrak{g}}\longrightarrow\mathcal{W}$ is obtained by associating to $p\in B_{\mathfrak{g}}$ to the unique $w\in\mathcal{W}$ such that $w^{-1}t_{0}=\tilde{p}(1)$, where $\tilde{p}$ is the unique lift of $p$ to a path in $\mathfrak{h}_{reg}$ such that $\tilde{p}(0)=t_{0}$.

Let $G$ be the complex, connected, and simply-connected Lie group with Lie algebra $\mathfrak{g}$, $T$ its torus with Lie algebra $\mathfrak{h}$, and $N(T)\subset G$ the normalizer of $T$ in $G$, so that $\mathcal{W}\cong N(T)/T$. We regard $B_{\mathfrak{g}}$ as acting on $V$ by choosing a homeomorphism $\sigma:B_{\mathfrak{g}}\longrightarrow N(T)$ compatible with the diagram in Figure 1.

By Brieskorn’s theorem (see [1, page 57]), $B_{\mathfrak{g}}$ is presented on generators $S_{1},S_{2},\cdots,S_{n}$, labelled by the simple reflections $s_{1},\cdots,s_{n}\in\mathcal{W}$, with relations

where $m_{ij}$ is the order of $s_{i}s_{j}$ in the Weyl group. One of the first works in which the relations from (3.1) appears is [7, Lemma 56].

Tits has given a simple construction of a canonical class of homomorphisms $\sigma$ that differ from each other via conjugation by an element of $T$, see [8].

This section will divide the proposition of Milson and Toledano [5, Proposition 2.14], into three new propositions to verify the relations (1.3)-(1.6) in our particular case, the Lie algebra $\mathfrak{sl}(n+1,\mathbb{C})$.

From this point on, we will be working with the Lie group of matrices $G=\operatorname{SL}(n+1;\mathbb{C})$, with $\mathfrak{g}=\mathfrak{sl}(n+1,\mathbb{C})$ its Lie algebra. For $\alpha_{i}$ a simple root as in section 2.3, we can define

where $I_{j}$ is the identity matrix in $\operatorname{GL}(j;\mathbb{C})$ and the matrix $0_{kl}\in\operatorname{Mat}_{k\times l}(\mathbb{C})$ is composed entirely of zeros.

It is easy to see that $G_{i}\cong\operatorname{SL}(2;\mathbb{C})$ and that the Lie algebra $\operatorname{Lie}(G_{i})$ is generated by

and

where $e,\,f,\,h\in\operatorname{Mat}_{2}(\mathbb{C})$ are as in Equation (2.7).

For any $i\in\{1,\cdots,n\}$, let $T_{i}\coloneqq\{\exp(xh_{i})|\,x\in\mathbb{C}\}\subset G_{i}$, for $1\leq i\leq n$, be an algebraic torus or more precisely:

Let $T\subset\operatorname{SL}(n+1;\mathbb{C})$ be the set of diagonal matrices with determinant equal to $1$.

Obviously, the Lie algebra of $T$ is equal to $\mathfrak{h}\subset\mathfrak{g}$, where $\mathfrak{h}$ is the subgroup of diagonal matrices which trace equal to zero.

For any $\sigma\in S_{n+1}$, we use $E_{\sigma}$ to denote $\mbox{sgn}(\sigma)E_{\sigma(1)}+\sum_{i=2}^{n+1}E_{\sigma(i)}$, where sgn$(\sigma)$ is the sign of $\sigma$ (which is equal to $1$ if $\sigma$ is even and is $-1$ if $\sigma$ is odd). It is easy to see that $\det\left(E_{\sigma}\right)=1$. Then, we have that the set $\{E_{\sigma}\mid\sigma\in S_{n+1}\}$ is a subset of $N(T)$.

On other hand, for any $\sigma,\tau\in S_{n+1}$, it holds true that:

Let $X\in N(T)$, and $\sigma\in S_{n+1}$ be such that $X=\sum_{i=1}^{n+1}x_{i}E_{\sigma(i)}$. Then, $XE_{\sigma}^{-1}$ is equal to $\operatorname{diag}(x_{1}\mbox{sgn}(\sigma),\,x_{2},\ldots,\,x_{n+1})$ and $\det(XE_{\sigma}^{-1})=\det(X)\det(E_{\sigma}^{-1})=1$. Thus, $XE_{\sigma}^{-1}\in T$, and $[X]=[E_{\sigma}]$, with $[X],\,[E_{\sigma}]\in N(T)/T$.