Exercise 23, § 2

Lie Groups and Lie Algebras, IV

Let $(W, S)$ be a Coxeter system and $k$ a commutative ring. Suppose that we are given, for all $s \in S$ , two elements $\lambda_s$ and $\mu_s$ of $k$ such that $\lambda_s = \lambda_{s'}$ and $\mu_s = \mu_{s'}$ whenever $s$ and $s'$ are conjugate in $W$ . Put $E = k^{(W)}$ and let $\{ e_w \}$ be the canonical basis of $E$ .

a) Show that $E$ has a unique algebra structure such that, for $s \in S$ and $w \in W$ ,

e_s \cdot e_w = \begin{cases} e_{sw} & \text{if } l(sw) > l(w) \\ \lambda_s e_w + \mu_s e_{sw} & \text{if } l(sw) < l(w) \end{cases}

(introduce the endomorphism $P_s$ of $E$ defined by the above formulas, where $e_s \cdot e_w$ is replaced by $P_s(w)$ , and the endomorphism $Q_s = j P_s j^{-1}$ , where $j$ denotes the automorphism of $E$ defined by $j(e_w) = e_{w^{-1}}$ ; show that $P_s Q_t = Q_t P_s$ for $s, t \in S$ by remarking that the conditions $l(swt) = l(w)$ and $l(sw) = l(wt)$ imply that $sw = wt$ ; then argue as in Exerc. 3 c)). The module $E$ , equipped with this algebra structure, will be denoted by $E_k((\lambda_s), (\mu_s))$ . Show that $E((0), (1))$ is the group algebra $k[W]$ of $W$ (Algebra, Chap. III, § 2, no. 6).

b) Show that the family of generators $(e_s)_{s \in S}$ and the relations

\begin{align*} e_s^2 &= \lambda_s e_s + \mu_s \quad \text{for } s \in S \\ (e_s e_t)^r &= (e_t e_s)^r \quad \text{for } s, t \in S \text{ such that } st \text{ is of finite even order } 2r \\ (e_s e_t)^r e_s &= (e_t e_s)^r e_t \quad \text{for } s, t \in S \text{ such that } st \text{ is of finite odd order } 2r + 1 \end{align*}

form a presentation of the algebra $E$ (argue as in the proof of Th. 1 of no. 6 of § 1).

Solution

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