Loop Lie algebras and their extensions

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 31 January 2010

Loop Lie algebras and their extensions

This section gives a presentation of the theory of loop Lie algebras. The main lines of the theory are exactly as in the classical case (see, for example, [Mac2, 4] and [Kac, ch. 7]) but, following recent trends (see [Ga2], [GK], [GR] and [Rou]) we treat the more general setting of the loop Lie algebra of a Kac-Moody Lie algebra.

Let ${𝔤}_0$ be a symmetrisable Kac-Moody Lie algebra with bracket ${\left[,\right]}_0:{𝔤}_0\otimes {𝔤}_0\to {𝔤}_0$ and invariant form ${⟨,⟩}_0:{𝔤}_0\otimes {𝔤}_0\to ℂ.$ The loop Lie algebra is $${𝔤}_0\left[t,t^{-1}\right]=ℂ\left[t,t^{-1}\right]{\otimes }_{ℂ}{𝔤}_0\text{ with bracket }{\left[t^{m}x,t^{n}y\right]}_0=t^{m+n}{\left[x,y\right]}_0,$$ for $x,y\in {𝔤}_0.$ Let $$𝔤={𝔤}_0\left[t,t^{-1}\right]\oplus ℂc\oplus ℂd,𝔤\text{'}={𝔤}_0\left[t,t^{-1}\right]\oplus ℂc,{\stackrel{-}{𝔤}}^{\text{'}}={𝔤}_0\left[t,t^{-1}\right]=\frac{𝔤\text{'}}{ℂc}$$ where the bracket on $𝔤$ is given by $$\left[t^{m}x,t^{n}y\right]=t^{m+n}{\left[x,y\right]}_0+{\delta }_{m+n,0}m{⟨x,y⟩}_{0}c,c\in Z\left(𝔤\right),\left[d,t^{m}x\right]=m{t}^{m}x.$$

By [Kac, Ex. 7.8], $𝔤\text{'}$ is the universal central extension of ${\stackrel{-}{𝔤}}^{\text{'}}.$ An invariant symmetric form on $𝔤$ is given by

$$⟨c,d⟩=1,⟨c,t^{m}y⟩=⟨d,t^{m}y⟩=0,⟨c,c⟩=⟨d,d⟩=0,$$

and

$$⟨t^{m}x,t^{n}y⟩=\left\{\begin{array}{r}{⟨x,y⟩}_0,\text{if }m+n=0,\\ 0,\text{otherwise,}\end{array}\right.$$ for $x,y\in {𝔤}_0,m,n\in \mathbb{Z}.$

Fix a Cartan subalgebra ${𝔥}_0$ of ${𝔤}_0$ and let

$$𝔥={𝔥}_0\oplus ℂc\oplus ℂd,𝔥\text{'}={𝔥}_0\oplus ℂc,\stackrel{-}{𝔥}\text{'}={𝔥}_0.$$

As in (2.2), let $h_1,\dots ,h_n,d_1,\dots ,d_l$ be a basis of ${𝔥}_0$ and let

$$\left\{h_1,\dots ,h_n,d_1,\dots ,d_l,c,d\right\}\text{ be a basis of}𝔥\text{ and}$$

$$\left\{{\omega }_1,\dots ,{\omega }_n,{\delta }_1,\dots ,{\delta }_l,{\Lambda }_0,\delta \right\}\text{ the dual basis in }{𝔥}^{*}$$

so that

$$\delta \left({𝔥}_0\right)=0,\delta \left(c\right)=0,\delta \left(d\right)=1,\text{and}{\Lambda }_0\left({𝔥}_0\right)=0,{\Lambda }_0\left(c\right)=1,{\Lambda }_0\left(d\right)=0.$$

Let $R$ be as in (2.9). As an $𝔥$-module

$$𝔤=\left(\underset{\alpha \in R,k\in \mathbb{Z}}{⨁}{𝔤}_{\alpha +k\delta }\right)\oplus \left(\underset{k\in {\mathbb{Z}}_{\ne 0}}{⨁}{𝔤}_{k\delta }\right)\oplus 𝔥,\text{where }𝔥={𝔥}_0\oplus ℂc\oplus ℂd,$$

$${𝔤}_{\alpha +k\delta }=t^k{𝔤}_{\alpha },{𝔤}_{k\delta }=t^k{𝔥}_0,\text{and}\tilde{R}=\left(R+\mathbb{Z}\delta \right)\cup {\mathbb{Z}}_{\ne 0}\delta$$

is the set of roots of $𝔤$.

Let $\alpha \in R_{\mathrm{re}}$ with $\alpha =w{\alpha }_i$ and fix choice of $e_{\alpha },f_{\alpha },h_{\alpha }$ in (2.18) (choose $\tilde{w}$). Then

$$e_{-\alpha +k\delta }=t^k{f}_{\alpha },f_{-\alpha +k\delta }=t^{-k}{e}_{\alpha },h_{-\alpha +k\delta }=-h_{\alpha }+k{⟨e_{\alpha },f_{\alpha }⟩}_{0}c,$$

span a subalgebra isomorphic to $𝔰{𝔩}_2.$ If ${𝔤}_0={𝔫}_0^{-}\oplus {𝔥}_0\oplus {𝔫}_0^{+}$ is the decomposition in (2.10) and $${𝔫}^{+}\text{ is the subalgebra generated by }{𝔫}_0^{+}\text{ and }{e}_{-\alpha +k\delta }\text{ for }\alpha \in R_{\mathrm{re}},k\in {\mathbb{Z}}_{>0},\text{ and}$$ $${𝔫}^{-}\text{ is the subalgebra generated by }{𝔫}_0^{-}\text{ and }{f}_{-\alpha +k\delta }\text{ for }\alpha \in R_{\mathrm{re}},k\in {\mathbb{Z}}_{>0},$$ then $$𝔤={𝔫}^{-}\oplus 𝔥\oplus {𝔫}^{+}\text{ with }{𝔫}^{+}={𝔫}_0^{+}\oplus \left(\underset{\alpha \in R\cup 0,k\in {\mathbb{Z}}_{>0}}{⨁}{𝔤}_{\alpha +k\delta }\right)\text{ and }{𝔫}^{-}={𝔫}_0^{-}\oplus \left(\underset{\alpha \in R\cup 0,k\in {\mathbb{Z}}_{<0}}{⨁}{𝔤}_{\alpha +k\delta }\right).$$

The elements $e_{\alpha +k\delta }$ and $f_{\alpha +k\delta }$

The elements $e_{-\alpha +k\delta }$ and $f_{-\alpha +k\delta }$ in (5.9) act localy nolpotently on $𝔤$ because $f_{\alpha }$ and $e_{\alpha }$ act locally nilpotently on ${𝔤}_0$. Thus

$${\tilde{s}}_{-\alpha +k\delta }=\exp \left(\mathrm{ad}{t}^k{f}_{\alpha }\right)\exp \left(-\mathrm{ad}{t}^{-k}{e}_{\alpha }\right)\exp \left(\mathrm{ad}{t}^k{f}_{\alpha }\right)$$

is a well defined automorphism of $𝔤$ and

$${\tilde{s}}_{-\alpha +k\delta }{𝔤}_{\beta }={𝔤}_{s_{-\alpha +k\beta }}\text{and}{\tilde{s}}_{-\alpha +k\delta }h=s_{-\alpha +k\delta }h,$$

for $h\in 𝔥$ and $\beta \in \tilde{R},$ where $s_{-\alpha +k\delta }:{𝔥}^{*}\to {𝔥}^{*}$ and $s_{-\alpha +k\delta }:𝔥\to 𝔥$ are given by

$$s_{-\alpha +k\delta }\lambda =\lambda -\lambda \left(h_{-\alpha +k\delta }\right)\left(-\alpha +k\delta \right)\text{ and }{s}_{-\alpha +k\delta }h=h-\left(-\alpha +k\delta \right)\left(h\right)h_{-\alpha +k\delta },$$

for $\lambda \in {𝔥}^{*}$ and $h\in 𝔥.$ The Weyl group of $𝔤$ is the subgroup of $\mathrm{GL}\left({𝔥}^{*}\right)$ ( or $\mathrm{GL}\left(𝔥\right)$) generated by the reflections $s_{-\alpha +k\delta },$ $$W_{\mathrm{aff}}=⟩s_{-\alpha +k\delta }|\alpha \in R_{\mathrm{re}},k\in \mathbb{Z}⟩$$

Noting that $𝔥*={𝔥}_0^{*}\oplus ℂ{\Lambda }_0\oplus ℂ\delta \text{ and }𝔥={𝔥}_0\oplus ℂc\oplus ℂd,$ use (5.12) to compute

$\begin{array}{rcl}{s}_{-\alpha +k\delta }\left(\stackrel{-}{\lambda }\right)& =& \stackrel{-}{\lambda }+\stackrel{-}{\lambda }\left(h_{\alpha }\right)\left(-\alpha +k\delta \right)\\ s_{-\alpha +k\delta }\left(\stackrel{-}{h}\right)& =& \stackrel{-}{h}+\alpha \left(\stackrel{-}{h}\right)\left(-h_{\alpha }+k{⟨e_{\alpha },f_{\alpha }⟩}_{0}c\right)\\ s_{-\alpha +k\delta }\left(l{\Lambda }_0\right)& =& l{\Lambda }_0-kl{⟨e_{\alpha },f_{\alpha }⟩}_0\left(-\alpha +k\delta \right)\\ s_{-\alpha +k\delta }\left(mc\right)& =& mc\\ s_{-\alpha +k\delta }\left(m\delta \right)& =& m\delta \\ s_{-\alpha +k\delta }\left(l\delta \right)& =& ld-kl\left(-h_{\alpha }+k{⟨e_{\alpha },f_{\alpha }⟩}_{0}c\right)\end{array}$

for $\stackrel{-}{\lambda }\in {𝔥}_0^{*},\stackrel{-}{h}\in {𝔥}_0,m,l\in ℂ.$ For $\alpha \in R_{\mathrm{re}}$ and $k\in \mathbb{Z}$

$$\text{define }{t}_{k{\alpha }^{\vee }}\in W_{\mathrm{aff}}\text{ by }{s}_{-\alpha +k\delta }=t_{k{\alpha }^{\vee }}{s}_{-\alpha },$$

and use (2.26) and (2.27) to compute

$\begin{array}{rcl}{t}_{k{\alpha }^{\vee }}\left(\stackrel{-}{\lambda }\right)& =& \stackrel{-}{\lambda }-\stackrel{-}{\lambda }\left(k{h}_{\alpha }\right)\delta \\ t_{k{\alpha }^{\vee }}\left(\stackrel{-}{h}\right)& =& \stackrel{-}{h}-k{\alpha }^{\vee }\left(\stackrel{-}{h}\right)c\\ t_{k{\alpha }^{\vee }}\left(l{\Lambda }_0\right)& =& l{\Lambda }_0+lk{\alpha }^{\vee }-l\frac{1}{2}{⟨k{h}_{\alpha },k{h}_{\alpha }⟩}_0\delta \\ t_{k{\alpha }^{\vee }}\left(mc\right)& =& mc\\ t_{k{\alpha }^{\vee }}\left(m\delta \right)& =& m\delta \\ t_{k{\alpha }^{\vee }}\left(l\delta \right)& =& ld+lk{h}_{\alpha }-l\frac{1}{2}{⟨k{h}_{\alpha },k{h}_{\alpha }⟩}_{0}c\end{array}$

Then $t_{k{\alpha }^{\vee }}{t}_{j{\beta }^{\vee }}\left(\stackrel{-}{\lambda }\right)=t_{k{\alpha }^{\vee }}\left(\stackrel{-}{\lambda }-\stackrel{-}{\lambda }\left(j{h}_{\beta }\right)\delta \right)=\stackrel{-}{\lambda }-\stackrel{-}{\lambda }\left(k,h_{\alpha },+,j,h_{\beta }\right)\delta$, and

This computation shows that $t_{k{\alpha }^{\vee }}{t}_{j{\beta }^{\vee }}=t_{j{\alpha }^{\vee }+k{\beta }^{\vee }}.$ Thus, if $W_0$ is the Weyl group of ${𝔤}_0$ and $Q^{*}=\mathbb{Z}\left\{{\alpha }_1^{\vee },\dots ,{\alpha }_n^{\vee }\right\}$ then

$$W_{\mathrm{aff}}=\left\{t_{{\lambda }^{\vee }}w|{\lambda }^{\vee }\in Q^{*},w\in W_0\right\}\text{with}{t}_{{\lambda }^{\vee }}{t}_{{\mu }^{\vee }}=t_{{\lambda }^{\vee }+{\mu }^{\vee }}\text{and}w{t}_{{\lambda }^{\vee }}=t_{w{\lambda }^{\vee }}w,$$

for $w\in W_0,{\lambda }^{\vee },{\mu }^{\vee }\in Q^{*}.$

Since $ℂ\delta$ is $W_{\mathrm{aff}}$-invariant, the group $W_{\mathrm{aff}}$ acts on ${𝔥}^{*}/ℂ\delta$ and $W_{\mathrm{aff}}$ acts on the set $\begin{array}{rcl}\left({𝔥}_0^{*}+{\Lambda }_0+ℂ\delta \right)/ℂ\delta & \tilde{\to }& {𝔥}_0^{*}\\ \stackrel{-}{\lambda }+{\Lambda }_0+ℂ\delta & \mapsto & \stackrel{-}{\lambda }\end{array}$

and the $W_{\mathrm{aff}}$-action on the right hand side is given by

$$s_{\alpha }\left(\stackrel{-}{\lambda }\right)=\stackrel{-}{\lambda }-\stackrel{-}{\lambda }\left(h_{\alpha }\right)\alpha \text{and}{t}_{k{\alpha }^{\vee }}\left(\stackrel{-}{\lambda }\right)=\stackrel{-}{\lambda }+k{\alpha }^{\vee },\text{for }\stackrel{-}{\lambda }\in {𝔥}_0.$$

Here ${𝔥}_0^{*}$ is a set with a $W_{\mathrm{aff}}$-action, the action of $W_{\mathrm{aff}}$ is not linear.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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