数理解析研究所講究録
第 1605 巻 2008 年 68-80

68Real forms and finite order automorphisms ofaffine Kac-Moody algebras-

an outline of a

new approach
Ernst Heintze

1

Introduction

The classification of real forms and finite order automorphisms of affine Kac-Moody algebras has been achieved by the efforts of many people. In particular the works of F.
Levstein [L] and G. Rousseau and his collaborators ([B], [BR], [Rl], [R2], [R3], $[B_{3}R]$ ,[BMR] have to be mentioned here, but see also [A], [Bat), [BP], [C], [JZ], Kob andother papers. The classification probably fills some hundred pages and took about 15
years to get completed.
The purpose of this note is to report on a simpler, quite elementary approach whichin addition gives more complete results. It moreover has the advantage to work in thesmooth as well as in the algebraic category, that is for affine Kac-Moody algebras which
are extensions of loop algebras consisting of smooth resp. algebraic loops.
While the above mentioned authors always worked in the algebraic setting we are mainlyinterested in the smooth case which is more appropriate for the purpose of geometry.
Actually our interest in these questions orginated from the study of symmetric spaces
related to affine Kac-Moody groups and hence from the classification of involutions of“smooth” affine Kac-Moody algebras ([HPTT], [H]). But it turns out that the results are
the same in both cases.
Our work started several years ago and, at in early stage, in collaboration with ChristianGroiS.
This is an expanded version of a talk given at the Symposium “Geometry related tothe theory of integrable systems“ at RIMS, Kyoto, September 2007. Details will appear
elsewhere.
$)$

$[$

$|$

69

2Smooth and algebraic affine Kac-Moody algebras
Instead of working with abstract affine Kac-Moody algebras we directly consider theirso called realizations. These are certain two dimensional extensions of (twisted) loop
algebras as follows.
Let be a simple Lie algebra over the fieldor and assume in addition to be$\sigma\inAut(g)$. Let
compact if
be an arbitrary automorphism, not necessarily of finite
order. Then we call$\mathbb{F}=\mathbb{R}$$g$$\mathbb{C}$$\mathfrak{g}$$\mathbb{F}=\mathbb{R}$$L(g, \sigma)$$:=\{u :\mathbb{R}arrow g|u(t+2\pi)=\sigma u(t), u\in C^{\infty}\}$a (twisted) loop algebra,
being the untwisted loop algebra.
is a
Lie algebra w.r. the pointwise bracket $[u, v]_{0}(t)$ $:=[u(t), v(t)]$ . If has finite order, say, then the $u\in L(g, \sigma)$ satisfy $u(t+2\pi l)=u(t)$ and are thus indeed loops. Usually
one changes the parameter in this case by the factor , i.e. replaces $u(t)$ by$:=u(lt)$ ,and embeds
in this way into $L(g)$ . But this has the slight disadvantage to depend$L(\mathfrak{g})$$L(g, \sigma)$$:=L(\mathfrak{g}, id)$$t$$\sigma$$\sigma^{l}=id$$l$$\tilde{u}(t)$$L(g, \sigma)$on (which not necessarily needs to be the order of but could be any multiple of it).
Moreover such an embedding does not exist if has infinite order. But we will see laterthat any $L(g, \sigma)$ is isomorphic to a twisted loop algebrawith of finite order.
$l$$\sigma$$\sigma$$L(g,\tilde{\sigma})$$\tilde{\sigma}$

One may weaken the differentiability condition and consider loops of Sobolev class
1. Everything in the following works equally well. But this is not so clear for the smallest
loop algebra, which is usually considered in algebra. This consists of the socalled algebraic loops which are by definition finite Laurent series of the form$H^{k},$$k\geq$$L_{alg}(g, \sigma)$$u(t)= \sum_{q\in \mathbb{Q}}u_{q}e^{iqt}$

with $u_{q}\in g$ (resp.
, whereifdenotes the complexification of g). The
periodicity condition $u(t+2\pi)=\sigma u(t)$ requires
to lie in the subalgebra $\{x\in g|\sigma^{k}x=x$for some} on which has finite order. In order to ensure surjectivity of theevaluation map $u\mapsto u(t)$ one is hence forced to assume
to be of finite order in thealgebraic case. Ifthen periodicity implies that $u(t)$ is actually of the form
$g_{\mathbb{C}}$$\mathbb{F}=\mathbb{R}$$g_{\mathbb{C}}$$u_{q}$$k\in \mathbb{N}$$\sigma$$\sigma$$\sigma^{l}=id$$u(t)= \sum_{|n|\leq N}u_{n}e^{int/l}$

Therefore we let$L_{alg}( g, \sigma)=\{u\in L(g, \sigma)|u(t)=\sum_{|n|\leq N}u_{n}e^{int/l}, N\in N, u_{n}\in g_{(\mathbb{C})}\}$.

The definition does not depend on , one only has to assume. The same remark as
above applies here: by changing the parameter by a factor one might embed
into $L_{alg}(g)$ $:=L_{a1g}(g, id)$ and this is usually done. But for our purposes the above
definition is more convenient.
$l$$\sigma^{l}=id$$l$$L_{alg}(g, \sigma)$

70

So far we have only considered the loop algebras. The affine Kac-Moody algebra is thefollowing 2-dimensional extension$\hat{L}(g, \sigma)=L(g, \sigma)+\mathbb{F}c+\mathbb{F}d$with

for all $u,$

and$v,$ $\in L(g, \sigma)$$[u, v]$$=[u, v]_{0}+(u^{l}, v)\cdot c$$[d, u]$$=u^{l}$$[c, x]$$=0$

where $(u,v)= \int_{0}^{2\pi}(u(t), v(t))_{0}dt$ . Here$x\in\hat{L}(g, \sigma)$$(,$$)_{0}$

denotesthe derivative of .
the Killing form of andis a Lie algebra. The construction could have been done
One easily checks that$:=L(g, \sigma)+\mathbb{F}c$ first, with brackets as above. This isin two steps by introducing$:=(u’, v)$ .
a one-dimensional central extension of $L(g, \sigma)$ defined by the cocyclewith .
is then a semidirect product ofand , respectively. $L(g, \sigma)$are
The derived algebra and the center ofof theis not a subalgebra of
but rather isomorphic to the quotientderived algebra by its center.
is defined in the same way and the above remarkstoThe extension ofalso apply in this case. In the following we merely consider
and $L(g, \sigma)$ and comeback to the algebraic case only in the last section.
$u’$

$g$

$u$

$\hat{L}(g, \sigma)$$\tilde{L}(g, \sigma)$$\omega(u, v)$$\hat{L}(g, \sigma)$$\tilde{L}(\mathfrak{g}, \sigma)$$\mathbb{F}$$\hat{L}(g, \sigma)$$\tilde{L}(g, \sigma)$$\mathbb{F}c$$\hat{L}(g, \sigma)’/\mathbb{F}c$$\hat{L}(g, \sigma)$$\hat{L}_{alg}(g, \sigma)$$L_{alg}(g, \sigma)$$\hat{L}(g, \sigma)$3

Isomorphisms between affine Kac-Moody algebras

An important step in our approach is the description of isomorphisms between affine
Kac-Moody algebras. They turn out to have a particularly simple form.
:
Any isomorphism :induces an isomorphism
between the loop algebras. Therefore we begin by studying these ffist. Simple examples
of isomorphisms :are given by$\hat{L}(\mathfrak{g}, \sigma)arrow\hat{L}(\tilde{g},\tilde{\sigma})$$\hat{\varphi}$$\varphi$$\varphi$$L(g, \sigma)arrow L(\tilde{g},\tilde{\sigma})$$L(g, \sigma)arrow L(\tilde{g},\tilde{\sigma})$$\varphi u(t)=\varphi_{t}(u(\lambda(t)))$::is a smooth curve ofis a diffeomorphism and(and similarly) satisfies the periodicity condition
isomorphisms. In order thatfor all we only have to require

where$\lambda$$\mathbb{R}arrow \mathbb{R}$$t\mapsto\varphi_{t}$$\varphi u$$\varphi u(t+2\pi)=\tilde{\sigma}\varphi u(t)$$garrow\tilde{g}$$\varphi^{-1}u$

$t$

(1)
(2)

$\lambda(t+2\pi)$$\varphi_{t+2\pi}$$=\lambda(t)+\epsilon 2\pi$$=\tilde{\sigma}\varphi_{t}\sigma^{-\epsilon}$

for some. Condition (1) means that covers a diffeomorphism of the circle$-1$(resp.
) if A and hence are orientation preserving (reversing).
andWe call such isomorphisms standard and to be of first (second) kind if $\epsilon=1(\epsilon=-1)$ .
$\overline{\lambda}$$\epsilon\in\{\pm 1\}$$\epsilon=1$$\lambda$$\lambda$

71

Theorem 3.1$\mathcal{A}ny$

isomorphism$\varphi$:$L(g, \sigma)arrow L(\tilde{g},\tilde{\sigma})$

is standard.

The theorem reduces questions about automorphisms of finite order immediately to finite
dimensions. It also shows that and have to be isomorphic. Therefore we will assume
from now on. But and can be different. The periodicity condition (2) gives the
only restriction$\tilde{g}$$\mathfrak{g}$$\tilde{g}=g$$\tilde{\sigma}$$\sigma$$\tilde{\sigma}=\varphi_{t+2\pi}\sigma^{\epsilon}\varphi_{t}$implying thatandare conjugate in Autg/Intg. Note thatis isomorphic
to the symmetry group of the Dynkin diagram and thus isomorphic to either 1,or(the symmetric group in three letters) and that hence each element is conjugate to its
inverse. Moreover the conjugacy class ofis determined by its order, which can be 1, 2
or 3.
$Aut\mathfrak{g}/Intg$$[\sigma]$$[\tilde{\sigma}]$$\mathbb{Z}_{2}$$S_{3}$$[\sigma]$

Conversely if
andare conjugate it is easy to find a smooth curve
satisfying (2). We thus have:$[\sigma]$$[\tilde{\sigma}]$$\varphi_{t}$of automorphism

Corollary 3.2
andare isomorphic if and only if andare conjugate
in Autg/Intg. In particular any twisted loop algebm is isomorphic to one with offinite$L(\mathfrak{g}, \sigma)$$L(g,\tilde{\sigma})$$[\sigma]$$[\tilde{\sigma}]$$\sigma$$|order$.

Remark 3.3 In connection with real forms section 5 it is interesting to note that Corollary 3.2 also holds in case a real non compact simple Lie algebra (by the same proof).
But in this case $Autg/Intg\cong 1,$(the dihedral group) or. Hence theorder ofin Autg/Intg is not enough in this case to distinguish conjugacy classes.
$)$

$($

$g$

$\mathbb{Z}_{2},$$\mathbb{Z}_{2}\cross \mathbb{Z}_{2},$$D_{4}$$S_{4}$$[\sigma]$

The proof of Theorem 3.1 consists of several steps. For simplicity let us assume. We then can define
andbyfor all $x\in g$ where denotes theconstant loop
. Now, the main point is to prove the existence of a function:
with$\sigma=\tilde{\sigma}=id$$\mathbb{F}=\mathbb{C}$$\varphi_{t}$$\varphi_{t}(x)=\varphi(\hat{x})(t)$$\hat{x}$$\hat{x}(t)\equiv x$$\lambda$$\mathbb{R}arrow \mathbb{R}$(3)$\varphi(f\cdot u)$$=(f\circ\lambda)\cdot\varphi(u)$for all $u\in L(g)$ and smooth -periodic :. In fact, if
is a basis of$u(t)=\Sigmaf_{i}(t)x_{i}$and, we then get
as desired. To
prove (3) we first show that for any fixed $u,$ and$a;=\varphi(fu)(t)$ and $b;=\varphi u(t)$ arelinearly dependent. This follows by observing$2\pi$$f$$\mathbb{R}arrow \mathbb{R}$$x_{1},$$\ldots,$$g$$x_{n}$$\varphi(u)(t)=\Sigma f_{i}(\lambda(t))\varphi_{t}(x_{i})=\varphi_{t}(u(\lambda(t)))$
$f$

$ad$

$a$

$t,$

$adx_{1}\ldots adx_{k}adb=adbadx_{1}\ldots adx_{k}ad$ $a$

for allandand then applying a classical theorem of Burnside to obtain
$ad$
$ad$
A $adb=adb$ A
for all $A\in End\tilde{g}$ . We next show $\varphi(fu)(t)=\alpha(f)\cdot\varphi u(t)$ for
all and but still fixed for some algebra homomorphism from the set of $2\pi\sim periodic$
smooth functions to . In the last step we prove $\alpha(f)=f(t^{*})$ for someand set.
$k\in \mathbb{N}$$x_{i}\in\overline{g}$

$a$

$u$

$a$

$f$

$t$

$\alpha$$\mathbb{C}$$\lambda(t):=t^{*}$$t^{*}\in \mathbb{R}$72

between affine Kac-Moody algeWe finally consider isomorphism :algebrathey are necessarily of thederivedthebras. Since they preserve the center andform$\hat{L}(g, \sigma)arrow\hat{L}(g,\tilde{\sigma})$$\hat{\varphi}$$=\mu_{1^{C}}$$\hat{\varphi}$(4)$\hat{\varphi}d=\mu_{2}d+u_{\varphi}+\nu_{\varphi}c$$\hat{\varphi}u=\varphi u+\alpha(u)\cdot c$:isis linear andare constants,
where$\varphiu(t)=$
the induced isomorphism between the loop algebras. From Theorem 1 we haveis a diffeomorphism with $\lambda(t+2\pi)=\lambda(t)+\epsilon 2\pi$
and :
where
. We call and to be of the first (second) kind if is of the first (second)andresp. $\epsilon=-1)$ .
kind, i.e. if$\mu_{1},$$\mu_{2},$$u_{\varphi}\in L(g,\tilde{\sigma}),$$\nu_{\varphi}\in \mathbb{F}$$\varphi_{t}(u(\lambda(t)))$$\lambda$$\varphi_{t}\in Aut\mathfrak{g}$$\epsilon\in\{\pm 1\}$$\hat{\varphi}$$\epsilon=1$$L(g, \sigma)arrow \mathbb{F}$$\alpha$$\varphi$$\mathbb{R}arrow \mathbb{R}$$\varphi$$\varphi$$($

is induced from then is linear, $i.e$ .
for some
is induced by an isomorphisms. Conversely, any such isomorphism
unique (up to the choice ofessentially
this
is
and
between the affine Kac-Moody algebmsin (4), which can be arbitmry).
extendingare precisely the onesMore precisely if $\varphi u(t)=\varphi_{t}(u(\epsilon t+t_{0}))$ then the
(4).
in
and
satisfying

Theorem 3.4 If$\epsilon\in\{\pm 1\},$$\varphi u(t)=\varphi_{t}(u(\epsilon t+t_{0}))$$\lambda$$\hat{\varphi}$$\varphi$$t_{0}\in \mathbb{R}$$\hat{\varphi}$$\varphi$$\nu_{\varphi}$$\hat{\varphi}$$\varphi$$\mu_{1}=\mu_{2}=\epsilon,$$\alpha(u)=-\epsilon(\varphi u, u_{\varphi})$$adu_{\varphi}=-\epsilon\varphi_{t}’\varphi_{t}^{-1}$Corollary 3.5 There is a bijection between automorphisms of finite order of$L(g, \sigma)$ .

$\hat{L}(g, \sigma)$andhas finite order. Conversely if has
In fact, if has finite order then the induced
finite order then there is precisely one of finite order extending namely the one withsplits as. The reason for this is thatfor some
and the second
on $L(g,$factor contains no elements of finite order.
$\hat{\varphi}$$\varphi$$\varphi$$\hat{\varphi}$$\varphi$$Aut\hat{L}(g, \sigma)$$\nu_{\varphi}=-\frac{\epsilon\Vert u_{\varphi}\Vert^{2}}{2}$$- \frac{\epsilon||u_{\varphi}||^{2}}{2}\}x\{\hat{\varphi}|\hat{\varphi}=id$4$\sigma)+\mathbb{F}c,\hat{\varphi}d=d+\nu_{\varphi}c$$\{\hat{\varphi}\in Aut\hat{L}(g, \sigma)|\nu_{\varphi}=$$\nu_{\varphi}\in \mathbb{F}\}$

Automorphisms of finite order

Rom the results of the last sections it foUows that classifying conjugacy classes of auis equivalent to classifying conjugacy classes oftomorphisms of finite order of$L(g,\sigma)$and the aim of this section is to describe such a
automorphisms of finite order of$\hat{L}(g, \sigma)$classification.
Thus let :be of finite order. We know that has the form $\varphi u(t)=$. After awithand $\lambda(t+2\pi)=\lambda(t)+\epsilon 2\pi$ for somewith $t_{0}=0$ if $\epsilon=-1)$ . This comes fromfirst conjugation we may assume
the fact that diffeomorphisms of the circle of finite order (like the one induced by ) areconjugate to a rotation or a reflection.
. A particularly simple case is the one whereThus we assumeis always conjugate to such an automorphism.
is constant and one may ask whether
We have studied those automorphisms in [HPTT]. The answer is the following.
$\varphi$$\varphi_{t}(u(\lambda(t)))$$L(g, \sigma)arrow L(\mathfrak{g}, \sigma)$$\varphi$$\epsilon\in\{\pm 1\}$$\varphi_{t+2\pi}=\sigma\varphi_{t}\sigma^{-\epsilon}$$\lambda(t)=\epsilon t+t_{0}$$($$\lambda$$\varphi u(t)=\varphi_{t}u(\epsilon t+t_{0})$$\varphi_{t}\equiv\varphi_{0}$$\varphi$

73

Theorem 4.1(i) Not every automorphism of $L(g, \sigma)$ of finite order is conjugate to one with$\varphi_{t}$con-stant.

(ii) But for every $\varphi\in Aut(L(g, \sigma)$ offinite order there exists
an isomorphismus :such that.
that is$L(g, \sigma)arrow L(g,\tilde{\sigma})$$\psi$$a$atogether withhas constant ,$\in Aut(g)$$\tilde{\varphi}:=\psi\varphi\psi^{-1}$$\tilde{\varphi}_{t}$$\tilde{\varphi}u(t)=\tilde{\varphi}_{0}(u(\epsilon t+t_{0}))$We
and quasiconjugate in the above situation to emphasize that has maybe
changed. But for automorphisms: $L(g, \sigma)arrow L(g, \sigma)$ on the same loop algebra,
quasiconjugate and conjugate are the same.
By the above results (quasi)conjugacy classes of automorphisms of finite order of $L(g, .)$andare classified by certain quadruples
withand$\sigma\in Autg$ modulo some equivalence relation. The equivalence relation of course reflects
the fact that certain quadruples correspond to the same quasiconjugacy class.
Although it is possible to prove Theorem 4.1 directly we choose a slightly different pathand associate first to any automorophism of finite order an “invariant”, not only to thosewith constant . We then prove that this is indeed invariant under quasiconjugations
and that it moreover distinguishes quasiconjugacy classes. Finally we show that eachpossible invariant is attained, even by an automorphism with constant. This proves
also Theorem 4.1, part (ii). Considering automorphisms of order of the firstand$(\epsilon=-1)$
kindfollowing
separately, we obtain more precisely thesecondresults.
$can_{\varphi}$$\sigma$$\tilde{\varphi}$$\varphi,\tilde{\varphi}$$\hat{L}(g, .))$$\epsilon\in\{\pm 1\},$$(\epsilon, t_{0}, \varphi_{0}, \sigma)$$t_{0}\in \mathbb{R}$$\varphi_{0},$$\varphi_{t}$$\varphi_{t}$$(\epsilon=1)$$q$

Theorem 4.2 Let
be a compact or $\omega mplex$ simple Lie algebm as above. Then thequasiconjugacy classes of $aut_{omo7}phisms$ of the first kind of order on the various $L(g, .)$$.))$ are in bijection with$:=\{(p, \rho, [\beta])/p\in \mathbb{Z},$ $0\leq p\leq q/2,$ $\rho\in Autg$orfroma list of representativs of conjugacy classes of $automo\varphi hisms$ of of order $(p, q)$ and
$g$

$q$

$($

$\hat{L}(g,$$3_{1}^{q}(\mathfrak{g})$$g$$\beta\in(Autg)^{\rho}\}$.

Here $(p, q)$ is the greatest common divisor of and anddenotes the conjugacy classof.
If $\varphi u(t)=\varphi_{t}u(t+t_{0})$ on $L(g, \sigma)$ is of order then we define its invariant as follows.
Necessarily
for some
and we may assume $0\leq p<q$ . Let $r:=(p, q),p^{l}$ $:=$
$p/r,$$:=q/r$ andwith $lp’+mq^{l}=1$ and $0\leq l<q’$ . Then $\varphi^{q’}u(t)=\rho_{t}(u(t))$
andfor some. Moreover has order and is thus of
the formwith from a list of order automorphisms (modulo conjugation)
Autg. We letandbe the invariant of . It is easily checkedthat this invariant does not change if is quasiconjugated by an isomorphism of the firstkind. But has to be replaced by $:=q-p$ $(p’=0 if p=0)$ after a quasiconjugation
by an isomorphism of the second kind, which explains the restriction $0\leq p\leq q/2$ in. In this way we get a mapping $hom$ quasiconjugacy classes tothe definition of. While injectivity of this mapping is the hard part surjectivity follows easily. In
fact, the invariant $(p,$may be realized as follows. Let
and
$L(g,
\sigma)$$\varphi u(t)=\varphi_{0}(u(t+p/q2\pi))$ with$m,p’,$
invariant,isofleavesabove.
Then
asorder and has invariant
.
$p$

$q$

$[\beta]$$\overline{\beta}\in\pi_{0}((Autg)^{\rho})=(Aut\mathfrak{g})^{\rho}/((Autg)^{\rho})_{0}$$q$$p\in \mathbb{Z}$$t_{0}=L^{\underline{2\pi}}q$$l,$ $m\in \mathbb{Z}$$q’$$\varphi^{l}u(t)=\Lambda_{t}(u(t+\frac{2\pi}{q}))$$\rho_{t},$$\rho_{t}=\alpha_{t}\rho\alpha_{t}^{-1}$$\Lambda_{t}\in Autg$$q$$\rho$$(p, \rho, [\alpha_{t+2\pi/q’}^{-1}\Lambda_{t}^{-1}\alpha_{t}])$$\alpha_{t}\in$$r$$\rho_{t}$$\varphi$$\varphi$$p^{l}$$p$$3_{1}^{q}(g)$$J_{1}^{q}(g)$$\rho,$$[\beta|)$$\sigma$

$l,$

$q$

$(\rho,p,$$[\beta|)$$q’$$\varphi$$:=\rho^{l}\beta^{q’},$$\varphi_{0}:=\rho^{m}\beta^{-p’}$74

To determine conjugacy classes of automorphisms of order on a fixed $L(g, \sigma)$ we only
have to restrict the invariants to those for which the above example is defined on a loop
algebra isomorphic to. By virtue of Corollary 3.2 this is equivalent tohavingthe same order as
in $\mathcal{A}utg/Intg$ .
In the case of involutions $(q=2)$ things are of course easier. Here $p=0$ or $p=1$ . In thelatter case $(p, q)=1$ , hence $\rho=id$ and these automorphisms are classified by conjugacy
classes of Autg/Intg. They are represented by $\varphi u(t)=\varphi_{0}(u(t+\pi))$ on
where$\varphi_{0}\in Autg$ runs through alist of representations of Autg/Intg (and may thus be chosen
to have order 1, 2 or 3). In particular they do not occur on $L(g, \sigma)$ if
is inner, butnot. In the former case the involutions with invariant $(0,$may be represented by$\varphi u(t)=\rho u(t)$ on $L(g, \beta)$ where $\rho\in Autg$ runs throughthe conjugacy classes of involutions
and $\beta\in(Autg)^{\rho}$ runs through the conjugacy class of $(Autg)^{\rho}/((Autg)^{\rho})_{0}$ . This lastgroup is known to be isomorphic to 1,
or
where
and
denote thedihedral and symmetric groups, respectively. Thus these involutions correspond to finite
dimensional symmetric spaces plus a certain extra information.
The situation for automorphisms of the second kind is described in the following Theorem.
Note that their order is necessarily even.
$q$$L(\mathfrak{g}, \sigma)$$[\rho^{l}\beta^{q’}]$$[\sigma]$$L(g, \varphi_{0}^{-2})$$\sigma^{2}$$\sigma$$\rho,$$\mathbb{Z}_{2},$$\mathbb{Z}_{2}x\mathbb{Z}_{2},$$D_{4}$$[\beta|)$$S_{4}$$D_{4}$$S_{4}$

Theorem 4.3 Let be as above. Then the quasiconjugacy classes of automorphisms oforder of the second kind on the various
are in bijection with$/\sim$ where the equivalence relation is genemted(Autg)2, order}byandfor all $\beta\in Autg$ with$g$

$L(\mathfrak{g}, .)$$q$$(\varphi_{+}, \varphi_{-})\sim(\varphi_{-}, \varphi_{+})$$\alpha^{-1}\beta\in((Autg)^{\varphi_{+}^{2}})_{0}$$\mathfrak{J}_{-1}^{q}(g)=\{(\varphi_{+}, \varphi_{-})\in$$\varphi_{\pm}^{2}=q/2$$|\varphi_{+}^{2}=\varphi_{-}^{2}$$(\varphi_{+}, \varphi_{-})\sim(\alpha\varphi_{+}\alpha^{-1}, \beta\varphi_{-}\beta^{-1})$$\alpha,$.

If $\varphi\in AutL(g, \sigma)$ has order and is already of the form $\varphi u(t)=\varphi_{t}(u(-t))$ thenhasorder $q/2$ and there exists
with. The periodicity condition
implies. Henceandhaveorder $q/2$ and satisfy. We thus can define
to be the invariant of andhence get a mapping from the quasiconjugacy classes to. Again, surjectivity follows
easily. In fact, the equivalence class
may be represented by the automorphisms$:=\varphi_{+}u(-t)$ andon $L(g, \sigma)$ with. (Note thatand hencesatisfies the periodicity condition).
Let us again look at the special case of involutions $(q=2)$ . Since$q$$\varphi_{t}\varphi_{-t}$$\varphi_{t}\varphi_{-t}=\alpha_{t}\varphi_{0}^{2}\alpha_{t}^{-1}$$\alpha_{t}$$\sigma\varphi_{t}\sigma$$(\varphi_{\pi}\sigma^{-1})^{2}=\varphi_{\pi}\varphi_{-\pi}$$\varphi_{t+2\pi}=$$\varphi+;=\alpha_{0}^{-1}\varphi_{0}\alpha_{0}$$\varphi_{+}^{2}=\varphi_{-}^{2}$$\varphi_{-}:=\alpha_{\pi}^{-1}\varphi_{\pi}\sigma^{-1}\alpha_{\pi}$$[\varphi_{+}, \varphi_{-}]$$\varphi$$J_{-1}^{q}(g)$$[\varphi_{+}, \varphi_{-}]$$\varphi u(t)$$\varphi$$\sigma$$:=\varphi_{-}^{-1}\varphi_{+}$$\varphi_{+}=\sigma\varphi_{+}\sigma$$\varphi$$J_{-1}^{2}=\{[\rho_{+}, \rho_{-}]|\rho_{\pm}^{2}=id\}$involutions of the second kind correspond essentially to pairs of compact symmetric spaces.
More precisely if
and is the compact simply connected Lie group with Lie algebra
thengives rise to the symmetric pairs $(G, K_{+})$ and $(G, K_{-})$ where $K\pm=G^{\rho\pm}$(assuming here $\rho\pm\neq id$ ). The action of $K_{+}\cross K_{-}$ on $G$ by $(k_{+}, k_{-}).g=k_{+}gk_{-}^{-1}$ is
hyperpolar, that is admits a flat section (actually a torus) which meets every orbit andalways orthogonally. The $K_{+}xK$-action on
is also called a Hermann action andKollross [K] has shown that essentially all hyperpolar actions on are Hermann actions.
Moreoverandare equivalent by the above equivalence relation if andonly if the corresponding Hermann actions are equivalent. Thus quasiconjugacy classes
of involutions of the second kind on the affine Kac-Moody algebras
essentiallycorrespond to equivalence classes of Hermann actions on .
$\mathbb{F}=\mathbb{R}$

$g$

$G$

$[\rho+, \rho_{-}]$

$G$

$G$

$(\rho+’\rho_{-})$$(\tilde{\rho}+,\tilde{\rho}_{-})$$\hat{L}(g, .)$

$G$

75

5

Real forms and involutions

Let be a complex simple Lie algebra. Real forms ofor $L(g, \sigma)$ are in bijection with
antilinear involutions that is with involutions satisfying(resp. $\varphi(ix)=$$-i\varphi(x))$ .
But antilinear automorphisms of finite order on these algebras can be treatedexactly the same way as linear ones.
In particular we only have to study the antilinear involutions on $L(g, \sigma)$ and these are likein the linear case of the form
, but now with
antilinear. Therefore
we can associate invariants to them as above and show that their quasiconjugacy classes
are parametrized by the setsof these invariants where is the order of and iseither 1 or $-1$ depending on whether is orientation preserving or reversing. The real
forms corresponding toare also called almost compact if
and almost split if$\epsilon=-1$ .
If $u\subset g$ is a compact -invariant real form of then lt $:=L(u, \sigma)$ is a real
form of $:=L(g, \sigma)$ which is also called a compact real form. It is the fixed point set of$\varphi u(t)=\omega(u(t)),$the conjugation of w.r.t . Automorphisms of order on can beextended to linear as well as to antilinear automorphisms on (of order if is even and$2q$ if
is odd) and the induced mappings on the sets of invariants tum out to be bijections.
For example in case $q=2$ one gets bijections
and. To explain these results we recall first the finite dimensional situation.
Up to conjugation the complex, simple Lie algebra has precisely one compact real form,
say . If is an involution of then the eigenspace decomposition
gives rise to
the real form
, which is non compact. In this way one gets a bijection between
conjugacy class of involutions on and conjugacy classes of non compact real forms of. Moreover each noncompact real form
has by construction a Cartan decomposition,that is a decomposition $g_{\mathbb{R}}=g+m$ into the eigenspaces of an involution such that $e+im$
is a compact Lie algebra and it turns out that this is unique up to conjugation.
Now the above results may be summerzied as follows.
$\hat{L}(g, \sigma)$$\mathfrak{g}$$\hat{\varphi}(ix)=-i\hat{\varphi}(x)$$\varphi u(t)=\varphi_{t}(u(\lambda(t)))$$\varphi_{t}$$\overline{3}_{\epsilon}^{q}(g)$$q$$\varphi$$\epsilon$$\lambda$$\epsilon=1$$\overline{J}_{\epsilon}^{q}(g)$$\sigma$$g$$\emptyset$$\omega$

$u$

$g$

$u$

$q$

$\emptyset$

$q$

$q$

$q$

$\mathfrak{J}_{1}^{2}(u)\cup 3_{1}^{1}(u)rightarrow\overline{\mathfrak{J}}_{1}^{2}(g),$$\mathfrak{J}_{-1}^{2}(g)rightarrow\overline{\mathfrak{J}}_{-1}^{2}(g)$$J_{\epsilon}^{2}(u)rightarrow J_{\epsilon}^{2}(g)$

$g$

$u$

$u$

$\rho$$u=f+\mathfrak{p}$$u^{*}=g+i\mathfrak{p}$

$u$

$g$

$g_{\mathbb{R}}$

Theorem 5.1 The situation for affine Kac-Moody algebras is exactly the same as thatfor finite dimensional simple Lie algebms. More precisely:(i) Each complex affine Kac-Moody algebm
has a ‘compact real form‘;$e.g$ . $u:=L(u, \sigma),$a -invariant compact real form, and this is unique up to
conjugation.
$\emptyset\wedge$$u\subset \mathfrak{g}$$:=\hat{L}(g, \sigma)$$t$$\sigma$(ii) Conjugacy classes of non compact real forms ofclasses of involutions of (and ).
$4\hat{1}$(iii) Each non compact real form of
to conjugation.

$\emptyset\wedge$

are in bijection with conjugacy$\hat{\emptyset}$$\hat{\emptyset}$

has a Cartan decomposition and this is unique up

Thus the classification of involutions ofgives also a classification of real forms. To
describe this more explicitly we first specialize our classification of finite order automorphisms to involutions.
$\hat{L}(g, \sigma)$

Theorem 5.2 Up to $quasi\omega njugation$ the involutions on the various $L(g, .)$ are given by

76la) $u(t)\mapsto\rho(u(t))$ on $L(g, \beta)$

on$lb)u(t)\mapsto\varphi(u(t+\pi))$2$)$$u(t)\mapsto\rho_{\dagger}(u(-t))$on$L(g, \varphi^{2})$$L(g, \rho_{-}\rho_{+})$,

where runs thmugh a list of conjugacy classes of involutions of $g,$
and represent(Autg) resp. andconjugacy classes ofrepresent equivalence classes$\{(\rho+, \rho_{-})\in(Autg)^{2}/\rho_{\pm}^{2}=id\}/\sim$ . Since all automorphisms can be chosen
to leave theofcompact real form of invartant, the above mappings classify also involutions on thevarious $L(u, .)$ .
$\beta$$\rho$$\pi_{0}((Autg)^{\rho}),$ $\pi_{0}$$u$$\varphi$$(\rho_{+}, \rho_{-})$$g$$(1, id, [\varphi])$ and
The invariants of these involutions are, respectively. The
involutions of type lb) correspond to the various isomorphism classes of the $L(g, \sigma)$ . If
has no outer autmorphism there exists precisely one such involution (given by $u(t)\mapsto$$u(t+\pi)$ on $L(g))$ and otherwise two unless
is of typewhich has three.
According to Theorem 5.1 the real forms (of the various $L(g,$ and up to quasiconjugation)
are thus the compact real forms $L(u, .)$ (corresponding to) and the fixed point setsofwhere is an involution $hom$ the list above. Here $\tilde{\omega}(u)(t)=\omega(u(t))$ , where
denotes conjugation w.r. . . From this we obtain the following classification.
$(0, \rho, [\beta]),$$[\rho+, \rho_{-}]$$g$$D_{4}$$g$$.)$$J_{1}^{1}(\iota c)$$\tilde{\omega}0\psi$$\psi$$\omega$$t$$u$

Theorem 5.3 The real forrns of the various $L(g, .)$ are up to quasiconjugation represented$by$runs through all conjugacy classes of real forms of andruns thmugh a list of representatives of conjugacy classes of.

la) $L(g_{R}, \sigma)\subset L(g, \sigma)$ where$\sigma\in Autg_{\mathbb{R}}$$g$$g_{\mathbb{R}}$$\pi_{0}(Autg_{\mathbb{R}})$$1b)\{u\in L(u, \varphi^{2})|u(t+\pi)=\varphi(u(t))\}+i\{u\in L(u, \varphi^{2})|u(t+\pi)=-\varphi(u(t))\}$,$\varphi\in Autu$

2$)$

represents a conjugacy class of $\pi_{0}(Autu)\cong\pi_{0}(Autg)$ .

$\{\Sigma u_{n}e^{int/\iota}\in L(g, \sigma)|u_{n}\in g^{\omega\rho+}\}$

sents an equivalence class ofwhere$\{(\rho+,$ $\rho_{-})\in$

The comesponding real forms of$lb)$ andin case 2).

$\hat{L}(\mathfrak{g}, .)$$\sigma=\rho_{-}\rho_{+}$(Autg)2

has order$l$$|\rho_{+}^{2}=\rho_{-}^{2}\}/\sim$

are obtained by adjoining

and$(\rho_{+}, \rho_{-})$.

$\mathbb{R}c+\mathbb{R}d$where
repre-

in case la) and$\mathbb{R}(ic)+\mathbb{R}(id)$Remarks 5.4(i)can always be chosen such that
Lemma 6.2.
$\rho\pm$$\rho_{-}\rho_{+}$

has finite order (in fact $\leq 4$ ), cf. also(ii) While the real forms in la) and lb) constitute the almost compact ones, those in 2)
are the almost split.

(iii) Cartan decompositons can be obtained in all cases above by intersecting the real
forms with curves in and $iu$ , respectively (assuming thatis -invariant in la) .
For example, if one considers in 2) only loops
which converge inthe whole puncutred planethen the two summands of the Cartan decomposition
consist of functions whose restriction to the real line is contained in
while
their restriction to the unit circle is contained in and $iu$ , respectively (and which
satisfy the appropriate periodicity condition).
$u$$g_{\mathbb{R}}$$\Sigma u_{n}\lambda^{n}$$)$$\omega$$:=\Sigma u_{n}e^{int/l}$$\mathbb{C}^{*}$$g_{\mathbb{R}}$$u$$:=g^{\omega\rho+}$

77

To illustrate Theorem 5.3 we consider the simplest case
. The algebra
has no outer automorphisms and up to conjugation only one involution, which may be$g=\mathfrak{s}\mathfrak{l}(2, \mathbb{C})$represented e.g. by $\tau=Ad(1 -1)$ or by$\mu$$\epsilon 1(2, \mathbb{C})$with $\mu A=-A^{t}$ . As compact real form

we may take $u=$ su(2). We let be the conjugation w.r. . , i.e.
. Inparticular,. Then the real forms of
are up to quasiconjugation$t$$\omega$$g^{\omega\mu}=51(2, \mathbb{R})$$u$$\omega A=-\overline{\mathcal{A}}^{t}$$L(51(2, \mathbb{C}))$the following.
la)$L(\mathfrak{s}u(2)),$$L(\epsilon 1(2, \mathbb{R})),$$L(\epsilon 1(2, \mathbb{R}), \tau)$lb) $\{u\in L(u)|u(t+\pi)=u(t)\}+i\{u\in L(u)|u(t+\pi)=-u(t)\}$2$)$$\{\Sigma u_{n}e^{int}\in L(\epsilon 1(2, \mathbb{C}))|u_{n}\in\epsilon u(2)\},$$\{\Sigma u_{n}e^{int}\in L(\epsilon 1(2, \mathbb{C}))|u_{n}\in 5[(2, \mathbb{R})\}$$\{\Sigma u_{n}e^{int/2}\in L(\mathfrak{s}\mathfrak{l}(2, \mathbb{C}), \mu)|u_{n}\in sl(2, \mathbb{R})\}$

.

, and

$(0, \mu, [id]),$ $(0, \mu, [\tau]),$ $(1, id, [id])$
They correspond to the invariants $(0,$ $id,$
and $[id, id],$, respectively. They are all real forms
of $L(sl(2, \mathbb{C}))$ except the third and the last one which are real forms of a twisted loop
algebra of. But an isomorphism :is for example$\psi_{t}=e^{adtX}$$\psiu(t)=\psi_{t}(u(t))$
given bywithsuch thatand this carries thecorresponding real form into one of $L(sl(2, \mathbb{C})$ .
$[id|)\in \mathfrak{J}_{1}^{1}(\mathfrak{s}\mathfrak{l}(2, \mathbb{C})),$$\in 3_{1}^{2}(51(2, \mathbb{C}))$$[\mu, \mu],$$[\mu, id]\in J_{2}^{2}(\epsilon 1(2, \mathbb{C}))$$\epsilon 1(2, \mathbb{C})$$\psi$$L(\epsilon 1(2, \mathbb{C}), \sigma)arrow L(\mathfrak{s}\mathfrak{l}(2, \mathbb{C}))$$\psi_{2\pi}=\sigma^{-1}$The algebraic case6

In contrast to the above smooth setting all authors so far (with the exception of [HPTT])
have considered the algebraic case, i.e. automorphisms and real forms ofand. This case is more rigid and thus more subtle. For example it is much harder
to find and “algebraic” isomorphism which conjugates two given smoothly conjugate automorphisms of. But it turns out that our methods and ideas also work in thissetting when suitably refined and combined with a result of Levstein [L]. In fact we have:$\hat{L}_{alg}(g, \sigma)$$L_{alg}(g, \sigma)$$\hat{L}_{alg}(g, \sigma)$

Theorem 6.1 Conjugacy classes of finite order automorphisms and real forms ofandare classified by the same invariants as in the smooth case and are thus inbijection to those of, respectively. Moreover the $non\omega mpact$ real forms
or
admit
a
decomposition
Cartanand this is unique up to $\omega njugation$ .
of$\hat{L}_{alg}(g, \sigma)$$L_{alg}(g, \sigma)$$\hat{L}(g, \sigma)$$L(g, \sigma)$$L_{alg}(g, \sigma)$

The proof follows along the same lines as above but has to be modified at several points.
To begin with, isomorphisms:
are not necessarily standard(assumingin the following). They are rather of the form
with standard$\varphi$$L_{alg}(g, \sigma)arrow L_{alg}(g,\tilde{\sigma})$$\mathbb{F}=\mathbb{C}$$\tilde{\varphi}\circ\tau_{r}$$\tilde{\varphi}$and$\tau_{r}(\sum_{|n|\leq N}u_{n}e^{int/t})=\sum_{|n|\leq N}u_{n}r^{n}e^{int/t}$

for some positive . The reason for this is that the algebra homomorphism :in the proof of Theorem 3.1 is not necessarily evaluation at some point when restricted
to the algebraic functions
but more generally of the form$r$$\alpha$$C^{\infty}(S^{1})arrow$$\mathbb{C}$$\{\sum_{|n|\leq N}a_{n}e^{int}\}$$\sum_{|n|\leq N}a_{n}z^{n}arrow$

78$\sum_{|n|\in N}a_{n}z_{0}^{n}=\sum_{|n|\leq n}a_{n}r^{n}e^{int^{*}}$

Fortunately, it turns out that these extra isomorphisms$\tau_{r}$donot affect the discussion of finite order automorphisms in an essential way. For example
automorphism of the first kind of finite order are always standard and those of the second
kind are conjugate to a standard one. Thus it is completely sound to work only withstandard isomorphisms. Since these can be viewed as special isomorphisms between thecorresponding smooth algebras we can apply the results of section 4 and associate to eachautmorphism of order of
an invariant and thus get a mappingfrom thesetof quasiconugacy classes of such automorphisms on thevarious $L_{alg}(g, .)$ into
, whereor-l depending on whether the automorphisms
are of the first or second kind. The goal is to prove bijectivity of these mappings.
Surjectivity follows essentially $hom$ what has been done in the smooth case. In fact,
the construction there yielded for each invariant a $\sigma\in Autg$ and an automorphism
of $L(g, \sigma)$ with this invariant of the form
. This also preserves, but in the algebraic case we have to ensure that has finite order. For example
ifthen the construction yieldedwhich is in general not of
finite order. But
can be replaced by an equivalent pair and the result follows
by showing first the existence of a-invariant compact real form of and then by
applying the next Lemma to.
$L_{alg}(g, \sigma)$$q$$I_{\epsilon}^{q}$$\mathcal{A}ut_{\epsilon}^{q}(L_{alg}(g, .))/Aut(L_{alg}(g, .))$$\epsilon=1$$\mathfrak{J}_{\epsilon}^{q}(g)$$\varphi$$u(t)\mapsto\varphi_{0}(u(\epsilon t+t_{0}))$$L_{alg}(g, \sigma)$$\varphi$$\sigma$$[\varphi+, \varphi_{-}]\in J_{-1}^{q}(g)$$\sigma$$:=\varphi_{-}^{-1}\varphi_{+}$$(\varphi+, \varphi_{-})$$u$$\varphi\pm$$g$$G:=\{\alpha\in(\mathcal{A}utg)^{\varphi_{\pm}^{2}}|\alpha(u)=n\}$Lemma 6.2 Let$g_{-}\in G$ withbe a compact Lie group andexists $h\in G_{0}$ such thatis of finite order.
$G$$g_{+},$$g_{+}^{2}=g_{-}^{2}$. Then there$(hg_{-}h^{-1})^{-1}\cdot g_{+}$

But the main problem is the injectivity of . Here our elementary methods do not suffice.
To solve it, we use the following basic result of Levstein.
$I_{\epsilon}^{q}$

Theorem 6.3 (Levstein [L], [R2]) Let be an automorphism ofThen
has a Cartan subalgebm which is invariant under .

$\hat{L}_{dg}(g, \sigma)$$\hat{\varphi}$of finite order.

$\hat{L}_{alg}(g, \sigma)$$\hat{\varphi}$

Since after a conjugation we may assume that leaves the standard Cartan subalgebra
invariant which consists of constant loops
we conclude that thefromis constant and this implies
for some $X\in g$ and,becauseassuming. In the next step we even get
rid of the-factor by a further quasiconjugation. Here it is essential to allow to
be replaced by someAutg. Thus we have also in the algebraic case the result thatany automorphism of finite order is quasiconjugate to one of the form,and
. The proof of the injectivity of is therefore considerably
simplified in that we have only to consider these very special automorphisms. Surprisingly,
it follows finally ffom the hyperpolarity ( . section 4) of the -action (for automorphismsof the first kind) and the Hermann action (for automorphisms of the second kind). If
is a compact connected Lie group and $\sigma\in AutG$ then the action of
on itself by$g.x$ $:=gx\sigma(g)^{-1}$ is called theaction. A maximal torus of $G^{\sigma}=\{g\in G/\sigma g=g\}$ meets
every orbit and always orthogonally (w.r. . any biinvariant metric).
$\hat{\varphi}$$+\mathbb{C}c+\mathbb{C}d$$u_{\varphi}$$\varphi_{t}=e^{adtX}\varphi_{0}$$\epsilon d+u_{\varphi}+\nu_{\varphi}c$$\varphi_{t}’\varphi_{t}^{-1}=-\epsilon adu_{\varphi}$$\hat{\varphi}d=$$\varphi_{0}\in Autg$$\varphi u(t)=\varphi_{t}(\epsilon t+t_{0}))$$($$e^{adtX}$$\sigma$$\tilde{\sigma}\in$$\hat{\varphi}c=\epsilon c,\hat{\varphi}d=\epsilon d$$\varphi u(t)=\varphi_{0}(u(\epsilon t+t_{0}))$$I_{\epsilon}^{q}$

$s$

$G$

$\sigma$$G$$\sigma$$t$

The same ideas also work for antilinear automorphisms of finite order and in the real case
. Therefore the results about real forms and Cartan decompositions carry overfrom the smooth to the algebraic setting without any difficulties.
$(\mathbb{F}=\mathbb{R})$79

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Math. Phys. 7, World Scientific (1989), 70–85$\mathcal{A}lmost$Ernst Heintze
Institut f\"ur Mathematik, Universit\"at Augsburg
Universit\"atsstrasse 14
D-86159 Augsburg, Germany
e-mail address: $heintze@ma^{f}h$ . uni-augsburg. $de$