144

数理解析研究所講究録
第 761 巻 1991 年 144-146

A decomposition of the adjoint representation of$U_{q}(\mathfrak{s}l_{2})$SUSUMU ARIKI
Department of Inforn\’iation Engineeringand Logistics
Tokyo University of Mercantile Marine
Etchujima Koto-ku, Tokyo 135, Japan

Quantum algebra. First, we introduce notation.

be an associalive algeb $ra/K=Q(q)(q$ is an
DEFINITION. Lel, anindeterminate.), defined by a system of general $ors;e,$their relalions:$U_{q}^{(l)}$$f,k^{1},k^{-\frac{1}{}}$$k^{\frac{1}{2}}k^{-\frac{1}{2}}=1$,$k^{f}ek^{-z}11=qe$$d$$k^{-\frac{1}{}}h^{\frac{1}{2}}=1$$k^{1}\tau fk^{-1}=q^{-1}f$$ef-fe= \frac{k^{2}-k^{-2}}{q^{2}-q^{-l}}$$As$usnal, we givea Hopf alge bra structure by $eq$ uippin il witln$U_{q}$$g$:$\Delta$$e e\otimes h^{-1}+k\otimes e$$f f\Phi h^{-1}+k\emptyset f$$k^{1} k^{1}\Phi k^{\iota}$$S$:$erightarrow-q^{-2}e,$$\epsilon$:$erightarrow 0,$$f$}$U_{q}^{(l)}$$U_{q}^{(m)}$$U_{q}^{(\iota)}$$k^{-1}f,$$-qf,$$frightarrow 0,$

DEFINITION.
(1)denoles the $su$ balgebra of(2)denol es lhe $sub$ algebra of$U_{q}^{(m)}$$-\rangle$$K=h^{2},$$K^{-1}$.

$k^{1}2\mapsto k^{-1}$$k^{1}\mapsto 1$$e,$

sysl $em$ of generat $ors;B,$ $F,$
REMARK. If we choose an, lh en and becomefor$ol1\iota er$$U_{q}^{(l)}$$\Delta$.

generaled by $f,$
generaled by $E=ek,$ $F=$$k,$$k^{-1}$$k^{-1}$$h^{1},$$S$$\Delta$:$B B\Phi 1+k^{2}\Phi B$$F F\otimes k^{-}+1QF$$S$

We pul,:$Brightarrow-k^{-}B,$ $F\mapsto-Fh$$C=fe+ \frac{qk+q^{-}k^{-}}{t^{q^{2}}-c^{-2})^{2}}$Typeset by-TEX$A_{\mathcal{M}}S$

145$A\iota boi_{I1}t$

Representation.

DBFINITION.

$U_{q}^{(\mathfrak{l})}$becomes a$U_{q}^{(t)}- r\mathfrak{n}odu1e$by$Ad(e)x=erk-q^{-2}ksee$$Ad(f)x=$

faeh $-q^{2}hrf$$( r\in U_{q}^{(t)} )$$Ad(k^{1})ae=k^{1}nk^{-1}$’

We den $ote$ it by $(Ad, U_{q}^{ad})$ , an we call it tlte adjoin representaiion.
$d$

DBFINITION. We$def_{I}\iota lesub_{I1}$$t$odules of$U_{q}^{ud}$as follo $ws$ :$V_{\alpha+:}=Ad(U_{q}^{(\dagger)})k^{\alpha+*}$$(\alpha\in Z)$$V_{l\alpha+1}=Ad(U_{q}^{(1)})k^{2\alpha+1}$$(\alpha\in Z)$$V_{\alpha}=Ad(U_{q}^{(1)})C^{-}$ “ $k^{2}$$(\alpha\in Z\leqq 0)$$V_{a}=Ad(U_{q}^{(l)})k^{-\alpha+}e^{\alpha}+Ad(U_{q}^{(t)})k^{-\infty+}f^{\alpha}$(ct E $Z>0$ )$V_{\iota oe}=\oplus,,\geq 0K[C]Ad(U_{q}^{(\mathfrak{l})})h^{-n}\epsilon$“DEFINITION.
(1)$X(d)=U_{q}^{(t)}/U_{q}^{(t)}\langle h^{i}- q^{t})$It has a aturally in duced
regular represen tation ofWe put,

$U_{l}^{(\mathfrak{l})}$$11$$U_{q}^{(\mathfrak{l})}$$v_{d}=1$

mod.

$t1_{t}e$$ble\iota I\iota odu$$\tau$IIBMAAK.

$\tau$

in duces an isom$n_{ecol11}1^{ooiti_{ol1}}$-module structure usirtg$U_{q}^{(1)}$(2) $L(d)$ isirreducile(3) is a K-algebr aaut omorphism
respec ti vely.
$orpl_{I}is\iota \mathfrak{n}$$(d\in Z)$($h$weiglrt$f,$
to $f,$$w1_{1}ic1_{1}\epsilon e\iota 1dse,$$U_{l}^{ad}=V_{\iota uc}\oplus(\oplus_{n\epsilon!I}V_{n})$(2)$g^{\oplus(\oplus_{n\epsilon zV_{\hslash}}}$).

$h^{\}}$

between $X(d)$ an $dX(-d)$(1)

2$q^{\ell}$$wit1_{1}t1\iota e1\iota igl_{1}e\epsilon t$$r_{1ltBonEM}$。

lefl#-q)of tlte AdJoint lLepresentation.

$U_{q}^{(1n)}=V_{1}$$tl_{1}e$$e,$.

$h^{-\}}$

146($)$U_{q}^{(\ell)}=V_{\iota}$$c^{\oplus t\oplus_{\mathfrak{n}\epsilon 2zV_{\hslash})}}$。

(4) Any irreducible $submodu$le ofis $con$ tained in$V_{n}(n\in\frac{1}{2}Z)s$(5)are indecomposable modules.
(6) $V_{2a}(\alpha\in Z_{>0})$ is isom orph $ic$ to$U_{q}^{oed}$$V_{l}$$e$。

.

$X(\alpha)\oplus X(-\alpha)/(-id\oplus\tau)(U_{q}^{(l)}f^{\alpha}v_{a})$are isomorph to $X(0)$ .
are mutually on-isomorph .
(7) $X(0)$ andis Rnitely gen erated an is in decom(8) If a direct summan of
posable, then it is isom orphic to a $L(d)$ , or a direc summan of$ic$$V\rho(\beta\not\in 2Z>0)s$$V_{2\alpha}s(\alpha\in Z_{>0})$
$d$

$ic$

$n$

$d$

$U_{q}^{ad}$$t$$X(0)^{\oplus}\oplus(\oplus_{j=1}^{r}V_{2j})$$d$.

REFERENCES(1). M. Jimbo, A q-difference analogue of $U(g)$ and the Yang.Baxter equation, Lett.
in Math. Phys. 10 (1985), 63-69.
(2). G. Lusztig, Quantum deformations of certtin simple modules over enveloping$alg$ ebras, Adv. in Math. 70 (1988), 237-249.
(The author is grateful to Dr. M. Noumi for showing him this problem.)3