数理解析研究所講究録 1227 巻 2001 年 209-217

209

Aconstruction of compact matrix
quantum groups and description of
the related-algebras*$\mathrm{C}^{*}$

JANUSZ WYSOCZA\’{N} SKI \dager
Institute of Mathematics, Wroclaw University
pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland
March 14, 2001

Abstract
Aconstruction of compact matrix quantum groups is
given. The construction is based on Woronowicz’s theory. Afundamental role in the construction is played by ageneralized determinant, related to permutation groups.
Description of thealgebras related to the quantum
groups is given in terms of irreducible ’-representations
on Hilbert spaces.
$\mathrm{C}’-$

1Introduction
In [SLW2] Woronowicz presented the following idea of compact
matrix quantum groups ( . the proof of Theorem 1.1). Let
be acompact group ofcomplex matrices. An
element $g\in G$ is then amatrix with entries $gjk$ and the entries’: $G\ni g\vdasharrow gjk=wjk(g)\in \mathrm{I}$ form acollection
functions$\{wjk : 1\leq j, \ \leq N\}$ ofcontinuous functions on the group. In terms of these functions we can describe various algebraic
properties of the group. The idea is that we can reflect algebraic
group properties as properties of the ’-algebra generated by thesefunctions,the complex conjugation.
Let us first consider the multiplication in . When two
matrices $h\in G$ are multiplied, the standard rule of multiplication of entries is expressed by the entries’ functions as$\mathrm{c}.\mathrm{f}$$G\subset lVI_{N}(\oplus)$$\mathrm{N}\mathrm{x}\mathrm{N}$$w_{jk}$$N^{2}$$G$$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}*\mathrm{i}\mathrm{s}$

$G$

$g,$

$w_{jk}(g \cdot h)=\sum_{\mathrm{r}=1}^{N}w_{j\mathrm{r}}(g)\cdot w_{tk}(h)=\sum_{r=1}^{N}(w_{j_{\Gamma}}\otimes w_{\mathrm{r}k})(g\otimes l_{l})$’Research partially supported by KBN grant-mail:jwys@math.uni.wroc.pl$\uparrow \mathrm{e}$$2\mathrm{P}03\mathrm{A}05415$.

210
reflects theHence the transformation
multiplication in . This transformation is therefore called c0multiplication.
Now let us consider the inverse in $G$ , which is the transformation $G\ni g\vdash*g^{-1}\in G$ . This can also be expressed in termsof the entries’ functions. Namely, by achange of the scalar prodwe can obtain unitary representation of , so theuct , inmatrixwill become the conjugate matrix. If astrictly
inversepositive matrix $M$ gives the change of the scalar product into
then
the new one $[,]$ , so that $[x, y]:=\langle Mx, y\rangle$ for$wjk(g*)=w_{kj}’\overline{(g)}$$g^{-1}=M^{-1}g*M$
.
is
Since
$M=g*Mg$ andacomplex conjugate combined with the transposition, it follows
that$\Phi(wjk)=\Sigma_{\mathrm{r}=1}^{N}wjf\otimes w_{\mathrm{r}k}$$G$$\langle$$\rangle$$G^{N}$$G$$x,$$y\in \mathcal{O}^{N}$$w_{jk}(g^{-1})= \sum_{r,s=1}^{N}(M^{-1})_{jr}w_{rs}(g*)(M)_{sk}=\sum_{r,s=1}^{N}(M^{-1})_{jt}\overline{w_{rs}(g)}(M)_{sk}$Hence the transformation(1.1)$\kappa(w_{\mathrm{j}k}):=\sum_{r,s=1}^{N}(M^{-1})_{jr}(M)_{sk}\overline{w_{rs}}$reflects taking the inverse in . This transformation istherefore called $co$ -inverse. The equality above shows, thatcan be expressed as alinear combination of complex conjugationsof of the entries’ functions, so it is an element of the ’-algebra
generated by these functions.
Let us now look at the properties of the group identity. Let$e\in G$ be the group identity, which is theidentity matrix.
Then for any $g\in G$ we have$G$$\kappa$$\kappa(wjk)$$\mathrm{N}\mathrm{x}\mathrm{N}$$\delta_{jk}=w_{jk}(e)=w_{jk}(gg^{-1})=\Sigma_{r=1}^{N}w_{jr}(g)w_{rk}(g^{-1})$$=\Sigma_{r=1}^{N}w_{jr}(g)\kappa(w_{rk})(g)=\Sigma_{r=1}^{N}(w_{j_{\Gamma}}\cdot\kappa(w_{rk}))(g)$This yields the equalities for the entries’ functions$\delta_{jk}\cdot I=\sum_{r=1}^{N}w_{jr}\kappa(w_{rk})=\sum_{r=1}^{N}\kappa(w_{jr})w_{rk}$These identities reflect the properties of the identity matrix
in the group , so they constitute the properties of the so called-unit. This way we see that, without having the group given
itself, we can “recover” it from the properties of aassociated c0structure. This -structure is what one calls the quantum group.
The notion of acompact matrix pseudogroup, later renamedfor compact matrix quantum group, was introduced by Woronowicz in [SLW2], to name aC’-algebraic structure which reflects$\mathrm{G}$$\mathrm{G}$$co$$\mathrm{c}\mathrm{o}$211group properties on the $C’-$ algebraic level. It consists of aC’algebra and an $N$ by $N$ matrix $u=(u_{jk})_{j,k=1}^{N}$ , with the ele-subalgebraA of , and withments $u_{jk}\in A$ generating a
the following additional structure:: $Aarrow A\otimes A$ , called the-homomorphism
1. a
multiplication, such that
$A$

$A$

$\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}*$$C^{*}$$\Phi$$\mathrm{c}+$(1.2)$\Phi(u_{jk})=\sum_{r=0}^{N}u_{j\mathrm{r}}\otimes u_{rk}$2. alinear anti-multiplicative mapping : $Aarrow A$ , called
for all elements $a\in A$ ,the -inverse, such that
and$\kappa$$\kappa(\kappa(a’)^{*})=a$$\mathrm{c}\mathrm{o}$(1.3)$\sum_{r=1}^{N}\kappa(u_{j_{\Gamma}})u_{\mathrm{r}k}=\delta_{jk}I$

$N$

(1.4)

$\sum u_{j\mathrm{r}}\kappa(u_{rk})=\delta_{jk}I$$r=1$

Let us mention that later in 1995 Woronowicz re-formulated
this definition in the following way. The compact quantum groupis apair $(A, \Phi)$ , consisting of aunital C’-algebra and a $C’-$homomorphism , such that:(1)diagram$A$$\Phi$$\mathrm{T}\mathrm{h}\mathrm{e}$$Aarrow^{\Phi}A\otimes A\downarrow A\otimes A^{\underline{d\otimes\varphi}},$$A\otimes A\otimes A$(1.5)

is commutative(2) The sets {(b@ $1)\Phi(c)$ : $c\in A$ } and { $(1\otimes b)\Phi(c)$ :.
are both dense inComparing the two definitions one may wonder, given thesecond definition, how to reconstruct the ’-subalgebra $A$ which
seems essential in the first definition. The answer comes from thetheory of unitary representations of compact quantum groups,
and says that this ’-subalgebra is generated by linear combination of matrix coefficients of the unitary representations of .
In [SLW3] Woronowicz provided ageneral method for constructing compact matrix pseudogroups. The method dependsof-element arrayon finding acomplex numbers, which is (left and right) non-degenerate. The
Theorem 1.4 of [SLW3] says that if a $C’-$ algebra , is generated
which satisfy:
elements
by
$b,$

$b,$

$c\in$$A\otimes A$

$A\}$

$A$

$\mathrm{n}$$N^{N}$$E=(E_{i_{1},\ldots,i_{N}})_{i_{1},\ldots,i_{N}=1,\ldots,N}$$A$$N^{2}$$u_{jk}$$\sum_{\mathrm{r}=1}^{N}u_{j\gamma}^{*}u_{rk}=\delta_{jk}I=\sum_{r=1}^{N}uj’.u_{rk}^{*}$

(1.6)

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(1.7)

$\sum_{k_{1,\ldots\prime}k_{N}}u_{j_{1}k_{1}}\ldots u_{j_{N}k_{N}}E_{k_{1\prime\cdots\prime}k_{N}}=E_{j_{1\prime}\ldots,j_{N}}I$

and if the array is non-degenerate, then $(A, u)$ is acompact
matrix quantum group, where $u=(ujk)_{j,k=1}^{N}$ . If for $\mu\in(0,1]$$N$ isone defines
if $\sigma(k)=i_{k}$ for $k=1,$
apermutation of $\{1, \ldots, N\}$ andotherwise, then as$(A,u)$ one gets the quantum group $S_{\mu}U(N)$ , called the twisted$SU(N)$ group. Here, for apermutationis the numberof inversions of the permutation , which is the number of pairs$(j,k)$ such that $j<k$ and $i_{j}=\sigma(j)>\sigma(k)=i_{k}$ . In this paper
we present, for $N=3$ , this construction for another function on
gives rise to another array .
$E$$E_{i_{1},\ldots,*_{N}}.=(-\mu)^{i(\sigma)}$$\ldots,$$E_{i_{1},\ldots,i_{N}}=0$$\sigma,$$i(\sigma)$$\sigma$$E$$\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s},\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{h}$2Compact quantum groups associated with cycles in permutations
In this section we describe the matrix quantum groups that arise,through the general receipt of Woronowicz, by considering thefunction related to the number of cycles on symmetric group.
We shall consider here the case of.
For asequence $(i,j, k)$ , with $\{i,j, k\}=\{1,2,3\}$ , we define
the function $c(i,j, k)$ as the number of cycles of the permutation$\mathrm{N}=3$$(\begin{array}{l}\mathrm{l},2,3i,j,k\end{array})$. For$t>\mathrm{O}$if
0if$t^{3-\mathrm{c}(i,j,k)}$$E_{i,j,k}=\{$

we define the array$E$in the following way:$\{i,j, k\}=\{1,2,3\}$$\{i,j,k\}\subseteq\{1,2,3\}$$\#\{i,j, k, \}\leq 2$(2.8)

Then the non-zero entries of the array
are $E_{1,2,3}=1$ ,$E_{1,3,2}=E_{2,1,3}=E_{3,2,1}=t$ and $E_{2,3,1}=E_{3,1,2}=t^{2}$ .
In the sequel we shall study the Hilbert space irreducible *representations of the $C’-$ algebra generated by the elements$\{ujk:j, k=1,2,3\}$ . The relations generating the algebra follow
from the general theory of the unitary representations compact
quantum groups. We shall skip these considerations in this exposition.
Let us say only, that the $C’-$ algebra , and hence the quantum group $(A, u)$ is generated by five elements $a,b,c,$ $d,v$ , which
satisfy the following relations:
$E$

$A$

$A$

(1) $av=va$(2) $cv=vc$(3) ac-l $tca=0$(4) $ac’+tc^{*}a=0$ (5) $cc’=c^{*}c$(6) $vv^{*}=v’ v=I$(7) $aa^{*}+t^{2}cc^{*}=I$ (8) a’a-l- $c^{*}c=I$

213
The -multiplication
on generators by$\mathrm{c}\mathrm{o}$$\Phi(a)=a\otimes a+tc’ v^{*}\otimes c$,$\Phi$

in the quantum group (A, u) is given$\Phi(c)=c\otimes a+a’ v^{*}\otimes c$,$\Phi(v)=v\otimes v$

.

(2.9)

The

-inverse$\mathrm{c}\mathrm{o}$$\kappa$

is defined by:$\kappa(a)=a^{*}v’,$ $\kappa(a^{*}v’)=a,$ $\kappa(c)=tc,$ $\kappa(c^{*}v’)=\frac{1}{t}c^{*}v’,$ $\kappa(v)=v$(2.10),
It follows from the relations (1) $-(8)$ that the elements$A$ of. Therefore, we
generate adense ’-subalgebra$G=(A,u)$quantum
acompactgroup,
ismatrixconclude thatwith the -multiplication given by (2.8) and the -inverse givenby (2.9).
$a,$$a^{*}v^{*},$

$c$

$A$

$c’ v^{*}$$\mathrm{c}\mathrm{o}$$\mathrm{c}\mathrm{o}$3Irreducible representations of the-algebra A$C^{*}$asWe shall now discuss representations of the C’-algebrabounded operators on Hilbert spaces. This will follow the construction of Woronowicz and [SLWI].
satisfy the relations
Let us notice, that the elements$SU_{q}(2)$,with $q=-t$ . Hence,
defining the quantum group$(A,u)$
However,thegroupquantumgroup.
to
this
equal
is
then
is different when the unitary is not identity.
We recall the construction from [SLWI] of the operators
which satisfy the relations of $SU_{q}(2)$ . The Hilbert space is$l_{2}(e_{n,k} : n\geq 0, -\infty<k<+\infty)$ , and the operators are defined on the orthogonal basis as follows:$A$$a,$ $c,$ $a^{*},$$c^{*}$$\mathrm{i}\mathrm{f}v=1$$\alpha,\gamma$$\alpha e_{n,k}=\sqrt{1-q^{2n}}e_{n-1,k},$ $(n\geq 1),$ $\alpha e_{0,k}=0,\gamma e_{n,k}=q^{2n}e_{n,k+1}$(3.11)In what follows we shall assume that -1 $<t=-q<$. Let $H$ be aseparable Hilbert space with ascalar product: $Aarrow B(H)$ be a(continuous, faithful), and letrepresentation and let $A=\pi(a),$ $C=\pi(c),$ $V=\pi(v)$ . Let us
also assume, that there is no $\pi(A)$ -invariant subspace of $H$ . Then
$A,$ $C,$.
satisfy the relations’,
commutes with all the other operators, and since
Since
there is no proper subspace of $H$ , invariant for all the operators,.
for some complex number
it must be$1$$\langle| \rangle$

$*-$

$\pi$

$A^{*}V$$C^{*}V^{*}$$1^{o}-8^{o}$$V$$V=\lambda I$$|\lambda|=1$

214
From the relations it also follows thatis an invariant subspace, and so is its orthogonal complement. Henceeither (1) $K_{0}=H$ or (2) $K_{0}=\{0\}$ . In the case (1) we have
$C=0$ , andimplies that is then unitary. Thus in this case,
since
and
commute, we have $A=\alpha I$ and $V=\lambda I$ , with$|\alpha|=|\lambda|=1$ .
It is evident that any pair of suchdefines
an irreducible representationof $A$ . Therefore, we have the$I\acute{\backslash }0=kerC$$A$$7^{o}$$A$$V$$\lambda$$\alpha,$$\pi_{\alpha,\lambda}$

following:

Proposition 3.1 Every pair , Aof complex numbers, $with|\alpha|=$, defines an $irreducible*$ -representationof A by:$\alpha$$|\lambda|=1$$\pi_{\alpha,\lambda}$$\pi_{\alpha,\lambda}(A)=\alpha$.I,$\pi_{\alpha,\lambda}(V)=\lambda$.I,(3.12)$\pi_{\alpha,\lambda}(C)=0$

Let us now consider the case (2) when $K_{0}=\{0\}$ trivial. Thenis invertible on $H$ . The kernel $H_{0}=kerA$ of is an invariant
subspace for $C,$ $C’,$
I and. Let us first observe thatthe kernel of is not the whole space $H$ . Indeed,
would
imply $CC^{*}=I=t^{2}CC’$ , which would not be possible for an
invertible and anon-zero with $|t|<1$ .
We are going to show that the kernel of is non-trivial. Let
us notice that, having trivial kernel,
would be invertible, as$C,C’$ . The proof of
its image is an invariant subspace for $A,$$ker(A)\neq\{0\}$ follows the idea used in [C-H-M-S], in the proof of
Theorem 4.4. First observe, that $P=CC^{*}$ is apositive operator
and since $A^{*}A=I-P$ is also positive, we have $0\leq P\leq I$ . Hencethe spectrum $Sp(P)$ of is contained in the interval $[0, 1]$ . Also,
zero is out of $Sp(P)$ , because is invertible. We claim that the
spectrum $Sp(P)$ contains apoint $\lambda<1$ . Otherwise, it would
consists of 1only, and thenwould be aprojection onto asubspace, on which $A^{*}A=0$ . Hence, the subspace would be{0}, and $P=0$ . Now, having a $\lambda\in Sp(P)$ with $0<\lambda<1$
it follows, that there is asequenceof unit vectors, for which. This implies that
–A. Hence,for
one can show that, so that$t^{-2}\lambda\in Sp(P)$ . It follows that $1\in Sp(P)$ isan eigenvalue. Taking
the associated eigenvector with $P\xi=\xi$ , one gets $A’ A\xi=$$(I-P)\xi=0$ , which contradicts the invertibility of .
In what follows we shall assume that $dimH_{0}\geq 1$ , so that
there are non-zero vectors in the kernel of . If $x\in H_{0}$ and$x\neq 0$ , then $C^{*}Cx=CC^{*}x=x-A’ Ax=x$ , so
is unitaryon. This implies that $AA’ x=(1-t^{2})x$ . Let us define$H_{n}:=(A’)^{n}H_{0}$ , then
$C$

$A$

$V=\lambda\cdot$$V^{*}$$A$$A=\mathrm{O}$

$C$

$t$

$A$

$A$

$A^{*},$

$P$

$C$

$P$

$\xi_{n}$$||P\xi_{n}-\lambda\xi_{n}||arrow 0$$||A\xi_{n}||arrow 1$$||P\eta_{n}-t^{-2}\lambda\eta_{n}||arrow 0$$\eta_{n}=\frac{A\xi_{\hslash}}{||A\xi_{n}||}$

$\xi$

$A$

$A$

$C$

$H_{0}$

Lemma 3.2 For all positive
the folloeoing hold:$intege|^{\backslash }sn\neq m$

and for all$x\in H_{0_{l}}$215
1. $A(A^{*})^{n}x=(1-t^{2n})(A^{*})^{n-1}x$ ,2. $A^{n}(A^{*})^{n}x= \prod_{n}^{k=1}(1-t^{2k})x$ ,S.

$H_{n}[perp] H_{m}$Proof:The proof in each of the three cases is inductive. We will use$AA^{*}=(1-t^{2})I+t^{2}A^{*}A$ , which easily follows from the relations
on $A,$ $C,$ . For the proof of (1) this relation gives the case$n=1$ . Then, for an $x\in H_{0}$ we have $A(A^{*})^{n+1}x=AA^{*}(A^{*})^{n}x=$$V$$(1-t^{2})x+t^{2}A’ A(A^{*})^{n}x=(1-t^{2})x+t^{2}(1-t^{2n})A^{*}(A^{*})^{n-1}x=$, from which (1) follows by induction. To proof
the equality (2) we write $A^{n}(A^{*})^{n}x=A^{n-1}[A(A^{*})^{n}]x$ and thenuse (1) to get $A^{n}(A^{*})^{n}x=(1-t^{2n})A^{n-1}(A^{*})^{n-1}x$ , which, throughfurther inductive expansion, gives the desired equation.
For the proof of (3) let us assume that $n<m$ and let $k=$
$m-n$ . Then for an $y\in H_{0}$ we have $A^{m}(A^{*})^{n}y=A^{k}[A^{n}(A^{*})n]y=$$f_{n}(t)\cdot A^{k}y=0$ , since $k\geq 1$ . Hereis the coefficient that appears in (2). Therefore, for arbitrary $x,$ $y\in H_{0}$ one can compute, and(3) follows.
$(1-t^{2n+2})(A^{*})^{n}x$$f_{n}(t)$$\langle(A^{*})^{m}x|(A^{*})^{n}y\rangle=\langle x|A^{k}[A^{n}(A^{*})^{n}]y\rangle=f_{n}(t)\cdot\langle x|A^{k}y\rangle=0$$\square$

It is also easy to observe that bothand C’ preserve the, for all positive integers , and that on each of thesubspaces
subspaces is ascalar multiple of aunitary operator.
$C$$H_{n}$$n$$C$

Proposition 3.3 For every positive integer
one has $CC^{*}y=t^{2n}y$$n$and for all $y\in H_{n}$Proof: This Proposition follows from the relation $CC^{*}A^{*}=$, applied times to $CC^{*}y=CC^{*}(A^{*})^{n}x$ , with $x\in H_{0}$ .
Since
is normal, it follows that the operator $D$ , defined on$y\in H_{n}$ by $Dy–( \frac{-1}{t})^{n}Cy$ is unitary on.
$t^{2}A^{*}CC^{*}$$\mathrm{n}$$C$$H_{n}$$\square$

It follows that the orthogonal direct sum
is anon$C,$
trivial subspace of $H$ , invariant for all the operators $A,$$H$
hence it must be equal to the whole space . Thus we have thefollowing$\oplus_{n\geq 0}H_{n}$$A^{*},$$C^{*},$

Proposition 3.4 The Hilbert space $H$ has the orthogonal decomposition$H=\oplus H_{n}n=0\infty$

preserved by $C,$
, and with the action of $A:H_{n}arrow H_{n-1}$
given by (1) of the Lemma (3.2).
$V,$$C^{*},$$V^{*}$$V,$ $V$216
We are now going to show that if $dimH_{0}\geq 2$ , then there
is anon-trivial orthogonal decomposition of $H$ into invariant
subspaces. Thus, irreducibility would imply $dimH_{0}=1$ .
Lemma 3.5 If $dimH_{0}\geq 2$ then there is a subspace $K\subset H$$V’$ .
$C,$
invariant for all the operators $A,$
Proof: Under the assumption $dimH_{0}\geq 2$ , for the unitary
we have anon-trivial orthogonal decomposition $H_{0}=$on, invariant for the unitary operator, and for is adjoint$C’$ .
Then, each of the subspaces $K_{n}:=(A^{*})^{n}K_{0}$ is also inand . Moreover, the orthogonal complevariant for both
. This can be seen
is justinment ofwith the help of the arguments that preceded the Proposition(3.4). The orthogonal sum $K=\oplus_{n\geq 0}K_{n}$ and its orthogonal
decompose $H$ into their direct sum
complement$H=K\oplus K$ ”, which is invariant for the considered operators.
$C^{*},$$A^{*},$

$V,$

$C$

$H_{0}$$K_{0}\oplus K_{0}^{[perp]}$$C$$C^{*}$$I\mathrm{f}_{n}^{[perp]}=(A’)^{n}K_{0}1$$H_{n}$$K_{n}$$K^{[perp]}=\oplus_{n\geq 0}K_{n}^{[perp]}$$\square$Corollary 3.6 If the representation is irreducible and if the, then the kernel
operator $A=\pi(a)$ has a non-trivial kernel
is one-dimensional. In this case, the representation has thefollowing form: $H=l_{f}^{2}$ and if $\{e_{n} : n\geq 0\}$ is the standard orthonormal basis, then $Ce_{n}=(-t)^{n}e_{n}=C’ e_{n},$ is the standard, andunilateral shift with the adjoint
for some.
complex
This way we have proved the following
Theorem 3.7 The irreducible ’-representations of the C’-algebra
A form the folloeuing two series:
with complex(1) One-dimensional characters$S^{1}=\{|z|=1\}$ , given by the formulae:$\pi$$H_{0}$

$\pi$

$A$

$Ve_{n}=\lambda e_{n}$$A^{*}$$|\lambda|=1$$\alpha,$$\pi_{\alpha,\lambda}$,,(2) Infinite dimensional representations
ing on the orthonormal standard basis$\pi_{\alpha,\lambda}(a)=\alpha$$\pi_{\alpha,\lambda}(c)=0$$\pi_{\lambda}(a)=A,$(3.13)
act-$\pi_{\alpha,\lambda}(v)=\lambda$$\pi_{\lambda}$$\{\delta_{n} :$A\delta_{n}=\sqrt{1-t^{2n}}\delta_{n-1},$

with$\lambda\in S_{f}^{1}$n\geq 0\}$

$A\delta_{0}=0$$\lambda\in$of$l^{2}$

as:,(3.14),,.
,
We end this section with description of the algebraic structure of the $C’-$ algebra $A$ . The unitary is in the center of $A$ ,and generates a $C’-$ algebra A2 isomorphic to the algebra $C(S^{1})$of continuous functions functions on the unit circle. Also, thewhich is isomorphic toelements $a,c$ generate the $C’-$ algebra$SU_{-t}(2)$$C’-$. Therefore thealgebra of the quantum groupthe$A$ is the tensor product $A=A_{1}\otimes A_{2}$ .
algebra$\pi_{\lambda}(c)=C$$C\delta_{n}=(-t)^{n}\delta_{n}$$\pi_{\lambda}(v)=V$$V\delta_{n}=\lambda\delta_{n}$$v$$A_{1}$

217
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