Matrix realizations of exceptional superconformal algebras

0.15 MB·English

by  Elena Poletaeva

Matrix realizations of exceptional superconformal algebras
Elena Poletaeva
Department of Mathematics, University of Texas-Pan American,
8
0 Edinburg, TX 78539
0 Electronic mail: [email protected]
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n
a
J
8 Abstract. We give a general construction of realizations of the contact superconformal algebras
2 K(2) and Kˆ′(4), and the exceptional superconformal algebra CK6 as subsuperalgebras of matrices

] over a Weyl algebra of size 2N ×2N, where N = 1,2 and 3. We show that there is no such a
h
realization for K(2N), if N ≥4.
p
-
h
t
a
m MSC: 17B65, 17B66, 17B68, 81R10
[
JGP SC: Lie superalgebras
1
v
Keywords: Superconformalalgebra,Poissonsuperalgebra,pseudodifferentialsymbols,
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9 Weyl algebra.
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4
.
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0
1. Introduction
8
0
v: Superconformal algebras are superextensions of the Virasoro algebra. They play
i an important rˆole in the string theory, conformal field theory and mirror symmetry,
X
and have been extensively studied by mathematicians and physicists. A supercon-
r
a
formal algebra is a simple complex Lie superalgebra, spanned by the coefficients of
a finite family of pairwise local fields a(z) = n∈Za(n)z−n−1, one of which is the
Virasoro field L(z) [3, 8, 9]. It can also be dePscribed in terms of vector fields and
symbols of differential operators.
An important class of superconformal algebras are the Lie superalgebras K(N) of
contact vector fields on the supercircle S1|N with even coordinate t and N odd coor-
dinates. The superalgebra K(N) is characterized by its action on a contact 1-form [3,
4, 9, 13]. It is spanned by 2N fields. These superalgebras are also known to physicists
as the SO(N) superconformal algebras [1, 2]. They are especially interesting, when
N is small. The universal central extension of K(2) is isomorphic to the “N = 2
superconformal algebra”. The superalgebra K′(4) has three independent central ex-
1
tension, one of which is given by the Virasoro 2-cocycle, and it is isomorphic to the
“big N = 4 superconformal algebra”, see [1, 2]. In this work we consider a differ-
ent non-trivial central extension Kˆ′(4) of K′(4). Note that K(N) has no non-trivial
central extensions, if N > 4 [13]. The superalgebra K(6) contains the exceptional
“N = 6 superconformal algebra” as a subsuperalgebra. It constitutes “one half” of
K(6), and it is also denoted by CK , see [3, 6, 9–12, 22–24].
6
In [17, 18], we proved that for every N ≥ 0, there exists an embedding of K(2N)
into the Poisson superalgebra P(2N) of pseudodifferential symbols on the supercircle
S1|N. P(2N) = P ⊗ Λ(2N), where P is the Poisson algebra of functions on the
cylinder T∗S1\S1, and Λ(2N) is the Grassmann algebra.
It is a remarkable fact that K(2), Kˆ′(4) and CK , for N = 1,2 and 3, respectively,
6
admit embeddings into the family P (2N) of Lie superalgebras of pseudodifferential
h
symbols on S1|N, which contracts to P(2N) [17, 18]. Such embeddings allow us to
obtain realizations of these superconformal algebras as subsuperalgebras of matrices
of size 2×2, 4×4, and 8×8, respectively, over a Weyl algebra W = Adi, where
i≥0
A = C[t,t−1] and d = ∂ , see [19, 20]. P
∂t
In [15] and [16] C. Martinez and E. I. Zelmanov obtained CK as a particular
6
case of superalgebras CK(R,d), where R is an associative commutative superalgebra
with an even derivation d. They also realized CK as a subsuperalgebra of matrices
6
of size 8×8 over W.
In this work we give a general construction of matrix realizations of K(2), Kˆ′(4)
and CK . Note that a semi-direct sum of the Lie algebra ø(2N,C) and the Heisenberg
6
Lie superalgebra heı(0|2N) can be embedded into the Clifford superalgebra C(2N)
and, correspondingly, it has a representation in the Lie superalgebra End(C2N−1|2N−1),
which is related to the spin representation of ø(2N + 1,C) in End(C2N), see [5, 21].
Thisrepresentation allowstorealizetheLiesuperalgebraspø(2|2N),whichpreserves a
non-degenerate super skew-symmetric form on a (2|2N)-dimensional superspace, as a
subsuperalgebra of End(W2N−1|2N−1). We prove that if N = 1,2 and 3, then spø(2|2N)
and the loop algebra of ø(2N,C) generate a subsuperalgebra of End(W2N−1|2N−1),
which is isomorphic to K(2), Kˆ′(4) and CK , respectively. If N ≥ 4, then the
6
generated superalgebra is the entire End(W2N−1|2N−1). Using this fact, we prove that
if N ≥ 4, then there is no embedding of K(2N) into End(W2N−1|2N−1).
In conclusion, we would like to point out that embeddings of superconformal
algebras into Lie superalgebras of pseudodifferential symbols on a supercircle and
into Lie superalgebras of matrices over a Weyl algebra (which are closely related
to each other) are only possible for superconformal algebras, which are in a sense,
exceptional, and they do not occur in the general case. This singles out exceptional
superconformal algebras from all superconformal algebras. It would be interesting to
give a rigorous mathematical formulation of this fact.
2
2. Preliminaries
Let Λ(2N) be the Grassmann algebra in 2N variables ξ ,...,ξ ,η ,...,η , and
1 N 1 N
let Λ(1,2N) = C[t,t−1]⊗Λ(2N) be the associative superalgebra with natural multi-
plication and with the following parity of generators: p(t) = ¯0, p(ξ ) = p(η ) = ¯1 for
i i
i = 1,...,N. Let W(2N) be the Lie superalgebra of all superderivations of Λ(1,2N).
Let ∂ , ∂ and ∂ stand for ∂ , ∂ and ∂ , respectively. By definition,
t ξi ηi ∂t ∂ξi ∂ηi
K(2N) = {D ∈ W(2N) | DΩ = fΩ for some f ∈ Λ(1,2N)}, (1)
whereΩ = dt+ N ξ dη +η dξ isadifferential1-form,whichiscalledacontactform,
i=1 i i i i
see [3, 4, 9, 13]P. Recall that K(2N) can be described in terms of pseudodifferential
symbols on S1|N, see [17, 18]. Consider the Poisson superalgebra of pseudodifferential
symbols
P(2N) = P ⊗Λ(2N), (2)
where the Poisson algebra P is formed by the formal series of the form
k
A(t,τ) = a (t)τi, (3)
i
i=X−∞
where k is some integer, a (t) ∈ C[t,t−1], and the even variable τ corresponds to ∂ ,
i t
see [14]. The Poisson super bracket is defined as follows:
N
{A,B} = ∂ A∂ B −∂ A∂ B +(−1)p(A)+1 (∂ A∂ B +∂ A∂ B). (4)
τ t t τ ξi ηi ηi ξi
Xi=1
Note that there exists an embedding
K(2N) ⊂ P(2N), N ≥ 0. (5)
Consider a Z-grading of the associative superalgebra
P(2N) = ⊕i∈ZP(i)(2N) (6)
defined by deg f = degf −1, where degf is defined by
Lie
deg t = deg η = 0 for i = 1,...,N,
i
(7)
deg τ = deg ξ = 1 for i = 1,...,N.
i
With respect to the Poisson super bracket,
{P (2N),P (2N)} ⊂ P (2N). (8)
(i) (j) (i+j)
3
Thus P (2N) is a subsuperalgebra of P(2N). We proved in [17] that P (2N) is
(0) (0)
isomorphic to K(2N). Note that K(2N) is spanned by 22N fields, one of which is
a Virasoro field. Recall that a Lie superalgebra is called simple, if it contains no
nontrivial ideals [7]. If N 6= 2, then K(2N) is simple. If N = 2, then K(4) is not
simple. In this case the derived Lie superalgebra K′(4) = [K(4),K(4)] is an ideal in
K(4) of codimension one, defined from the exact sequence
0 → K′(4) → K(4) → Ct−1τ−1ξ ξ η η → 0, (9)
1 2 1 2
and K′(4) is simple. Thus K′(4) is spanned by 16 fields inside P(4). Each field
consists of elements, which are indexed by n, where n runs through Z. These fields
are
L = tn+1τ, Q = tn+1τη η , Xi = tn+1τη ,
n n 1 2 n i
(10)
Yi = tnξ , Rji = tnη ξ , Zi = tnη η ξ , i,j = 1,2,
n i n j i n 1 2 i
G0 = tn−1τ−1ξ ξ , Gi = tn−1τ−1ξ ξ η , i = 1,2,
n 1 2 n 1 2 i
(11)
G3 = ntn−1τ−1ξ ξ η η , n 6= 0.
n 1 2 1 2
Note that K′(4) has three independent central extensions [13]. K(6) contains the
exceptional superconformal algebra CK as a subsuperalgebra [3, 6, 9–12, 22–24].
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CK has no non-trivial central extensions [3]. In [18] we obtained a realization of
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CK in terms of pseudodifferential symbols on S1|3, and proved that CK is spanned
6 6
by 32 fields inside K(6) ⊂ P(6). Each field consists of elements indexed by n, where
n runs through Z. Explicitly, CK is spanned by the following 20 fields:
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L = tn+1τ, Gi = tn+1τη , where i = 1,2,3,
n n i
G˜i = tnξ −ntn−1τ−1η ξ ξ , where i = 1,j = 2 or i = 2,j = 3 or i = 3,j = 1,
n i j i j
Tij = tnη ξ −ntn−1τ−1η η ξ ξ , where i,j,k ∈ {1,2,3} and i 6= j 6= k,
n i j k i k j
(12)
Jij = tn+1τη η , where 1 ≤ i < j ≤ 3,
n i j
J˜ij = tn−1τ−1ξ ξ , where 1 ≤ i < j ≤ 3,
n i j
I = tn+1τη η η ,
n 1 2 3
and the following 12 fields, where i = 1,j = 2,k = 3 or i = 2,j = 3,k = 1 or
i = 3,j = 1,k = 2:
Ti = −tn(η ξ +η ξ )+ntn−1τ−1η η ξ ξ ,
n j j k k j k j k
Si = −tnη (η ξ +η ξ )+ntn−1τ−1η η η ξ ξ ,
n i j j k k i j k j k
S˜i = tn−1τ−1(η ξ −η ξ )ξ ,
n j j k k i
Ii = tn−1τ−1η ξ ξ ,
n i j k
Note that L is a Virasoro field.
n
4
3. Lie superalgebras of matrices over a Weyl algebra
By definition, a Weyl algebra is
W = Adi, (13)
Xi≥0
where A is an associative commutative algebra and d : A → A is a derivation of A,
with the relations
da = d(a)+ad, a ∈ A, (14)
see [15, 16]. Set
A = C[t,t−1], d = ∂ . (15)
t
Let End(Wm|n) be the complex Lie superalgebra of matrices of size (m+n)×(m+n)
over W. Let spø(2|2N) be a Lie superalgebra, which preserves an even non-degenerate
super skew-symmetric form on the (2|2N)-dimensional superspace.
Lemma 3.1. For each N ≥ 1 there exists an embedding
spø(2|2N) ⊂ End(W2N−1|2N−1). (16)
Proof. Let V = Span(ξ ,...,ξ ,η ,...,η ). Let heı(0|2N) be the Heisenberg
1 N 1 N
Lie superalgebra: heı(0|2N)¯1 = V with the non-degenerate symmetric bilinear form
(ξi,ηi) = (ηi,ξi) = 1, and heı(0|2N)¯0 = CC, where C is a central element in heı(0|2N).
Let C(2N) be the Clifford superalgebra with generators ξ ,η and relations
i i
ξ ξ = −ξ ξ , η η = −η η , η ξ = δ −ξ η , i,j = 1,...,N. (17)
i j j i i j j i i j i,j j i
Let
ι : ø(2N,C)+⊃ heı(0|2N) → C(2N), (18)
where ø(2N,C) ∼= Λ2(V), be an embedding given by
ι(ξ ξ ) = ξ ξ , ι(η η ) = η η , ι(ξ η ) = ξ η , i 6= j,
i j i j i j i j i j i j
1 (19)
ι(ξ η ) = ξ η − , ι(ξ ) = ξ , ι(η ) = η , ι(C) = 1.
i i i i i i i i
2
Note that C(2N) ∼= End(C2N−1|2N−1). The elements ξ act by multiplication on the
i
superspace Λ(ξ ,...,ξ ), and η acts as ∂ . Hence there exists an embedding
1 N i ξi
ρ : ø(2N,C)+⊃ heı(0|2N) → End(C2N−1|2N−1). (20)
Note that if we consider V as an even vector space, then formulas (19) define an
embedding of ø(2N +1,C) ∼= Λ2(V)⊕V into the Clifford algebra C(2N), and corre-
spondingly, the spin representation of ø(2N +1,C) in End(C2N), see [5, 21].
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Let
End(C2N−1|2N−1) = End ⊕End ⊕End , (21)
−1 0 1
where End is the set of even complex matrices, and End and End are the sets of
0 −1 1
odd upper triangular matrices and odd lower triangular matrices, respectively. Let
X ∈ V. Then ρ(X) = ρ(X) +ρ(X) , where ρ(X) ∈ End . Define
−1 1 ±1 ±1
ρ(X)± ∈ End(W2N−1|2N−1) (22)
by setting
ρ(X)± = ρ(X)± +ρ(X)±, (23)
−1 1
where
ρ(X)± = t±1ρ(X) ,
−1 −1
1 (24)
ρ(X)± = (tdt±1 ∓ t±1)ρ(X) .
1 2 1
Define g = g¯0 ⊕g¯1 by setting
g¯1 = ρ(V)±,
(25)
g¯0 = ρ(ø(2N,C))⊕sl(2),
where sl(2) = Span(E,H,F), and E,H and F are the following diagonal matrices in
End(W2N−1|2N−1):
1 1 1
E = i (tdt2 − t2)12N−1 | (t2dt− t2)12N−1 ,
2 2 2
(cid:16) (cid:17)
1 1 1 (26)
F = i (tdt−2 + t−2)12N−1 | (dt−1 + t−2)12N−1 ,
2 2 2
(cid:16) (cid:17)
H = (td)12N−1|2N−1,
so that the standard commutation relations hold:
[H,E] = 2E, [H,F] = −2F, [E,F] = H.
Then g ∼= spø(2|2N).
(cid:3)
Let ˜ø(2N,C) = Span(tnρ(X) | X ∈ ø(2N,C), n ∈ Z). Thus ˜ø(2N,C) is isomorphic to
the loop algebra of ø(2N,C).
Theorem 3.2. If N = 1, then spø(2|2) and ˜ø(2,C) generate K(2),
if N = 2, then spø(2|4) and ˜ø(4,C) generate Kˆ′(4),
if N = 3, then spø(2|6) and ˜ø(6,C) generate CK ,
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if N ≥ 4, then spø(2|2N) and ˜ø(2N,C) generate End(W2N−1|2N−1).
Proof. Case N = 1. It is easy to see that spø(2|2)¯1 = Span(ρ(ξ1±),ρ(η1±)), where
0 0 0 0
ρ(ξ+) = , ρ(ξ−) = ,
1 (cid:18) t 0 (cid:19) 1 (cid:18) t−1 0 (cid:19)
(27)
0 t2d+ 1t 0 d− 1t−1
ρ(η+) = 2 , ρ(η−) = 2 .
1 (cid:18) 0 0 (cid:19) 1 (cid:18) 0 0 (cid:19)
−tn 0
Hence spø(2|2) and ˜ø(2,C) = Span( ) generate the following subsuperal-
(cid:18) 0 tn (cid:19)
gebra of End(W1|1):
g = Span(L ,H ,G ,G˜ | n ∈ Z),
n n n n
where
tn+1d+ntn 0 −tn 0
L = , H = ,
n (cid:18) 0 tn+1d (cid:19) n (cid:18) 0 tn (cid:19)
(28)
0 tn+1d+ ntn 0 0
G = 2 , G˜ = .
n (cid:18) 0 0 (cid:19) n (cid:18) tn 0 (cid:19)
The isomorphism
σ : K(2) ⊂ P(2) → g
is given by
n 1
σ(tn+1τ) = L + H , σ(tnξ η ) = H ,
n n 1 1 n
2 2 (29)
σ(tnξ ) = G˜ , σ(tn+1τη ) = G .
1 n 1 n
Note that L is a Virasoro field.
n
In the cases when N = 2 and 3, we will use an embedding of Kˆ′(4) and CK ,
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respectively, intoadeformationofP(2N). LetP (2N) = P ⊗C(2N). Theassociative
1 1
multiplication in the vector space P = P is determined as follows (see [14]):
1
1
A(t,τ)◦B(t,τ) = ∂nA(t,τ)∂nB(t,τ). (30)
n! τ t
Xn≥0
The product of A = A ⊗X and B = B ⊗Y, where A ,B ∈ P , and X,Y ∈ C(2N),
1 1 1 1 1
is given by
AB = (A ◦B )⊗(XY). (31)
1 1
The Lie super bracket in P (2N) is [A,B] = AB −(−1)p(A)p(B)BA. P (2N) is called
1 1
the Lie superalgebra of pseudodifferential symbols on S1|N, see [17, 18].
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Case N = 2. We proved in [17] that Kˆ′(4) is spanned inside P (4) by the 12 fields
1
given in (10) and 4 fields
G0 = τ−1 ◦tn−1ξ ξ , Gi = τ−1 ◦tn−1ξ ξ η , i = 1,2,
n 1 2 n 1 2 i
(32)
G3 = nτ−1 ◦tn−1ξ ξ η η +tn.
n 1 2 1 2
Note that L is a Virasoro field. The central element in Kˆ′(4) is G3 = 1, and the
n 0
corresponding 2-cocycle is
c(L ,G3) = −nδ ,
n k n+k,0
c(Xi,Gj) = (−1)jδ , 1 ≤ i 6= j ≤ 2, (33)
n k n+k,0
c(Q ,G0) = δ .
n k n+k,0
Notethatthis2-cocycleisdifferentfromtheVirasoro2-cocycle. LetVµ = tµC[t,t−1]⊗
Λ(ξ ,ξ ), where µ ∈ C\Z. Define a representation of Kˆ′(4) in Vµ accordingly to the
1 2
formulas (10) and (32). In particular, τ−1 is identified with an antiderivative, and the
central element acts by the identity operator. Consider the following basis in Vµ:
v0 (µ) = tm+µ, vi (µ) = tm+µξ , i = 1,2,
m m i
tm+µ (34)
v3 (µ) = ξ ξ , m ∈ Z.
m m+µ 1 2
Explicitly, the action of Kˆ′(4) on Vµ is given as follows:
L (vi (µ)) = (m+µ)vi (µ), i = 0,1,2,
n m m+n
L (v3 (µ)) = (n+m+µ)v3 (µ),
n m m+n
Xi(vi (µ)) = (m+µ)v0 (µ), i = 1,2,
n m m+n
X1(v3 (µ)) = v2 (µ),
n m m+n
X2(v3 (µ)) = −v1 (µ),
n m m+n
Q (v3 (µ)) = −v0 (µ),
n m m+n
Yi(v0 (µ)) = vi (µ), i = 1,2,
n m m+n
(35)
Y1(v2 (µ)) = (n+m+µ)v3 (µ),
n m m+n
Y2(v1 (µ)) = −(n+m+µ)v3 (µ),
n m m+n
Rii(v0 (µ)) = v0 (µ), i = 1,2,
n m m+n
Rii(vj (µ)) = vj (µ), Rij(vi (µ)) = −vj (µ), i 6= j = 1,2,
n m m+n n m m+n
Z1(v2 (µ)) = −v0 (µ), Z2(v1 (µ)) = v0 (µ),
n m m+n n m m+n
G0(v0 (µ)) = v3 (µ), Gi(vi (µ)) = v3 (µ), i = 1,2,
n m m+n n m m+n
G3(vi (µ)) = vi (µ), n 6= 0, i = 0,1,2,3.
n m m+n
8
These formulas remain valid for µ = 0. Thus we obtain a representation of Kˆ′(4) in
the superspace C[t,t−1]⊗Λ(ξ ,ξ ) with a basis
1 2
{v0 ,v3 ;v1 ,v2 },
m m m m
where
v0 = tm, v3 = tmξ ξ , vi = tmξ , i = 1,2, m ∈ Z.
m m 1 2 m i
We have
L (vi ) = tn+1dvi , i = 0,1,2, L (v3 ) = tdtnv3
n m m n m m
Xi(vi ) = tn+1dv0 , i = 1,2, X1(v3 ) = tnv2 ,
n m m n m m
X2(v3 ) = −tnv1 , Q (v3 ) = −tnv0 ,
n m m n m m
Yi(v0 ) = tnvi , i = 1,2, Y1(v2 ) = tdtnv3 ,
n m m n m m
Y2(v1 ) = −tdtnv3 , Rii(v0 ) = tnv0 , i = 1,2, (36)
n m m n m m
Rii(vj ) = tnvj , Rij(vi ) = −tnvj , i 6= j = 1,2,
n m m n m m
Z1(v2 ) = −tnv0 , Z2(v1 ) = tnv0 ,
n m m n m m
G0(v0 ) = tnv3 , Gi(vi ) = tnv3 , i = 1,2,
n m m n m m
G3(vi ) = tnvi , n 6= 0, i = 0,1,2,3.
n m m
This gives a realization of Kˆ′(4) as a subsuperalgebra of End(W2|2). Note that
spø(2|4) ⊂ Kˆ′(4) ⊂ End(W2|2), (37)
where spø(2|4)¯1 is spanned by the following elements:
1 1
ρ(ξ )± = Y1 ∓ G2 , ρ(ξ )± = Y2 ± G1 ,
1 ±1 2 ±1 2 ±1 2 ±1
(38)
1 1
ρ(η )± = X1 ± Z2 , ρ(η )± = X2 ∓ Z1 .
1 ±1 2 ±1 2 ±1 2 ±1
˜ø(4,C) is generated by Rn12,Rn21,G0n and Qn. When these elements act on spø(2|4)¯1,
they generate the 8 fields Xi,Yi,Gi and Zi, where i = 1,2, which span Kˆ′(4) , and
n n n n ¯1
hence Kˆ′(4) is generated by spø(2|4) and ˜ø(4,C).
Case N = 3. We proved in [18] that CK is spanned inside P (6) by the 8 fields: L ,
6 1 n
Gi, I , and Jij, given in (12), and the following 24 fields, where n ∈ Z:
n n n
12 fields:
G˜i = tnξ −nτ−1 ◦tn−1ξ ξ η , where i = 1,j = 2 or i = 2,j = 3 or i = 3,j = 1,
n i i j j
Tij = tnη ξ −nτ−1 ◦tn−1ξ ξ η η , where i,j,k ∈ {1,2,3} and i 6= j 6= k,
n i j k j k i
J˜ij = τ−1 ◦tn−1ξ ξ , where 1 ≤ i < j ≤ 3,
n i j
(39)
9
and the following 12 fields, where i = 1,j = 2,k = 3 or i = 2,j = 3,k = 1 or
i = 3,j = 1,k = 2:
Ti = −tn(η ξ +η ξ )+nτ−1 ◦tn−1ξ ξ η η +tn,
n j j k k j k j k
Si = −tnη (η ξ +η ξ )+nτ−1 ◦tn−1ξ ξ η η η +tnη ,
n i j j k k j k i j k i
(40)
S˜i = τ−1 ◦tn−1(ξ ξ η −ξ ξ η ),
n j i j k i k
Ii = τ−1 ◦tn−1ξ ξ η .
n j k i
Let Vµ = tµC[t,t−1]⊗Λ(ξ ,ξ ,ξ ), where µ ∈ C\Z. Define a representation of CK in
1 2 3 6
Vµ according to the formulas (12) and (39–40). Consider the following basis in Vµ:
tm+µ
vi (µ) = tm+µξ , vˆi (µ) = ξ ξ , 1 ≤ i ≤ 3,
m i m m+µ j k
tm+µ
v4 (µ) = tm+µ, vˆ4 (µ) = − ξ ξ ξ ,
m m m+µ 1 2 3
where m ∈ Z and (i,j,k) is the cycle (1,2,3) in the formulas for vˆi (µ). Explicitly,
m
the action of CK on Vµ is given as follows:
6
L (vi (µ)) = (m+µ)vi (µ), L (vˆi (µ)) = (m+n+µ)vˆi (µ),
n m m+n n m m+n
Gi(vi (µ)) = (m+µ)v4 (µ), Gi(vˆ4 (µ)) = −(m+n+µ)vˆi (µ),
n m m+n n m m+n
Gi(vˆk (µ)) = vj (µ), Gi(vˆj (µ)) = −vk (µ),
n m m+n n m m+n
G˜i(v4 (µ)) = vi (µ), G˜i(vˆi (µ)) = −vˆ4 (µ),
n m m+n n m m+n
G˜i(vk (µ)) = −(m+n+µ)vˆj (µ), G˜j(vj (µ)) = (m+µ)vk (µ),
n m m+n n m m+n
Tij(vi (µ)) = −vj (µ), Tij(vˆj (µ)) = vˆi (µ),
n m m+n n m m+n
Ti(vi (µ)) = −vi (µ), Ti(v4 (µ)) = −v4 (µ), (41)
n m m+n n m m+n
Ti(vˆi (µ)) = vˆi (µ), Ti(vˆ4 (µ)) = vˆ4 (µ),
n m m+n n m m+n
Si(vi (µ)) = −v4 (µ), Si(vˆ4 (µ)) = −vˆi (µ),
n m m+n n m m+n
S˜i(vk (µ)) = −vˆj (µ), S˜i(vj (µ)) = −vˆk (µ),
n m m+n n m m+n
Ii(vi (µ)) = vˆi (µ), I (vˆ4 (µ)) = v4 (µ),
n m m+n n m m+n
Jij(vˆk (µ)) = −v4 (µ), Jij(vˆ4 (µ)) = vk (µ),
n m m+n n m m+n
J˜ij(v4 (µ)) = vˆk (µ), J˜ij(vk (µ)) = −vˆ4 (µ),
n m m+n n m m+n
where (i,j,k) is the cycle (1,2,3). These formulas remain valid for µ = 0. Thus we
obtain a representation of CK in the superspace C[t,t−1]⊗Λ(ξ ,ξ ,ξ ) with a basis
6 1 2 3
{vˆi ,v4 ;vi ,vˆ4 }, i = 1,2,3,
m m m m
10