The Automorphism Group Functor of the N= 4 Lie Conformal Superalgebra

Home , Automorphism group

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1 ±1 d ± m d := (k[t ], dt ) := ( lim k[t ], dt ) D → D −→m will lead to the classification of twistedb loop conformal superalgebras based on . AIn order to completely classify twisted loop conformal superalgebras based on one of the N = 2 and N = 4 conformal superalgebras , the au- A tomorphism group Aut( )( ) has been determined in [10]. The automorphism group functorsA ofD the N = 1, 2, 3 conformal superalgebras were b studied in [1] as well. Interestingly, for the N = 1, 2, 3 Lie conformal superalgebra N , Aut( N ) has a subgroup functor GrAut( N ) such that K K K GrAut( N )( ) = Aut( N )( ) for = (R, δ) with R an integral do- K R K R R main, and GrAut( N ) is the group functor obtained by lifting the k– K group scheme ON of N N–orthogonal matrices via the forgetful functor × 1The k–differential ring plays the role of a simply-connected cover of . D D b 2 fgt : k-drng k-rng, =(R, δ) R, i.e., → R 7→ fgt ON GrAut( N ): k-drng k-rng grp. K −→ −→ We call a functor from k-drng to grp forgetfully representable if it is obtained by lifting a k-group scheme using the forgetful functor. It has been conjecture in [1] that the automorphism group functor Aut( ) of the N = 4 Lie conformal superalgebra also satisfies similar properties:F F (1) It has a subgroup functor GrAut( ) such that GrAut( )( ) = Aut( )( ) if =(R, δ) such that RFis an integral domain.F R F R R (2) GrAut( ) is forgetfully representable by F SL SL (k) 2 × 2 , I , I h− 2 − 2i where SL is the group scheme of 2 2 matrices with determinant 1, 2 × and SL2(k) is the constant group scheme. In this paper, we will show that the assertion (1) is true in Section 4, but the assertion (2) fails as we shall see in Remark 3.6. Nonetheless, the subgroup functor GrAut( ) is still presented by certain affine group F schemes. The key ingredient of this paper is to study the relationship between GrAut( ) and certain affine group schemes (c.f. Section 3). Finally, we will provideF an example to show that the definition of the subgroup functor GrAut( ) of a Lie conformal superalgebra depends on the choice of generators ofA in Section 5. A A

Notation: Throughout this paper, Z, Z+, and Q denote the sets of integers, non-negative integers, and rational numbers, respectively. We will always assume k is an algebraically closed ﬁeld of characteristic zero, i is √ 1 k, and k[∂] is the polynomial ring over k in one variable − ∈ ∂. We will use ǫijl to denote the sign of a cycle (ijl). For a commutative associative k-algebra R, Mat2(R) will denote the associative algebra of all 2 2–matrices with entries in R. × We will also use the standard notation of the group scheme µ2, which is a functor from k-rng to grp assigning each object R in k-rng the group of 2 square roots of unity in R, i.e., µ2(R)= r R r =1 . { ∈ | } 2 Preliminaries

In this section, we will review some basic concepts of Lie conformal superalgebras over k, their base change with respect to an extension of k–diﬀerential

3 rings, and their automorphism group functors. We then describe the N =4 Lie conformal superalgebra using generators and relations. We also introduce some notation to simplify our computations for automorphism groups.

Deﬁnition 2.1 ([9, deﬁnition 2.7]). A Lie conformal superalgebra over k is a Z/2Z–graded k[∂]–module = ¯ ¯ equipped with a k–bilinear operation A A0⊕A1 n : k for each n Z satisfying the following axioms: −( )− A ⊗ A→A ∈ + (CS0) a b = 0 for n 0, (n) ≫ (CS1) ∂(a) b = na b, (n) − (n−1) j+n (j) (CS2) a n b = p(a, b) ( 1) ∂ (b n j a), ( ) − j∈Z+ − ( + ) mP m (CS3) a(m)(b(n)c)= j=0 j (a(j)b)(m+n−j)c + p(a, b)b(n)(a(m)c), P (j) j p(a)p(b) where ∂ = ∂ /j!, j Z+ and p(a, b)=( 1) , p(a) (resp. p(b)) is the parity of a (resp. b). ∈ −

We also use the conventional λ-bracket notation: 1 [a b]= λna b, λ n! (n) nX∈Z+ where λ is a variable and a, b . ∈A Given a Lie conformal superalgebra over k and a k–differential ring A =(R, δ), we can form a new Lie conformal superalgebra k (the base R A ⊗ R change of with respect to k ) as follows: The underlying Z/2Z–graded A → R k–vector space of k is k R, the k[∂]–module structure is given by ∂ = ∂ 1+1A ⊗ δ,R andA the ⊗n-th product for each n Z is given by: A⊗kR A ⊗ ⊗ ∈ + (a r) (b s)= (a b) δ(j)(r)s, ⊗ (n) ⊗ (n+j) ⊗ jX∈Z+ where a, b ,r,s R and δ(j) = δj/j!. ∈A ∈ It is pointed out in [10] that k is not only a Lie conformal superalgebra over k, but also an –LieA ⊗ conformalR superalgebra (c. f. [10, definition 1.3]). The –structureR yields the definition of its –automorphisms: R 2 R an –automorphism of k is an automorphism of the Z/2Z–graded R– R A ⊗ R module k R which preserves all n-th products and commutes with ∂ k . A⊗ A⊗ R All –automorphisms of k form a group, denoted by Aut ( k ). R A⊗ R R-conf A⊗ R 2Throughout this paper, an automorphism of a Z/2Z–graded R–module is required to be an automorphism of degree 0,¯ i.e., it preserves the Z/2Z–gradation.

4 These automorphism groups are functorial in . Hence, one deﬁnes a functor from the category of k–diﬀerential rings to theR category of groups:

Aut( ): k-drng grp, Aut ( k ). A → R 7→ R-conf A ⊗ R It is called the automorphism group functor of . A In this paper, we focus on the N = 4 Lie conformal superalgebra . As F a Z/2Z–graded k[∂]–module, = ¯ ¯, where F F0 ⊕F1 1 2 3 ¯ = k[∂]L k[∂]T k[∂]T k[∂]T , F0 ⊕ ⊕ ⊕ 1 2 1 2 ¯ = k[∂]G k[∂]G k[∂]G k[∂]G . F1 ⊕ ⊕ ⊕ The λ–bracket on is given by F i i [LλL]=(∂ +2λ)L, [LλT ]=(∂ + λ)T , p 3 p p 3 p [LλG ]= ∂ + 2 λ G , LλG = ∂ + 2 λ G , i j l p q p q [T λT ]= iǫijlT ,[G λG ]= G λG = 0, 2 2 i p 1 i q i p 1 i q [T λG ]= 2 σpqG , T λG = 2 σqpG , − q=1 q=1 P 3 P p q i i G λG =2δpqL 2(∂ +2λ) σ T , − pq iP=1 where i, j =1, 2, 3, p, q =1, 2 and σi, i =1, 2, 3 are the Pauli spin matrices:

0 1 0 i 1 0 σ1 = , σ2 = − , σ3 = . 1 0 i 0 0 1 −

The even part ¯ has a sub Lie conformal algebra F0 = k[∂]T1 k[∂]T2 k[∂]T3, B ⊕ ⊕ 3 which is isomorphic to the current Lie conformal algebra Cur(sl2(k)). Analogous to the cases where N =1, 2, 3, the automorphism group functor Aut( ) has a subgroup functor GrAut( ) deﬁned as follows: ﬁx the F F k–vector space

1 2 3 1 2 1 2 V = spank L, T , T , T , G , G , G , G , (1) n o 3Every ﬁnite-dimensional Lie superalgebra g over k associates a current Lie conformal superalgebra Cur(g) = k[∂] k g with the n-th product: a(0)b = [a,b] and a(n)b = 0 for n > 1 [9, Example 2.7]. ⊗

5 and deﬁne a subgroup of Aut( )( ) for each k–diﬀerential ring =(R, δ): A R R

GrAut( )( )= φ Aut( )( ) φ(V k R) V k R . F R { ∈ F R | ⊗ ⊆ ⊗ } This construction is functorial in . To simplify computations for theseR automorphism groups, we write ∂ = ∂ for short and introduce the following notation4: for r R, X = F⊗kR b x y a b ∈ sl (R) and M = Mat (R), we set z x ∈ 2 c d ∈ 2 − L(r) := L r, ⊗ T(X) := T1 (y + z)+ iT2 (y z)+2T3 x, ⊗ 1 ⊗ − 2 ⊗ G(M) := G1 d + G a G2 b + G c. ⊗ ⊗ − ⊗ ⊗ Then every element of V (resp. V R) can be written as ⊗ L(r)+T(X)+G(M), where r k,X sl (k), M Mat (k) (resp. r R,X sl (R), M ∈ ∈ 2 ∈ 2 ∈ ∈ 2 ∈ Mat2(R)). Furthermore, the λ-bracket on k satisﬁes the following relations: for r, r′ R, X,X ,X sl (R), andF ⊗M,R N Mat (R), ∈ 1 2 ∈ 2 ∈ 2 ′ ′ ′ [L(r)λL(r )]=(∂ +2λ)L(rr )+2L(δ(r)r ),

[L(r)λT(X)]=(∂ + λ)T(rX)+T(δ(r)X),

[T(X1)λT(X2)] = T([X1,X2]), 3 3 [L(r) G(M)] = ∂ + λ G(rM)+ G(δ(r)M), λ 2 2

[T(X)λG(M)] = G(XM), † † † [G(M)λG(N)] = 2L(tr(MN ))+(∂ +2λ)T(MN NM ) − + 2T(δ(M)N † Nδ(M †)). − where tr : Mat (R) R is the trace map, and 2 → a b † d b := − (2) c d c a − is the standard sympletic involution on Mat2(R).

4These notations allow us to rewrite all commutative relations of using matrix mul- tiplication, which lead to nice formulas for describing the automorphisFms of F

6 3 The group functor GrAut( ) F We will compute the group GrAut( )( )fora k–diﬀerential ring =(R, δ) in this section. The main resultsF willR be Theorem 3.3, PropositionsR 3.4 and 3.5. Lemma 3.1. For an arbitrary k–diﬀerential ring =(R, δ), there is a group R homomorphism

ι : SL (R) SL (R ) GrAut( )( ), (A, B) θA,B, R 2 × 2 0 → F R 7→ where R = ker δ, and θA,B is the –automorphism of k defined by 0 R F ⊗ R −1 θA,B(L) =L+T(δ(A)A ), −1 θA,B(T(X)) = T(AXA ), −1 θA,B(G(M)) = G(AMB ), for X sl2(k), M Mat2(k). In addition, the homomorphism ιR is functorial in ∈ . ∈ R Proof. These formulas define a homomorphism θA,B of R–modules V k R ⊗ → V k R. It is extended to a homomorphism of R–modules k k ⊗ F ⊗ R→F⊗ R which preserves the Z/2Z–grading and satisfies ∂ θA,B = θA,B ∂. This map is also denoted by θ . ◦ ◦ A,B b b To show θA,B is a homomorphism of Lie conformal superalgebras, by Lemma 3.1 (ii) in [10], it suffices to show

θA,B([ξ 1λη 1]) = [θA,B(ξ 1)λθA,B(η 1)] (3) ⊗ ⊗ ⊗ ⊗ for all ξ, η V . This can be accomplished by a direct computation. ∈ For instance, let M, N Mat (k), then ∈ 2 θA,B([G(M)λG(N)]) † † † = θA,B 2L(tr(MN ))+(∂ +2λ)T(MN NM ) − = 2L(tr( MN †)) + 2T(tr(MN †)δ(A)A−1) +(∂ +2λ)T(A(MN † NM †)A−1), − [θA,B(G(b M))λθA,B(G(N))] −1 −1 = [G(AMB )λG(ANB )] = 2L(tr(AMB−1(ANB−1)†)) +(∂ +2λ)T(AMB−1(ANB−1)† ANB−1(AMB−1)†) − + 2T(δ(AMB−1)(ANB−1)† ANB−1δ(AMB−1)†). − For the standard sympletic involution as deﬁned in (2), we have

7 (MN)† = N †M † for M, N Mat (R). • ∈ 2 MN † +NM † = tr(MN †)I for M, N Mat (R), where I is the identity • ∈ 2 matrix. A−1 = A† for A SL (R). • ∈ 2 Hence, AMB−1(ANB−1)† = AMB−1BN †A−1 = AMN †A−1. Noting that δ(B) = 0 and δ(A−1)= A−1δ(A)A−1, we obtain − 2(δ(AMB−1)(ANB−1)† ANB−1δ(AMB−1)†) − = 2(δ(A)MB−1BN †A−1 ANB−1BM †δ(A−1)) − = δ(A)(MN † NM †)A−1 + A(MN † NM †)δ(A−1) − − + δ(A)(MN † + NM †)A−1 A(MN † + NM †)δ(A−1) − = δ(A(MN † NM †)A−1)+tr(MN †)(δ(A)A−1 Aδ(A−1)) − − = δ(A(MN † NM †)A−1) + 2tr(MN †)δ(A)A−1. − It follows that

θA,B([G(M)λG(N)]) = [θA,B(G(M))λθA,B(G(N))]. Similarly, it is easy to verify that the equation (3) holds for all ξ, η ∈ V . Hence, θA is a homomorphism of –Lie conformal superalgebras. Then R θA,B θ −1 −1 = θ −1 −1 θA,B = id implies that θA,B is an automorphism ◦ A ,B A ,B ◦ F of the –Lie conformal superalgebra k . Hence, we obtain θA,B R F ⊗ R ∈ GrAut( )( ) since θA,B(V k R) V k R. F R ⊗ ⊆ ⊗ To prove (A, B) θA,B deﬁnes a group homomorphism, it suﬃces to 7→ show θA1A2,B1B2 = θA1,B1 θA2,B2 for A1, A2 SL2(R), B1, B2 SL2(R0). By [10, Proposition 3.1], this◦ can be easily done∈ by verifying ∈

θA A ,B B (ξ 1) = θA ,B θA ,B (ξ 1) 1 2 1 2 ⊗ 1 1 ◦ 2 2 ⊗ for ξ V . For instance, ∈ −1 θA1A2,B1B2 (L) = L+T(δ(A1A2)(A1A2) ) −1 −1 = L+T((δ(A1)A2 + A1δ(A2))A2 A1 ) −1 −1 −1 =L+T(δ(A1)A1 + A1δ(A2)A2 A1 ), −1 θA1,B1 θA2,B2 (L) = θA1,B1 (L+T(δ(A2)A2 )) ◦ −1 −1 −1 =L+T(δ(A1)A1 )+T(A1δ(A2)A2 A1 ). In summary, we obtain a group homomorphism ι : SL (R) SL (R ) GrAut( )( ), R 2 × 2 0 → F R for each k–differential ring which is functorial in . R R 8 Lemma 3.2. For every k–differential ring =(R, δ), R ker(ιR) ∼= µ2(R0), where R0 = ker δ and µ2(R0) is diagonally embedded into SL2(R) SL2(R0). Additionally, these isomorphisms are functorial in . × R Proof. Let (A, B) ker(ι ), where A SL (R) and B SL (R ). Since ∈ R ∈ 2 ∈ 2 0 θA,B = id, −1 T(X)= θA,B(T(X)) = T(AXA ), for all X sl2(k). Hence, AX = XA for all X sl2(k). It follows that A = aI for∈ some a µ (R). ∈ 2 ∈ 2 Since −1 G(M)= θA,B(G(M)) = G(AMB ), for all M Mat (k), we obtain aM = AM = MB for all M Mat (k), and ∈ 2 ∈ 2 hence B = aI and a R as B SL (R ). Therefore, (A, B)=(aI , aI ) 2 ∈ 0 ∈ 2 0 2 2 for a µ2(R0). Conversely,∈ for a µ (R ), it can be checked that (aI , aI ) ker(ι ). ∈ 2 0 2 2 ∈ R Hence, ker(ιR) ∼= µ2(R0). Summarizing Lemmas 3.1 and 3.2, we obtain the following theorem: Theorem 3.3. For every k–differential ring = (R, δ), there is an exact R sequence of groups

ιR 1 µ (R ) SL (R) SL (R ) GrAut( )( ), (4) → 2 0 → 2 × 2 0 −→ F R where R = ker δ. Furthermore, the exact sequence is functorial in . 0 R

In general, ιR fails to be surjective. However, it has properties analogous to the universal surjectivity of the quotient morphisms of k–group schemes. More precisely, we have the following Propositions 3.4 and 3.5. We call an extension = (R, δR) = (S, δS) of k–differential rings étale if the homomorphismR R S of→ rings S is étale. → Proposition 3.4. Let =(R, δR)bea k–differential ring with R an integral R domain. For every ϕ GrAut( )( ), there is an étale extension =(S, δS) of , an element A ∈SL (S), andF R an element B SL (S ) suchS that R ∈ 2 ∈ 2 0

ιS (A, B)= θA,B = ϕS , where S0 = ker δS, and ϕS is the image of ϕ under the induces group homomorphism GrAut( )( ) GrAut( )( ). F R → F S

9 Proof. We ﬁrst write ϕ(L) = L(r)+T(X ), where r R,X sl (R). Then 0 ∈ 0 ∈ 2

ϕ([LλL]) = (∂ +2λ)(L(r)+T(X0)), 2 [ϕ(L)λϕ(L)] = (∂b +2λ)(L(r )+T(rX0)).

2 We deduce from ϕ([LλL]) = [ϕ(L)bλϕ(L)] that r = r and rX0 = X0. Since R is an integral domain, r =0or1. If r = 0, we obtain X0 = rX0 = 0, and so ϕ(L) = 0. This contradicts the injectivity of ϕ. Hence, r = 1, i.e.,

ϕ(L)= L+T(X0). (5)

i Next we write ϕ(T(σ )) = L(ri)+T(Xi) for ri R,Xi sl (R), i =1, 2, 3. ∈ ∈ 2 Thus,

i ϕ([LλT(σ )]) = (∂ + λ)(L(ri)+T(Xi)), i [ϕ(L)λϕ(T(σ ))] = (∂b +2λ)L(ri)+(∂ + λ)T(Xi) + λT(riX0)+T(riδ(X0)) + T([X0,Xi]).

i i We deduce from ϕ([LλT(σ )]) = [ϕ(L)λϕ(T(σ ))], i = 1, 2, 3 that ri = 0 and i δ(Xi) = [X0,Xi], i = 1, 2, 3. It follows that ϕ(T(σ )) = T(Xi), i = 1, 2, 3.

Hence, ϕ( k ) k and ϕ B⊗kR is an automorphism of k . B ⊗ R ⊆ B ⊗ R | 1 2 B ⊗3 R Recall that ¯ has a subalgebra = k[∂]T k[∂]T k[∂]T , which F0 B ⊕ ⊕ is isomorphic to Cur(sl (k)). Thus k = Cur(sl (k)) k as –Lie 2 B ⊗ R ∼ 2 ⊗ R R conformal superalgebras. By Corollary 3.17 of [10], there is an R-linear automorphism ϕ of the Lie algebra sl (k) k R = sl (R) such that ϕ(T(X)) = 2 ⊗ 2 T(ϕ(X)) for all X sl (R). ∈ 2 It is known that SL2(R) acts on sl2(k) k R functorially via conjugation and there is a quotient morphism π : SL ⊗Aut(sl (k)) of k-group schemes. 2 → 2 From [2, III, 1, 2.8 and 3.2], there exist an étale extension S of R and A SL (S) such§ that ∈ 2 ϕ (X)= AXA−1, X sl (S). (6) S ∈ 2 Since R S is an étale homomorphism, by [7, Chapter 0, Corollary 20.5.8], → there is a unique k–derivation δS of S extending δR. Hence, = (S, δS) is an étale extension of . S R Now we consider the image ϕS of ϕ in GrAut( )( ). From (5) and (6), we obtain F S

i i −1 ϕS (L)= L+T(X0), and ϕS (T(σ )) = T(Aσ A ), i =1, 2, 3.

i i Then we deduce from [ϕS (L)λϕS (T(σ ))] = (∂ + λ)ϕS (T(σ )) that

i −1 i −1 δS(Aσ A ) = [X0, Aσ A b], i =1, 2, 3.

10 −1 Hence, a direct computation shows that X0 = δS(A)A . Let ψ = ϕ θ−1 . Then ψ GrAut( )( ) and ψ = id. Next we S ◦ A,I ∈ F S |F0¯⊗kS consider ψ ¯ kS. Suppose |F1⊗ ψ(G(M)) = G(ν(M)), M Mat (S), ∈ 2 where ν : Mat (S) Mat (S) is a bijective S–linear map. Now we deduce 2 → 2 from ψ([T(X)λG(M)]) = [ψ(T(X))λψ(G(M))] that X ν(M) = ν(XM) for all X sl (S) and M Mat (S). Then a · ∈ 2 ∈ 2 straightforward computation shows that there is an element B GL2(S) −1 ∈ such that ν(M)= MB for all M Mat2(S). Next we show B SL (S ). Let∈M, N Mat (k), then ∈ 2 0 ∈ 2 ψ([G(M) G(N)]) = 2T(MN † NM †), (1) − [ψ(G(M)) ψ(G(N))] = 2T(MB−1(NB−1)† NB−1(MB−1)†) (1) − = 2T(det(B−1)(MN † NM †)). − It follows that det(B−1)=1, so B SL (S). ∈ 2 For M Mat (k), we consider ∈ 2 1 3 ψ([G(M) L]) = ∂ + λ G(MB−1), λ 2 2 b 1 3 −1 3 −1 [ψ(G(M))λψ(L)] = ∂ + λ G(MB )+ G(Mδ(B )). 2 2 2

−1 These yield that δ(B ) = 0, and hence B SL2(S0). Therefore, ψ = −1 ∈ ϕ θ = θI,B, i.e., ϕ = θI,B θA,I = θA,B. S ◦ A,I S ◦ Proposition 3.5. Let = (R, δ) be a k–differential ring. If R is an inte- R 1 gral domain and the étale cohomology set Hét(R, µ2) is trivial, then ιR is surjective. Proof. Given ϕ GrAut( )( ), as in the proof of Proposition 3.4, its ∈ F R restriction to k yields an R–linear automorphism ϕ of the Lie algebra B ⊗ R 1 2 3 sl (k) k R, where = k[∂]T k[∂]T k[∂]T = Cur(sl (k)). It is known 2 ⊗ B ⊕ ⊕ ∼ 2 that there is an short exact sequence of k–group schemes

1 µ2 SL Aut(sl (k)) 1, → → 2 → 2 → which yields a long exact sequences:

1 1 µ2(R) SL (R) Aut(sl (k))(R) H (R, µ ) . → → 2 → 2 → ´et 2 →··· 11 1 Hence, the triviality of H´et(R, µ2) yields that ϕ is the image of an element A SL (R). i.e., ∈ 2 ϕ(T(X)) = AXA−1,X sl (R). ∈ 2 Then the proof is completed by similar arguments as in Proposition 3.4.

Remark 3.6. We consider the k-differential ring ′ = (k[tq q Q], 0). Since D | ∈ k[tq q Q] is an integral domain and H1 (D, µ ) is trivial (see [3, Theo- ét b2 rem| 2.9]),∈ we deduce from Proposition 3.5 that b SL (D) SL (D) GrAut( )( ′)= 2 × 2 , F D bI2, I2 b b h− − i which shows that the group functor GrAut( ) fails to be forgetfully repre- F sentable by SL2×SL2(k) . h−I2,−I2i Although the group functor GrAut( ) is not forgetfully representable, it F is strongly related to the group scheme SL2. In the remaining of this section, we will discuss the representability of group functors involved in the exact sequence (4). Recall that both SL2 and µ2 are affine group schemes over k (representable functors from k-rng to grp). More precisely,

SL = Homk-rng(k[SL ], ) and µ = Homk-rng(k[µ ], ), 2 2 − 2 2 − where their coordinates rings are k[SL2]= k[x11, x12, x21, x22]/(x11x22 x12x21 1) and k[µ ]= k[x]/(x2 1), respectively. − − 2 − Analogous to the case of N =1, 2, 3 Lie conformal algebras [1], both SL2 and µ2 yield forgetfully representable functors:

fgt SL SLfgt : k-drng k-rng 2 grp, =(R, δ) SL (R). 2 −→ −−→ R 7→ 2 fgt µ µfgt : k-drng k-rng 2 grp, =(R, δ) µ (R). 2 −→ −→ R 7→ 2 However, the group functor =(R, δ) SL (R ), R = ker δ fails to be R 7→ 2 0 0 forgetfully representable. Theorem 3.3 suggests that we should consider the functor5:

cst : k-drng k-rng, =(R, δ) R = ker(δ). → R 7→ 0 5For a k–diﬀerential ring = (R,δ), R = ker δ is called the ring of constants of [4]. R 0 R

12 It also yields two group functors:

cst cst SL2 SL2 : k-drng k-rng grp, =(R, δ) SL2(R0). −→ µ−−→ R 7→ µcst : k-drng cst k-rng 2 grp, =(R, δ) µ (R ). 2 −→ −→ R 7→ 2 0 cst cst Furthermore, SL2 and µ2 are representable as functors from k-drng to grp in the following sense: k[SL2] together with the zero derivation form a k–diﬀerential ring (k[SL2], 0). Similarly, (k[µ2], 0) is a k–diﬀerential ring. Then it is easy to observe that

cst cst SL = Homk-drng ((k[SL ], 0), ) and µ = Homk-drng ((k[µ ], 0), ) . 2 2 − 2 2 − fgt cst Proposition 3.7. µ2 = µ2 .

fgt 2 Proof. Let =(R, δ)be a k–diﬀerential ring and r µ2 ( ), then r = 1. Thus 2rδ(r)=R 0, and so rδ(r) = 0, from which we obtain∈ δ(Rr)= rrδ(r) = 0, cst fgt i.e., r R0. This yields that r µ2(R0) = µ2 ( ). Therefore, µ2 ( ) µcst( ∈). The reverse inclusion is∈ obvious. R R ⊆ 2 R 4 The group functor Aut( ) F In this section, we consider the relationship between Aut( ) and GrAut( ). F F Theorem 4.1. Let = (R, δ) be a k–diﬀerential ring with R an integral R domain. Then GrAut( )( )= Aut( )( ). F R F R

Proof. Let ϕ Aut( )( ). It suﬃces to show ϕ(V k R) V k R. ∈ F R ⊗ ⊆ ⊗ Recall that = k[∂]T1 k[∂]T2 k[∂]T3. If we write B ⊕ ⊕

Mi i m ϕ(T(σ )) = ∂ L(rim)+ fi, mX=0 b rim R, fi k R, i =1, 2, 3, then ∈ ∈ B ⊗ i i ϕ([T(σ )λT(σ )]) = 0, i i [ϕ(T(σ ))λϕ(T(σ ))]

Mi m n = ( λ) (∂ + λ) ((∂ +2λ)L(rimrin)+L(δ(rim)rin)) + hi, − m,nX=0 b

13 where hi k[λ] k k R. By comparing the degree and coeﬃcients of λ in ∈ ⊗ B ⊗ i i i i ϕ([T(σ )λT(σ )]) = [ϕ(T(σ ))λϕ(T(σ ))], i =1, 2, 3,

2 we obtain Mi = 0 and riMi = 0, i = 1, 2, 3. Thus riMi = 0 i = 1, 2, 3 since i R is an integral domain, i.e., ϕ(T(σ )) k R. Since = Cur(sl2(k)), by i 1∈ B ⊗2 3 B ∼ Corollary 3.17 in [10], ϕ(T(σ )) (kT kT kT ) k R V k R. More ⊆ ⊕ ⊕ ⊗ ⊆ ⊗ precisely, there is an R–linear automorphism of the Lie algebra α : sl2(R) sl (R), σi Xi, i =1, 2, 3 such that ϕ(T(σi)) = T(Xi), i =1, 2, 3. → 2 7→ A similar argument using ϕ([LλL]) = [ϕ(L)λϕ(L)] yields that

M m ϕ(L) =L+ ∂ T(Ym), mX=0 b i i for Ym sl (R). Then ϕ([LλT(σ )]) = [ϕ(L)λϕ(T(σ ))] implies ∈ 2 i [Ym,X ]=0, for i =1, 2, 3,m> 0.

Applying the automorphism α−1, we obtain

−1 i [α (Ym), σ ]=0, for i =1, 2, 3,m> 0.

i −1 Since σ , i = 1, 2, 3 is a basis of sl2(k) and α (Ym) sl2(R), we deduce {−1 } m ∈ that α (Ym) = 0 for m > 0, and so Y = 0,m > 0. Hence, ϕ(L) = L+T(Y ), i.e., ϕ(L) V k R. 0 ∈ ⊗ Next we consider the odd part. By considering

ϕ([LλG(M)]) = [ϕ(L)λϕ(G(M))], for M Mat (k), we obtain ϕ(G(M)) V k R. Hence, ϕ GrAut( )( ). ∈ 2 ∈ ⊗ ∈ F R

Remark 4.2. The assumption of integral domain in Theorem 4.1 is not super- fluous. As an example, we consider the k–differential ring =(k[τ]/(τ 2), 0). R In this situation, the –conformal superalgebra k has the automorphism ϕ defined by R F ⊗ R

ϕ(∂ℓa r)= ∂ℓa r + ∂ℓ+1a τr,¯ ⊗ ⊗ ⊗ 2 for ℓ Z+, a V,r k[τ]/(τ ), where V is the k–vector space given in (1) andτ ¯∈is the canonical∈ ∈ image of τ in k[τ]/(τ 2). However, ϕ is not contained in GrAut( )( ), which shows that Aut( )( ) = GrAut( )( ). An F R F R 6 F R analogous example for the N = 2 conformal superalgebra has been given in [1, Remark 3.1].

14 Corollary 4.3. Let = (R, δ) be a k–diﬀerential ring such that R is an R1 integral domain and H´et(R, µ2) is trivial. Then

SL2(R) SL2(R0) Aut( )( ) ∼= × , F R µ2(R0) where R0 = ker δ. Remark 4.4.

1 Since k is an algebraically closed ﬁeld of characteristic zero, H (k, µ2) • ´et is trivial. Hence,

SL2(k) SL2(k) Autk ( )= Aut( )(k, 0) = × . -conf F F ∼ ( I , I ) h − 2 − 2 i

For the k-differential ring = (k[tq q Q], d ), the étale cohomology • D | ∈ dt set H1 (D, µ ) is trivial ([3, Theorem 2.9]) and D = ker( d ) = k. ét 2 b 0 dt Hence, b b

SL2(D) SL2(k) AutDb-conf( k )= Aut( )( )= × , F ⊗ D F D ( bI2, I2) b b h − − i which is the result of Proposition 3.69 in [10].

5 Comments on the functor GrAut( ) A Let be an arbitrary Lie conformal superalgebra over k such that is a free Ak[∂]–module of finite rank, then there is a finite-dimensional k–vectorA space W such that = k[∂] k W as k[∂]–modules. Hence, we can define a A ⊗ subgroup of Aut( )( ) for each k–differential ring =(R, δ): A R R

GrAut( , W )( )= φ Aut( )( ) φ(k k W k R) k k W k R . A R { ∈ A R | ⊗ ⊗ ⊆ ⊗ ⊗ } This construction is also functorial in . Hence, GrAut( , W ) is also a functor from the category of k–diﬀerentialR rings into the categoryA of groups. However, W is not uniquely determined by . In other words, the group functor GrAut( , W ) may depend on the choiceA of W . According toA [9, section 5.10], the N =1, 2, 3 Lie conformal superalgebra N can be realized as follows: N = k[∂] k Λ(N), where Λ(N) is the K K ⊗

15 Grassman superalgebra in N variables, ∂ acts on N in the natural way, and the n-th product for each n Z is deﬁned by K ∈ + N 1 1 |f| f g = f 1 ∂ fg + ( 1) (∂if)(∂ig) (0) 2| | − ⊗ 2 − Xi=1 1 f g = ( f + g ) 2 fg, (1) 2 | | | | −

f(n)g =0, n > 2, where f,g Λ(N) are homogenous polynomials in ξ , ,ξN of degree f ∈ 1 ··· | | and g respectively, and ∂i is the derivative with respect to ξi, i =1, , N. | | ··· For the N =1, 2, 3 Lie conformal superalgebras N , the particular choice that W = Λ(N) has been considered in [1]. In thisK situation,

GrAut( N , Λ(N))( )= Aut( N )( ), N =1, 2, 3, K R K R for every k–diﬀerential ring =(R, δ) with R an integral domain. R For the N = 4 Lie conformal superalgebra , we take W = V to be the k–vector space as in (1) in Section 2. It is knownF from Theorem 4.1 that

GrAut( ,V )( )= Aut( )( ) F R F R for every k–differential ring =(R, δ) with R an integral domain. However, for a given Lie conformalR superalgebra over k and an arbitrary A choice of W such that = k[∂] k W , GrAut( , W )( ) Aut( )( ) may fail to be an equality evenA if R⊗is the field k. A R ⊆ A R For instance, considering the N = 2 Lie conformal superalgebra , we K2 have known from [1] that W = Λ(2) is a good choice to define GrAut( 2, Λ(2)). However, is also realized as follows (see [9, section 5.10]): K K2

= k[∂] k (Der(Λ(1)) Λ(1)), K2 ⊗ ⊕ where Λ(1) = k kξ is the Grassman superalgebra in one variable and ⊕ Der(Λ(1)) is the superalgebra of all derivations of Λ(1). The n-th products for n Z on are given by ∈ + K2 p(a)p(f) a n b = δn [a, b], a f = a(f), a n f = δn ( 1) fa, if n > 1, ( ) 0 (0) ( ) − 1 − f g = ∂(fg), f g = 2δj fg, if n > 1, (0) − (n) − 1 where a, b Der(Λ(1)),f,g Λ(1), and p(a) (resp. p(b)) is the parity of a (resp. b). ∈ ∈

16 If we choose d d W = W ′ := Der(Λ(1)) Λ(1) = k kξ (k kξ), ⊕ dξ ⊕ dξ ⊕ ⊕

′ ′ then 2 = k[∂] k W . For such a k–vector subspace W 2, the subgroup K ⊗ ′ ⊆K functor GrAut( 2, W ) is not necessarily equal to Aut( 2) when evaluating at a k-diﬀerentialK ring = (R, δ) with R an integral domain.K Indeed, the Lie conformal superalgebraR has the automorphism φ : deﬁned K2 K2 → K2 by

d d d d d φ(1) = 1 ∂ξ , φ(ξ)= , φ = ξ, φ ξ = ξ . − dξ dξ dξ dξ − dξ

Although (k, 0) is a k–differential ring and k is a field, the automorphism φ ′ ′ is not contained GrAut( 2, W )(k). Hence, W is not a suitable choice to define GrAut( ). K K2 Acknowledgments

The author is grateful to Professor Arturo Pianzola for his valuable sugges- tions and to Valerie Budd for her careful reading of the manuscript. He thanks the referees for their helpful comments. Finally, the author appreci- ates the support of the University of Alberta and the Chinese Scholarship Council.

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