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

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The Automorphism Group Functor of the N= 4 Lie Conformal Superalgebra The automorphism group functor of the N = 4 Lie conformal superalgebra Zhihua Chang∗ Department of Mathematics, Bar-Ilan University, Ramat-Gan, 52900, Israel. September 25, 2018 Abstract In this paper, we study the structure and representability of the automorphism group functor of the N = 4 Lie conformal superalgebra over an algebraically closed field k of characteristic zero. Keywords: N = 4 Lie conformal superalgebra, automorphism group, twisted form 2000 MSC: 17B40, 14L15, 17B65 1 Introduction Let k be an algebraically closed field of characteristic zero and A a finite- dimensional (associative, Lie, Jordan...) algebra over k. The automorphism group Aut(A) of A is an affine group scheme of finite type over k, which is a representable functor from the category k-rng of commutative associative k– arXiv:1302.3897v4 [math.RA] 11 Aug 2014 algebras to the category grp of groups, assigning a commutative associative k–algebra R the group of R–linear automorphisms of A k R. The geometric properties of Aut(A) are strongly related⊗ to the algebraic properties of A. For instance, if Aut(A) is smooth and connected, then all ±1 twisted loop algebras based on A are trivial, i.e., isomorphic to A k k[t ] as algebras over the ring k[t±1] (see [12]). ⊗ In the case of Lie algebras, the centreless core of an affine Kac-Moody Lie algebra is a twisted loop algebra based on a finite-dimensional simple Lie ∗[email protected] 1 algebra g over k (see [8, Chapters 7 and 8]). Based on the point of view taken in [12], such a twisted loop algebra is a twisted form of the untwisted loop ±1 ±1 algebra g k k[t ] with respect to an ´etale extension of k[t ]. These twisted forms are⊗ classified in [12] by investigating the non-abelian ´etale cohomology of the k–group scheme Aut(g). The case of finite-dimensional simple Lie superalgebras was studied in [5] and [6] as well. In [10], in order to address the results about the N = 2, 4 supercon- formal algebras of [13], an analogous theory of twisted forms of differential conformal superalgebras is developed. To obtain the “correct” definition of conformal superalgebra over a ring, k–differential rings were considered as a replacement of commutative associative k–algebras. A k–differential ring is a pair = (R, δ) consisting of a commutative associative k–algebra R and a k–linearR derivation δ : R R. k–differential rings form a category, → denoted by k-drng, where a morphism f : = (R, δR) = (S, δS) is a R → S homomorphism of k-algebras f : R S such that f δR = δS f. For a → ◦ ◦ conformal superalgebra over k, its automorphism group functor Aut( ) is a functor from k-drngA to the category of groups (denoted by grp). WeA will review the precise definition of a Lie conformal superalgebra over k and its automorphism group functor in Section 2. Unlike the case of usual finite-dimensional algebras, for an arbitrary Lie conformal superalgebra , Aut( ) fails to be representable. However, it is A A still significant for studying the Lie conformal superalgebra : it is pointed A out in [10] that the isomorphism classes of twisted forms of k with A ⊗ R respect to a faithfully flat extension of k–differential rings are pa- 1 R → S rameterized by the non-abelian cohomology set H ( / , Aut( )R). In par- ticular, a consideration on the extension1 of k–differentialS R rings:A 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 auto- morphism 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 su- peralgebra 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 be- tween GrAut( ) and certain affine group schemes (c.f. Section 3). Finally, we will provideF an example to show that the definition of the subgroup func- tor 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 superal- gebras 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 super- algebra over k, but also an –LieA ⊗ conformalR superalgebra (c. f. [10, defi- nition 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 .
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