arXiv:math/0701399v2 [math.QA] 17 Jan 2007 NONCOMMUTATIVE LOOPS OVER LIE ALGEBRAS AND LIE GROUPS ARKADY BERENSTEIN AND VLADIMIR RETAKH Abstract. The aim of this paper is to introduce and study Lie algebras over noncommutative rings. For any Lie algebra g sitting inside an associative algebra A and any associative algebra F we introduce and study the F -loop algebra (g, A)(F ), which is the Lie subalgebra of F ⊗ A generated by F ⊗ g. In most examples A is the universal enveloping algebra of g. Our description of the loop algebra has a striking resemblance to the commutator expansions of F used by M. Kapranov in his approach to noncommutative geometry. To each F -loop algebra (g, A)(F ) we associate a “noncommutative algebraic” group which naturally acts on (g, A)(F ) by conjugations and conclude the paper with a number of examples of such groups. Contents 0. Introduction 1. Commutator expansions and identities 2. N -Lie algebras and N -loop Lie algebras 3. Upper bounds of N -loop Lie algebras 4. Perfect pairs and achievable upper bounds 5. N -groups 5.1. From N -Lie algebras to N -groups and generalized K1 -theories 5.2. N -loop groups for compatible pairs References 2 4 9 16 21 27 28 31 40 Date: January 14, 2007. The authors were supported in part by the NSF grant DMS #0501103 (A.B.), and by the NSA grant H98230-06-1-0028 (V.R.). 1 2 A. BERENSTEIN and V. RETAKH 0. Introduction The aim of this paper is to introduce and study algebraic groups and Lie algebras over noncommutative rings. A naive definition of a Lie algebra/group as a bimodule over a noncommutative algebra F over a field k does not bring any interesting example beyond GLn (F ) and the corresponding Lie algebra gln (F ) = Mn (F ) = F ⊗ gln (k). Even the special Lie algebra sln (F ) = [gln (F ), gln(F )] (which is the Lie subalgebra of all matrices in gln (F ) whose traces belong to the commutator [F , F ]) is not an F -bimodule. Similarly, the special linear group SLn (F ) is not defined by equations but rather by congruences given by the Dieudonne determinant (see [1]). This is why the “straightforward” approach to classical groups over rings started by J. Dieudonne in [5] and continued by O. T. O’Meara and others (see [7]) does not lead to algebraic groups. Also, unlike in the “commutative case, these methods does not employ rich structural theory of Lie algebras. To explain our approach we start with the fundamental inclusion which plays a pivotal role in the algebraic K-theory (0.1) En (F ) ⊂ GLn (F ) where En (F ) is the subgroup generated by matrices 1 + tEij , t ∈ F , i 6= j. Here Eij ’s are elementary matrices. It is well-known and widely used that En (F ) is normal in GLn (F ). It turns out that both groups En (F ) and GLn (F ) are completely determined by the Lie algebra sln (F ). To be more specific, assume that F is an algebra over a field k of characteristic zero, and denote A := Mn (k) = gln (k) and g := sln (k). Then sln (F ) (for n ≥ 2) can be viewed as as a Lie subalgebra in F ⊗ A generated by F ⊗ g; and gln (F ) can be viewed as the Lie algebra structure on F ⊗ A and, at the same time, the normalizer of sln (F ). Then the group En (F ) (viewed as the Lie group of sln (F )) is generated by all elements g ∈ GLn (F ) of the form g = 1 + t ⊗ E where E ∈ g is a nilpotent, t ∈ F ; and the group GLn (F ) (which can be viewed as the Lie group of gln (F )) is the group of units of F ⊗ A and, at the same time, is characterized by the property that X · sln (F ) · X −1 ⊂ sln (F ) for any X ∈ GLn (F ). Motivated by this observation, we propose to generalize inclusions of type (0.1) to all algebraic groups over noncommutative rings by looking for a structural theory of Lie algebras over noncommutative rings. Noncommutative loops over Lie algebras and Lie groups 3 Contrary to the “straightforward” approach to classical groups over (noncommutative) algebras, we start this paper by introducing “noncommutative” Lie algebras as noncommutative loops over Lie algebras, or N -loop Lie algebras. For any pair (g, A), where g is a Lie subalgebras of the associative algebra A and any algebra F we define N -loop Lie algebra (g, A)(F ) to be the Lie subalgebra of F ⊗ A generated by F ⊗ g. In other words, (g, A)(F ) can be viewed as an F -envelope of g in F ⊗ A. In many important examples A is an algebra of endomorphisms of a g-module or the universal enveloping algebra of g but we do not restrict ourselves by special cases. We explicitly compute (g, A)(F ) for a large class of algebras including semisimple and (generalized) Kac-Moody Lie algebras (Theorem 4.3). The formula for (g, A)(F ) has a striking resemblance to the commutator expansions of F used by M. Kapranov in [8] and then by M. Kontsevich and A. Rosenberg in [9] as an important tool in noncommutative geometry. Generalizing Theorem 4.3, we provide a similar computation of the noncommutative loop Lie algebra (g, A)(F ) for all perfect pairs in the sense of Definition 4.1. We also get a compact formula (Theorem 4.9) for (sl2 (k), A)(F ) which, apparently, has physical implications. We define the “noncommutative” loop group or, in short, the N -loop group Gg,A (F ) to be the set of all invertible X ∈ F ⊗ A such that X · (g, A)(F ) · X −1 = (g, A)(F ). This is our generalization of GLn (F ) as the ambient group in the inclusion (0.1). For g = sl2 (k) we explicitly compute the “Cartan subgroup” of Gg,A (F ) (Theorem 5.12) in terms of difference derivatives. We expect this result to generalize to all semisimple Lie algebras and give rise to noncommutative root systems. At the same time there are several normal subgroups in Gg,A (F ) which can play a role of En (F ). These subgroups include the commutator subgroup of Gg,A (F ) and normal subgroups ES,g,A (F ) generated by 1 + S, where S is a certain set of nilpotents in (g, A)(F ) invariant under the adjoint action of Gg,A (F ). The quotients of groups Gg,A (F ) over these normal subgroups can be viewed as generalizations of functors F 7→ K1 (F ) = GLn (F )/En (F ). Such quotients plays a role of generalized “noncommutative” K1 -functors. (We will describe some functorial choices of S and introduce the corresponding K1 -functors in Section 5.1). This paper is continuation of our study of algebraic groups over noncommutative rings and their representations started in [2]. Part of our results was published in [3]. In the next paper we will focus on “reductive groups over noncommutative rings”, their geometric structure and representations. 4 A. BERENSTEIN and V. RETAKH The paper is organized as follows. • Section 1 contains some preliminary results on ideals in associative algebras F generated by k-th commutator spaces of F . Several key results are based on the Jacobi-Leibniz type identity (1.5). • In Section 2 we introduce N -Lie algebras and their important subclass: N -loop Lie algebras (g, A)(F ) over Lie algebras g. As our first examples, we describe algebras (g, A)(F ) for all classical Lie algebras. • Section 3 contains upper bounds for N -loop Lie algebras. • Section 4 contains our main result for Lie algebras (g, A)(F ): upper bounds for algebras (g, A)(F ) coincide with them for a large class of compatible pairs (g, A) including all such pairs for semisimple Lie algebras g. • In Section 5 we introduce N -groups defined by N -Lie algebras, important classes of their normal subgroups similar to subgroups En (F ) and the corresponding K1 functors. We also consider useful examples of N -subgroups and their “Cartan subgroups” attached to the standard representations of classical Lie algebras g and to various representations of g = sl2 (k) Our description of these subgroups is based on a new class of algebraic identities for noncommutative difference derivatives (Lemma 5.25) which are of interest by themselves. Acknowledgements. The authors would like to thank M. Kapranov for very useful discussions and encouragements during the preparation of the manuscript and C. Reutenauer for explaining an important Jacobi type identity for commutators. The authors are grateful to Max-Planck-Institut für Mathematik for its hospitality and generous support during the essential stage of the work. Throughout the paper, Alg will denote the category which objects are algebras (not necessarily with 1) over a field k of characteristic zero and morphisms are algebra homomorphisms; and N will stand for a sub-category of Alg. Also Alg1 will denote that sub-category of Alg which objects are unital k-algebras over k and arrows are homomorphisms of unital algebras. 1. Commutator expansions and identities Throughout the section we will fix an arbitrary sub-category N of Alg. Given an object F N , for k ≥ 0 define the k-th commutator space F (k) of F recursively as F (0) = F , F (1) = F ′ = [F , F ], F (2) = F ′′ = [F , F ′], . . . , F (k) = [F , F (k−1) ], . . . , where for any subsets S1 , S2 of F the notation [S1 , S2 ] stands for the linear span of all Noncommutative loops over Lie algebras and Lie groups 5 commutators [a, b] = ab − ba, a ∈ S1 , b ∈ S2 . The above recursive definition defines the canonical map πk : F ⊗k ։ F (k) . For any subset S of F set S (k) = πk (S) for k > 0 and S (0) is the k-linear span of S in F . Using this notation, for any subset S ⊂ F define the space S (•) by X (1.1) S (•) = S (k) k≥0 The following result is obvious. Lemma 1.1. For any S ⊂ F the subspace S (•) is the Lie subalgebra of F generated by S. Following [8] and [9], define the subspaces Ikℓ (F ) by: X F (λ1 ) F (λ2 ) · · · F (λℓ ) , Ikℓ (F ) = λ where the summation goes over all λ = (λ1 , λ2 , . . . , λℓ ) ∈ (Z≥0 )ℓ such that λi = k. i=1 Denote also (1.2) ℓ P Ik≤ℓ (F ) := X Ikℓ (F ), Ik (F ) := Ik≤∞ = ′ 1≤ℓ′ ≤ℓ X Ikℓ (F ) . ℓ≥1 Clearly, F Ikℓ (F ), Ikℓ (F )F ⊂ Ikℓ+1 (F ). Therefore, Ik (F ) is a two-sided ideal in F . It is also easy to see that Ik (F ) = Ikk (F ) + F · Ikk (F ). Lemma 1.2. For each k, ℓ ≥ 1 one has: ≤ℓ ℓ (a) Ikℓ (F ) ⊂ Ik−1 (F ), Ik≤ℓ (F ) ⊂ Ik−1 (F ). ≤ℓ ℓ ℓ (b) [F , Ik−1(F )] ⊂ Ik (F ), [F , Ik−1(F )] ⊂ Ik≤ℓ (F ). ≤ℓ ≤ℓ+1 (c) Ik≤ℓ+1 (F ) = F [F , Ik−1 (F )] + [F , Ik−1 (F )]. Proof. To prove (a) and (b), we need the following obvious recursion for Ikℓ (F ): X ℓ−1 (1.3) Ikℓ (F ) = F (i) Ik−i (F ) i≥0 ′ (with the natural convention that Ikℓ′ (F ) = 0 if k ′ < 0). Then we prove (a) by induction in ℓ. If ℓ = 1, the assertion becomes F (k) ⊂ F (k−1) . Iterating this inclusion and using the inductive hypothesis, we obtain X X ℓ−1 ℓ−1 F (i) Ik−i (F ) = F Ikℓ−1(F ) + Ikℓ (F ) = F (i) Ik−i (F ) ⊂ i≥0 i>0 6 A. BERENSTEIN and V. RETAKH ℓ−1 ⊂ F Ik−1 (F ) + X ℓ−1 F (i−1) Ik−i (F ) = i>0 X ℓ−1 ℓ F (i) Ik−i−1 (F ) = Ik−1 (F ) . i≥0 This proves (a). Prove (b) also by induction in ℓ. If ℓ = 1, the assertion becomes [F , F (k−1)] ⊂ F (k) , which is obvious. Using the inductive hypothesis, we obtain X ℓ ℓ−1 [F , Ik−1 (F )] = [F , F (i−1) Ik−i (F )] ⊂ i≥1 ⊂ X ℓ−1 ℓ−1 (F ) + F (i−1) [F , Ik−i (F )] ⊂ [F , F (i−1) ]Ik−i X ℓ−1 F (i) Ik−i (F ) = Ikℓ (F ) . i≥0 i≥1 This proves (b). ≤ℓ+1 ≤ℓ (F )]+[F , Ik−1 (F )] by (b). Therefore, Prove (c). Obviously, Ik≤ℓ+1 (F ) ⊃ F [F , Ik−1 it suffices to prove the opposite inclusion ≤ℓ ≤ℓ+1 Ik≤ℓ+1 (F ) ⊂ F [F , Ik−1 (F )] + [F , Ik−1 (F )] . We will use the following obvious consequence of (1.3): X ≤ℓ Ik≤ℓ+1 (F ) = F (i) Ik−i (F ) . i≥0 Therefore, it suffices to prove that (1.4) ≤ℓ+1 ≤ℓ ≤ℓ (F )] + [F , Ik−1 (F )] F (i) Ik−i (F ) ⊂ F [F , Ik−1 for all i ≥ 0, ℓ ≥ 1, k ≥ 1. We prove (1.4) by induction in all pairs (ℓ, i) ordered lexicographically. Indeed, suppose that the assertion is proved for all (ℓ′ , i′ ) < (ℓ, i). The base of induction is when ℓ = 1, i = 0. Indeed, Ik≤1 (F ) = F (k) for all k and ≤2 (1.4) becomes F F (k) ⊂ F [F , F (k−1)] + [F , Ik−1 (F )], which is obviously true since [F , F (k−1)] = F (k) . If ℓ ≥ 1, i > 0, we obtain, using the Leibniz rule, the following inclusion: ≤ℓ ≤ℓ ≤ℓ ≤ℓ F (i) Ik−i (F ) = [F , F (i−1) ]Ik−i (F ) ⊂ [F , F (i−1) Ik−i (F )] + F (i−1) [F , Ik−i (F )] . ≤ℓ ≤ℓ ≤ℓ Therefore, F (i) Ik−i (F ) ⊂ [F , F (i−1) Ik−i (F )] + F (i−1) Ik+1−i (F ) by (b). Finally, using the inductive hypothesis for (ℓ, i − 1) and taking into account that ≤ℓ+1 (i−1) ≤ℓ F Ik−i (F ) ⊂ Ik−1 (F ), and, therefore, ≤ℓ ≤ℓ+1 [F , F (i−1) Ik−i (F )] ⊂ [F , Ik−1 (F )] , we obtain the inclusion (1.4). Noncommutative loops over Lie algebras and Lie groups 7 If ℓ ≥ 2, i = 0, then using the inductive hypothesis for all pairs (ℓ − 1, i′ ), i′ ≥ 0, we obtain: ≤ℓ−1 ≤ℓ Ik≤ℓ (F ) = F [F , Ik−1 (F )] + [F , Ik−1 (F )] . Multiplying by F on the left and using the distributivity of multiplication of subspaces in F · A, we obtain: ≤ℓ−1 ≤ℓ ≤ℓ F Ik≤ℓ (F ) = F 2 [F , Ik−1 (F )] + F [F , Ik−1 (F )] = F [F , Ik−1 (F )] ≤ℓ−1 ≤ℓ because F 2 ⊂ F and Ik−1 (F ) ⊂ Ik−1 (F ). This immediately implies (1.4). Part (c) is proved. The lemma is proved.  Lemma 1.3. For any k ′ , k ≥ 0, and any ℓ, ℓ′ ≥ 1 one has: ′ ≤ℓ+ℓ′ ≤ℓ′ ≤ℓ ℓ+ℓ′ (a) Ikℓ (F )Ikℓ′ (F ) ⊂ Ik+k ′ (F ), Ik (F )Ik ′ (F ) ⊂ Ik+k ′ (F ). ′ ′ ≤ℓ+ℓ′ −1 ℓ+ℓ′ −1 (F )]. (F )], [Ik≤ℓ (F ), Ik≤ℓ (b) [Ikℓ (F ), Ikℓ′ (F )] ⊂ [F , Ik+k ′ (F )] ⊂ [F , Ik+k ′ ′ Proof. Part (a) follows from the obvious fact that (F (λ1 ) F (λ2 ) · · · F (λℓ1 ) )(F (µ1 ) F (µ2 ) · · · F (µℓ2 ) ) ⊂ Ikℓ1 +ℓ2 (F ) , where k = λ1 + λ2 + · · · + λℓ1 + µ1 + µ2 + · · · + µℓ2 . Prove (b). First, we prove the first inclusion for ℓ = 1. We proceed by induction in k. The base of induction, k = 0, is obvious because I01 (F ) = F . Assume that the assertion is proved for all k1 < k, i.e., we have: ′ ′ [F (k1 ) , Ikℓ′ (F )] ⊂ [F , Ikℓ1+k′ (F )] . Then, using the fact that F (k) = [F , F (k−1)] and the Jacobi identity, we obtain: ′ ′ [F (k) , Ikℓ′ (F )] = [[F , F (k−1) ], Ikℓ′ (F )] ⊂ ′ ′ ⊂ [F , [F (k−1) , Ikℓ′ (F )]] + [F (k−1) , [F , Ikℓ′ (F )]] ⊂ ′ ′ ′ ⊂ [F , [F , Ikℓ′+k−1 (F )]] + [F (k−1) , Ikℓ′ +1 (F )] ⊂ [F , Ikℓ′+k (F )] by the inductive hypothesis and Lemma 1.2(b). This proves the first inclusion of (b) for ℓ = 1. Furthermore, we will proceed by induction in ℓ. Now ℓ > 1, assume that the assertion is proved for all ℓ1 < ℓ, i.e., we have the inductive hypothesis in the form: ′ ℓ1 +ℓ −1 (F )] [Ikℓ1 (F ), Ikℓ′ (F )] ⊂ [F , Ik+k ′ ′ for all k ≥ i ≥ 0. 8 A. BERENSTEIN and V. RETAKH We need the following useful analogue of Jacobi and Leibniz identities in F : (1.5) [ab, c] + [bc, a] + [ca, b] = 0 for all a, b, c ∈ F (The identity was communicated to the authors by C. Reutenauer and was used in a different context in the recent paper [6]). ′ ℓ−1 Using (1.3) and (1.5) with all a ∈ F (i) , b ∈ Ik−i (F ), c ∈ Ikℓ′ (F ), we obtain for all i ≥ 0: ′ ′ ′ ℓ−1 ℓ−1 ℓ−1 (F ), F (i)Ikℓ′ (F )] ⊂ (F )Ikℓ′ (F )] + [Ik−i [F (i) Ik−i (F ), Ikℓ′ (F )] ⊂ [F (i) , Ik−i ℓ+ℓ −1 ℓ−1 ℓ +1 ℓ+ℓ −1 [F (i) , Ik+k (F )] ′ −i (F )] + [Ik−i (F ), Ik ′ +i (F )] ⊂ [F , Ik+k ′ ′ ′ ′ by the the already proved (a) and inductive hypothesis. This finishes the proof of the first inclusion of (b). The second inclusion of (b) also follows.  Generalizing (1.3), for any subset S of F denote by Ikℓ (F , S) the image of S under the canonical map F ⊗kℓ ։ Ikℓ (F ), i.e., Ikℓ (F , S) = (1.6) X S (λ1 ) S (λ2 ) · · · S (λℓ ) . λ In particular, Ik1 (F , S) = S (k) and I0ℓ = S ℓ . The following result is obvious. Lemma 1.4. Let F be an object of N and S ⊂ F . Then: (a) For any k ≥ 0, ℓ ≥ 2 one has Ikℓ (F , S) = k X ℓ−1 S (i) Ik−i (F , S) . i=0 (b) For any k ′ , k ≥ 0, and any ℓ, ℓ′ ≥ 1 one has: ℓ ℓ ℓ+ℓ −1 ℓ+ℓ Ikℓ (F , S)Ikℓ′ (F , S) ⊂ Ik+k ′ (F , S), [Ik (F , S), Ik ′ (F , S)] ⊂ Ik+k ′ +1 (F , S) . ′ ′ ′ ′ In particular, (1.7) ℓ+1 ℓ S (i) Ikℓ (F , S) ⊂ Ik+i (F , S), [S (i) , Ikℓ (F , S)] ⊂ Ik+i+1 (F , S) . Noncommutative loops over Lie algebras and Lie groups 9 2. N -Lie algebras and N -loop Lie algebras Given objects F and A of Alg, we refer to a morphism ι : F → A in Alg as a F -algebra structure on A (we will also refer to A an F -algebra). Note that each F -algebra structure on A turns A into an algebra in the category of F -bimodules if A is equipped with actions F ⊗F A → A, A ⊗F F → A given by the multiplication of A and ι(F ). Definition 2.1. An N -Lie algebra is a triple (F , L, A), where F is an object of N , A is an F -algebra, and L is an F -Lie subalgebra of A, i.e., if L is a Lie subalgebra (under the commutator bracket) of A invariant under the adjoint action of F on A given by (f, a) 7→ ι(f ) · a − a · ι(f ) for all f ∈ F , a ∈ A. A morphism (F1 , L1 , A1) → (F2 , L2 , A2) of N -Lie algebras is a pair (ϕ, ψ), where ϕ : F1 → F2 is a morphism in N and ψ : A1 → A2 such that ψ(L1 ) ⊂ L2 and ψ ◦ ι1 = ι2 ◦ ϕ. Denote by LieAlgN the category of N -Lie algebras. For an N -Lie algebra (F , L, A), let Lij (F , L, A) := (F , L(ij) , A), 0 ≤ i ≤ j ≤ 3, where L(00) = L, L(ij) = L(ii) ∩ L(jj) for 0 ≤ i < j ≤ 3, and L(ii) , i = 1, 2, 3 are given by: • L(11) is the normalizer Lie algebra of L in A. • L(22) is the Lie subalgebra of A generated by ι(F ) ⊂ A and by the semigroup S = {s ∈ A : s · L = L · s}. • L(33) is the Lie subalgebra of A generated by G(ι(F )) ⊂ A, where G is the stabilizer of L in the group Autk (A), i.e., (2.1) G = {g ∈ Autk (A) : g(L) = L} . The following result is obvious. Lemma 2.2. The above three correspondences define: (a) The functors: Lij : LieAlgN → LieAlgN , 0 ≤ i ≤ j ≤ 3 such that L00 = L01 = Id. (b) The “injective” natural transformations: Lij ֒→ Ljj , Lij ֒→ Lii , 0 ≤ i < j ≤ 3 and L0j ֒→ L1j , j = 1, 2, 3; and Id ֒→ L11 . Therefore, we can construct a number of new N -Lie algebras by repeatedly applying the functors Lij to a given N -Lie algebra. 10 A. BERENSTEIN and V. RETAKH Remark 2.3. The functor L33 a “purely noncommutative” construction, because for any object (F , L, A) of LieAlgN such that F is commutative and all automorphisms of A are inner, one obtains L(33) = ι(F ) and therefore, L33 (F , L, A) = (F , ι(F ), A). Denote by π the natural (forgetful) projection functor LieAlgN → N such that π(F , L, A) = F and π(ϕ, ψ) = ϕ. Definition 2.4. A noncommutative loop Lie algebra (N -loop Lie algebra) is a functor s : N → LieAlgN such that π ◦ s = IdN (i.e., s is a section of π). Note that if N has only one object F , then the N -loop Lie algebra is simply any object of LieAlgN of the form (F , L, A). In this case, we will sometimes refer to the Lie algebra L an F -Lie algebra. In principle, we can construct a number of N -Lie algebras by twisting a given one with functors Lij from Lemma 2.2. However, the study of such “derived” N -Lie algebras is beyond the scope of the present paper. In what follows we will suppress the tensor sign in expressions like F ⊗ A and write F · A instead. Note that for any object A of Alg1 and any object F of Alg the product F ⊗ A is naturally an F -algebra via the embedding F ֒→ F · A (f 7→ f · 1). The following is a first obvious example of N -loop Lie algebras. Lemma 2.5. For any object algebra A of Alg1 and any object F of N define the object sA (F ) = (F , F · A, F · A) of LieAlgN . Then the association F 7→ sA (F ) defines a noncommutative loop Lie algebra sA : N → LieAlgN . The main object of our study will be a refinement of the above example. Given an object A of Alg1 , and a subspace g ⊂ A such that [g, g] ⊂ g (i.e., g is a Lie subalgebra of A), we say that (g, A) is a compatible pair. For any compatible pair (g, A) and an object F of N , denote by (g, A)(F ) the Lie subalgebra of the F · A = F ⊗ A (under the commutator bracket) generated by F · g, that is, (g, A)(F ) = (F · g)(•) in the notation (1.1). Proposition 2.6. For any compatible pair (g, A) the association F 7→ (F , (g, A)(F ), F · A defines the N -loop Lie algebra (g, A) : N → LieAlgN . Noncommutative loops over Lie algebras and Lie groups 11 Proof. It suffices to show that any arrow ϕ in N , i.e., any algebra homomorphism ϕ : F1 → F2 defines a homomorphism of Lie algebras (g, A)(F1 ) → (g, A)(F2 ). We need the following obvious fact. Lemma 2.7. Let A1 , A2 be objects of Alg and let ϕ : A1 → A2 be a morphism in Alg. Let S1 ⊂ A1 and S2 ⊂ A2 be two subsets such that ϕ(S1 ) ⊂ S2 . Then the (•) restriction of ϕ to the Lie algebra S1 (in the notation (1.1)) is a homomorphism of (•) (•) Lie algebras S1 → S2 . Indeed, applying Lemma 2.7 with Ai = Fi · A, Si = Fi · g, i = 1, 2, ϕ = f ⊗ idA : F1 · A → F2 · A, the trivial extension of F , we obtain a Lie algebra homomorphism (g, A)(F1) = (F1 · g)(•) → (F2 · g)(•) = (g, A)(F2). It remains to construct the action of F on L = (g, A)(F ) = (F · g)(•) . Indeed, S = F · g is invariant under the adjoint action of F on F · A. Therefore, L = (F · g)(•) is also invariant under this action of F . The proposition is proved.  If F is commutative, then (g, A)(F ) = F · g is the F -loop algebra. Therefore, if F is an arbitrary object of N , the Lie algebra (g, A)(F ) deserves a name of the N -loop Lie algebra associated with the compatible pair (g, A). If A = U(g), the universal enveloping algebra of g, then we will abbreviate g(F ) := (g, U(g))(F ). Another natural choice of A is the algebra End(V ), where V is a faithful g-module. In this case, we will sometimes abbreviate (g, V )(F ) := (g, End(V ))(F ). The following result provides an estimation of (g, A)(F ) from below. Set X (2.2) hgi = gk , k≥1 i.e., hgi is the associative subalgebra of A, maybe without unit, generated by g. Proposition 2.8. Let (g, A) be a compatible pair and F be an object of N . Then: (a) F (k) · gk+1 ⊂ (g, A)(F ) and F F (k) · [g, gk+1 ] ⊂ (g, A)(F ) for all k ≥ 0. (b) If g is abelian, i.e., g′ = [g, g] = 0, then X (2.3) (g, A)(F ) = F (k) · gk+1 . k≥0 (c) If [F , F ] = F (i.e., F is perfect as a Lie algebra), then (g, A)(F ) = F · hgi. Proof. Prove (a). We need the following technical result. 12 A. BERENSTEIN and V. RETAKH m the k-linear Lemma 2.9. Let (g, A) be a compatible pair. For all m ≥ 2 denote by gf span of all powers g m , g ∈ g. Then for any m ≥ 2 one has (2.4) m + (gm−1 ∩ gm ) = gm gf Proof. Since gi−1 g′ gm−i−1 ⊂ gm−1 for all i ≤ m − 1, we obtain the following congruence for any c = (c1 , · · · , cm ) ∈ (k× )m and x = (x1 , . . . , xm ) ∈ g×m : X m m cλ xλ mod (gm−1 ∩ gm ) , (c1 x1 + · · · cm xm ) ≡ λ λ where the summation is over all partitions λ = (λ1 , . . . , λm ) of m and we abbreviated cλ = cλ1 1 · · · cλmm and xλ = xλ1 1 · · · xλmm . Varying c = (c1 , · · · , cm ) ∈ (k× )m , the above m + (gm−1 ∩ gm ). In congruence implies that that each monomial xλ belongs to gf m + (gm−1 ∩ gm ). Taking into particular, taking λ = (1, 1, . . . , 1), we obtain gm ⊆ gf m ⊆ gm , we obtain (2.9). The lemma is proved. account that gf  We also need the following useful identity in F · A: (2.5) [sE, tF ] = st · [E, F ] + [s, t] · F E = ts · [E, F ] + [s, t] · EF for any s, t ∈ F , E, F ∈ A. We will prove the first inclusion (a) by induction in k. If k = 0, one obviously has (0) 1 F g = F · g ⊂ (g, A)(F ). Assume now that k > 0. Then for g ∈ g we obtain using (2.5): [F · g, F (k−1) · g k ] = [F , F (k−1)] · g k+1 = F (k) · g k+1 k+1 ⊂ (g, A)(F ) (in the notation of Lemma 2.9). Using which implies that F (k) · gg Lemma 2.9, we obtain k+1 ≡ F (k) · gk+1 F (k) · gg mod F (k) · (gk ∩ gk+1 ) . Taking into account that F (k) · (gk ∩ gk+1 ) ⊂ F (k−1) · gk ⊂ (g, A)(F ) by the inductive hypothesis (where we used the inclusion F (k) ⊂ F (k−1) ), the above implies that F (k) · gk+1 also belongs to (g, A)(F ). This proves the first inclusion of (a). To prove the second inclusion, we compute: [F · g, F (k−1) · gk ] ≡ F F (k−1) · [g, gk ] mod F (k) · gk+1 . Therefore, using the already proved inclusion F (k) · gk+1 ⊂ (g, A)(F ), we see that F F (k−1) · [g, gk ] also belongs to (g, A)(F ). This finishes the proof of (a). Noncommutative loops over Lie algebras and Lie groups 13 Prove (b). Clearly, (a) implies that (g, A)(F ) contains the right hand side of (2.3). Therefore, it suffices to prove that the latter space is closed under the commutator. Indeed, since g is abelian, one has [F (k1 ) · gk1 +1 , F (k2 ) · gk2 +1 ] = [F (k1 ) , F (k2 ) ] · gk1 +k2 +2 ⊂ ⊂ F (k1 +k2 +1) · gk1 +k2 +2 ⊂ (g, A)(F ) because [F (k1 ) , F (k2) ] ⊂ F (k1 +k2 +1) . This finishes the proof of (b). Prove (c). Since F ′ = F , the already proved part (a) implies that F ·gk ⊂ (g, A)(F ) for all k ≥ 1, therefore, F · hgi ⊆ (g, A)(F ). But since hgi is an associative subalgebra of A containing g, we obtain an opposite inclusion (g, A)(F ) ⊆ F · hgi. This finishes the proof of (c). The proposition is proved.  Remark 2.10. Proposition 2.8(c) shows that the case when [F , F ] = F is not of much interest. This happens, for example, when F is a Weyl algebra or the quantum torus. In these cases a natural anti-involution on F can be taken into account. We will discuss it in a separate paper. Definition 2.11. We say that a compatible pair (g, A) is of finite type if there exists m > 0 such that g + g2 + · · · + gm = A, and we call such minimal m the type of (g, A). If such m does not exists, we say that (g, A) is of infinite type. Note that (g, A) is of type 1 if and only if g = A, which, in its turn, implies that (g, A)(F ) = F · A for all objects F of N . Note also that if hgi = A and A is finite-dimensional over k, then (g, A) is always of finite type. Proposition 2.12. Assume that (g, A) is of type 2, i.e., g 6= A and g + g2 = A. Then (2.6) (g, A)(F ) = F · g + F ′ · A + F F ′ · [A, A] , where F ′ = [F , F ]. Proof. Proposition 2.8(a) guarantees that F · F · g + F ′ · g2 + F F ′ · [g, g2 ] ⊂ (g, A)(F ) . Clearly, F · g + F ′ · g2 = F · g + F ′ · A (because F ′ ⊂ F ). Let us now prove that [g, A] = [A, A]. Obviously, [g, A] ⊆ [A, A]. The opposite inclusion immediately 14 A. BERENSTEIN and V. RETAKH follows from the following one [g2 , g2 ] ⊆ [g, g3 ], which, in its turn follows from (1.5): taking any a ∈ g, b ∈ g, c ∈ g2 in (1.5), we obtain [ab, c] ⊂ [g, g3 ]. Using the equation [g, A] = [A, A] we obtain F F ′ · [A, A] ⊂ (g, A)(F ). This proves that (g, A)(F ) contains the right hand side of (2.6). To finish the proof, it suffices to show that the latter space is closed under the commutator. Indeed, abbreviating A′ = [A, A], we obtain [F · g, F ′ · A] ⊂ F F ′ · [g, A] + [F , F ′] · Ag ⊂ F F ′ · A′ + F ′ · A ⊂ (g, A)(F ) , [F · g, F F ′ · A′ ] ⊂ F 2 F ′ · [g, A′ ] + [F , F F ′] · A′ g ⊂ F F ′ · A′ + F ′ · A ⊂ (g, A)(F ) , [F ′ · A, F ′ · A] ⊂ (F ′ )2 · A′ + [F ′ , F ′ ] · A2 ⊂ F F ′ · A′ + F ′ · A ⊂ (g, A)(F ) , [F ′ · A, F F ′ · A′ ] ⊂ F ′F F ′ · [A, A′ ] + [F ′ , F F ′] · A′ A ⊂ (g, A)(F ) because F ′ F F ′ · [A, A′ ] ⊂ F F ′ · A′ ⊂ (g, A)(F ) and [F ′ , F F ′] · A′ A ⊂ F ′ · A ⊂ (g, A)(F ). Finally, [F ′ F · A′ , F F ′ · A′ ] ⊂ (F ′ F )2 · [A′ , A′ ] + [F ′ F , F F ′] · (A′ )2 ⊂ (g, A)(F ) because (F ′ F )2 · [A′ , A′ ] ⊂ F ′F · A′ ⊂ (g, A)(F ) and [F ′ F , F F ′] · [A, A]2 ⊂ F ′ · A ⊂ (g, A)(F ). The proposition is proved.  For any k-vector space V and any object F of N we abbreviate sl(V, F ) := (sl(V ), End(V ))(F ) and gl(V, F ) := (End(V ), End(V ))(F ) = F · End(V ). Corollary 2.13. Let V be a finite-dimensional k-vector space such that dim V > 1. Then (g, A) = (sl(V ), End(V )) is of type 2 and sl(V, F ) = F ′ · 1 + F · sl(V ) . Hence sl(V, F ) is the set of all X ∈ gl(V, F ) such that T r(X) ∈ F ′ = [F , F ] (where T r : gl(V, F ) = F · End(V ) → F is the trivial extension of the ordinary trace End(V ) → k). Proof. Let us prove that the pair (g, A) = (sl(V ), End(V )) is of type 2, i.e., sl(V ) + sl(V )2 = End(V ). It suffices to show that 1 ∈ sl(V )2 . To prove it, choose a basis e1 , . . . , en in V so that V ∼ = Mn (k). = sln (k) and A = End(V ) ∼ = kn , sl(V ) ∼ Indeed, for any indices i 6= j both Eij and Eji belong to sl(V ), therefore, Eij Eji = P Eii ∈ sl(V )2 . Therefore, 1 = ni=1 Eii also belongs to sl(V )2 . Applying Proposition 2.12 and using the obvious fact that [A, A] = sl(V ), we obtain sl(V, F ) = F · sl(V ) + F ′ · A + F F ′[A, A] = F · sl(V ) + F ′ · 1 . Noncommutative loops over Lie algebras and Lie groups 15 This proves the first assertion. The second one follows from the obvious fact that the trace T r : F · End(V ) → F is the projection to the second summand of the direct sum decomposition F · End(V ) = F · sl(V ) + F · 1 . This proves the second assertion. The corollary is proved.  We can construct more pairs of type 2 as follows. Let V be a k-vector space and Φ : V × V → k be a bilinear form on V . Denote by o(Φ) the orthogonal Lie algebra of Φ, i.e., o(Φ) = {M ∈ End(V ) : Φ(M(u), v) + Φ(u, M(v)) = 0 ∀ u, v ∈ V } . Denote by K = KΦ ⊂ V the sum of the left and the right kernels of Φ (if Φ is symmetric or skew-symmetric, then K is the left kernel of Φ). Finally, denote by End(V, K) the parabolic subalgebra of End(V ) which consists of all M ∈ End(V ) such that M(K) ⊂ K. Clearly, o(Φ) ⊂ End(V, K), i.e., (o(Φ), End(V, K)). For and any object F of N we abbreviate o(Φ, F ) := (o(Φ), End(V, K))(F ). Corollary 2.14. Let V be a finite-dimensional k-vector space and Φ be a symmetric or skew-symmetric bilinear form on V . Then (o(Φ), End(V, K)) is of type 2. In particular, (2.7) o(Φ, F ) = F · o(Φ) + F ′ · 1 + F ′ · 1K + (F F ′ + F ′ ) · sl(V, K) . Here sl(V, K) is the set of all M in End(V, K) such that T r(M) = 0 and T r(MK ) = 0, where MK : K → K is the restriction of M to K and 1K ∈ End(V, K) is any element such that k · 1 + k · 1K + sl(V, K) = End(V, K). If K = 0, we set 1K = 0. Proof. First prove that (g, A) = (o(Φ), End(V, K)) is of type 2. We “complexify” the involved objects, i.e., replace both V and K with V = k · V = k ⊗ V , K = k · K etc, where k is the algebraic closure of k. Using the obvious fact that U + U ′ = U +U ′ and U · U ′ = U · U ′ for any subspaces of End(V ) and o(Φ) = o(Φ), we see that it suffices to show that the pair (o(Φ), End(V , K) is of type 2. Furthermore, without loss of generality we consider the case when K = 0, i.e., the form Φ is non-degenerate. If Φ is symmetric, one can choose a basis of V so that n n V ∼ = k , and Φ is the standard dot product on k . In this case o(Φ) is on (k), the Lie algebra of orthogonal matrices, which is generated by all elements Eij −Eji where Eij is the corresponding elementary matrix. Using the identity (Eij −Eji )2 = −(Eii +Ejj ) for i 6= j, we see that on (k)2 contains all diagonal matrices. Furthermore, if i, j, k 16 A. BERENSTEIN and V. RETAKH are pairwise distinct indices then (Eij − Eji )(Ejk − Ekj ) = Eik . Thus we have shown that on (k)2 = Mn (k) = End(V , K). Therefore, on (k)2 = Mn (k) = End(V, K). This proves the assertion for the symmetric Φ. If Φ is skew-symmetric and non-degenerate, then n = 2m and one can choose a n basis of V such that V is identified with k and o(Φ) is identifies the symplectic Lie algebra sp2m (k). Recall that a basis in sp2m (k) can be chosen as follows. It consists of elements Eij − Ej+m,i+m , Ei,m+j + Ej,m+i , Em+i,j + Em+j,i , for i, j ≤ m. Using the identity (Ei,i+m + Ei+m,i )2 = Eii + Ei+m,i+m and the fact that (Eii − Ei+m,i+m ) ∈ sp2m (k), we see that all diagonal matrices belong sp2m (k) + sp2m (k)2 . Also, the identity (Eii − Ei+m,i+m )(Eij − Ej+m,i+m ) = Eij for i 6= j implies that Eij ∈ sp2m (k) for all i, j ≤ m. Similarly, one can prove that Eij ∈ sp2m (k) for i, j ≥ m. Furthermore, the identity (Eii −Ei+m,i+m )(Eiℓ +Ei+m,ℓ−m ) = Eiℓ −Ei+m,ℓ−m implies that sp2m (k) + sp2m (k)2 contains all Eik for i ≤ m, k > m and for i > m, k ≤ m. Thus we have shown that sp2m (k) + sp2m (k)2 = Mn (k) = End(V , K). Therefore, sp2m (k) + sp2m (k)2 = Mn (k) = End(V, K). This proves the assertion for the skewsymmetric Φ. Prove (2.7) now. We abbreviate A = End(V, K). Obviously, [A, A] = sl(V, K) and, if K 6= {0}, then k · 1 + sl(V, K) is of codimension 1 in A, i.e, 1K always exists. Therefore, applying Proposition 2.12, we obtain o(Φ, F ) = F ·o(Φ)+F ′ ·A+F F ′[A, A] = F ·o(Φ)+F ′ ·1+F ′ ·1K +(F F ′ +F ′)·sl(V, K) . This finishes the proof of Corollary 2.14.  3. Upper bounds of N -loop Lie algebras ^ For any compatible pair (g, A) define two subspaces (g, A)(F ) and (g, A)(F ) of F · A by: X ^ Ik (F ) · [g, gk+1 ] + [F , Ik−1 (F )] · gk+1 , (3.1) (g, A)(F ) = F · g + k≥1 where Ik (F ) is defined in (1.3); and X (3.2) (g, A)(F ) = F · g + Ikℓ11 +1 Ikℓ22 +1 · [Jℓk11 +1 , Jℓk22 +1 ] + [Ikℓ11 +1 , Ikℓ22 +1 ] · Jℓk22 +1 Jℓk11 +1 , where the summation is over all k1 , k2 ≥ 0, ℓ1 , ℓ2 ≥ 0, and we abbreviated Ikℓ := Ikℓ (F ), Jkℓ := Ikℓ (A, g) in the notation (1.6). Noncommutative loops over Lie algebras and Lie groups 17 ^ We will refer to (g, A)(F ) as the upper bound of (g, A)(F ) and to (g, A)(F ) as refined upper bound of (g, A)(F ). ^ It is easy to see that the assignments F 7→ (g, A)(F ) and F 7→ (g, A)(F ) are ^ functors (g, A) and (g, A) from N to the category V ectk of k-vector spaces. The following result is obvious. Lemma 3.1. If (g, A) is a compatible pair of type m (see Definition 2.11), then (3.3) ^ (g, A)(F ) = F · g + m−1 X Ik (F ) · [g, gk+1 ] + [F , Ik−1(F )] · gk+1 . k=1 The following is the main result of this section, which explains this terminology ^ and proves that both (g, A) and (g, A)(F ) define N -loop Lie algebras N → LieAlgN . Theorem 3.2. For any compatible pair (g, A) and any object F of N one has: ^ (a) The subspace (g, A)(F ) is a Lie subalgebra of F · A. (b) The subspace (g, A)(F ) is a Lie subalgebra of F · A. ^ (c) (g, A)(F ) ⊆ (g, A)(F ) ⊆ (g, A)(F ). Proof. Prove (a). Using (2.5), we obtain ^ [F · g, F · g] ⊂ F 2 · [g, g] + [F , F ] · g2 ⊂ (g, A)(F ) because F 2 ⊂ F , [g, g] ⊂ g, and I0 (F ) = F . Furthermore, ^ [F · g, Ik (F ) · [g, gk+1 ]] ⊂ F Ik (F ) · [g, [g, gk+1 ]] + [F , Ik (F )] · [g, gk+1 ]g ⊂ (g, A)(F ) because F Ik (F ) ⊂ Ik (F ), [g, [g, gk+1 ]] ⊂ [g, gk+1 ], and [g, gk+1 ]g ⊂ gk+2 . Finally, abbreviating Jk := [F , Ik−1(F )], we obtain: ^ [F · g, Jk · gk+1 ] ⊂ F · Jk · [g, gk+1 ] + [F , Jk ] · gk+2 ⊂ (g, A)(F ) because, taking into the account that Jk ⊂ Ik−1 (F ) since Ik−1 (F ) is a two-sided ideal in F , we have F Jk ⊂ F Ik−1(F ) ⊂ Ik−1 (F ) and, taking into account that Jk ⊂ Ik (F ) by Lemma 1.2(b) taken with ℓ = ∞, we have [F , Jk ] ⊂ [F , Ik (F )]. Furthermore, we need the following obvious consequence of (1.5). Lemma 3.3. For any compatible pair (g, A) one has [gk+1 , gm+1 ] ⊂ [g, gk+m−1 ] for any k, m ≥ 1. 18 A. BERENSTEIN and V. RETAKH Therefore, for any k, m ≥ 1 one has: [Ik (F ) · gk+1 , Im (F ) · gm+1 ] ⊂ ^ ⊂ Ik (F )Im (F ) · [gk+1 , gm+1 ] + [Ik (F ), Im(F )] · gk+m+2 ⊂ (g, A)(F ) because Ik (F )Im (F ) = Ik+m (F ) by Lemma 1.3(a), [gk+1 , gm+1 ] ⊂ [g, gk+m−1 ] by Lemma 3.3, and [Ik (F ), Im(F )] ⊂ [F , Ik+m−1(F )] by Lemma 1.3(b) taken with ℓ = ∞. Therefore, taking into account that ^ [Ik , Im ] ⊂ [Ik (F ) · gk+1 , Im (F ) · gm+1 ] ⊂ (g, A)(F ) for Ir stands for any of the spaces Ir (F ) · [g, gr+1 ], [F , Ir−1(F )] · gr+1 we finish the proof of (a). Prove (b). For any subsets X and Y of an object A of Alg and ε ∈ {0, 1} denote  X · Y if ε = 0 X •ε Y := [X, Y ] if ε = 1 We need the following result. Lemma 3.4. Let Γ be an abelian group and let A and F be objects of Alg. Assume that Aα ⊂ F and Bα ⊂ A are two families labeled by Γ such that (3.4) Aα •ε Aβ ⊆ Aα+β+ε·v , Bβ •ε Bα ⊆ Bα+β−ε·v for all α, β ∈ Γ, ε ∈ {0, 1}, where v is a fixed element of Γ. Then for any α0 ∈ Γ the subspace X (Aα •1−ε Aβ ) · (Bβ+v •ε Bα+v ) h = Aα0 · Bα0 +v + α,β∈Γ,ε∈{0,1} is a Lie subalgebra of F · A = F ⊗ A. Proof. The equation (2.5) implies that [A · B, A′ · B ′ ] ⊂ (A •1−δ A′ ) · (B ′ •δ B) for each δ ∈ {0, 1}. Therefore, (i) Taking A = Aα •1−ε Aβ , B = Bβ+v •ε Bα+v , A′ = Aα′ •1−ε′ Aβ ′ , B ′ = Bβ ′ +v •ε′ Bα′ +v , and taking into the account that A ⊆ Aα′′ , A′ ⊆ Aβ ′′ , B ⊆ Bα′′ +v , and B ′ ⊆ Bβ ′′ +v by (3.4), where α′′ = α + β + (1 − ε) · v and β ′′ = α′ + β ′ + (1 − ε′ ) · v, we obtain for each δ ∈ {0, 1}: [A · B, A′ · B ′ ] ⊂ (Aα′′ •1−δ Aβ ′′ ) · (Bβ ′′ +v •δ Bα′′ +v ) ⊂ h . Noncommutative loops over Lie algebras and Lie groups 19 (ii) Taking A = Aα0 , B = Bα0 +v , A′ = Aα′ •1−ε′ Aβ ′ , B ′ = Bβ ′ +v •ε′ Bα′ +v , and taking into the account that A′ ⊆ Aβ ′′ and B ′ ⊆ Bβ ′′ +v by (3.4), where β ′′ = α′ +β ′ +(1−ε′ )·v, we obtain for each δ ∈ {0, 1}: [A · B, A′ · B ′ ] ⊂ (Aα0 •1−δ Aβ ′′ ) · (Bβ ′′ +v •δ Bα0 +v ) ⊂ h . (ii) Taking A = A′ = Aα0 , B = B ′ = Bα0 +v , we obtain for each δ ∈ {0, 1}: [A · B, A′ · B ′ ] ⊂ (Aα0 •1−δ Aα0 ) · (Bα0 +v •δ Bα0 +v ) ⊂ h . The lemma is proved.  Taking in Lemma 3.4: Γ = Z2 , α = (k, ℓ + 1) ∈ Z2 , v = (1, −1),   I ℓ+1 (F ) if k, ℓ ≥ 0 I k+1 (A, g) if k, ℓ ≥ 0 k ℓ Aα = , Bα+v = , 0  otherwise 0 otherwise Lemma 1.4 implies that (3.4) holds for all α, β ∈ Z2 , ε ∈ {0, 1}. Therefore, applying Lemma 3.4 with α0 = (0, 1), we finish the proof of the assertion that (g, A)(F ) is a Lie subalgebra of F · A. This finishes the proof of (b). Prove (c). The first inclusion (g, A)(F ) ⊂ (g, A)(F ) is obvious because F · g ⊂ (g, A)(F ) and (g, A)(F ) is a Lie subalgebra of F · A. Let us prove the second inclusion (g, A)(F ) ⊂ (g, A)(F ) of (c). Rewrite the result of Lemma 1.3(b) (with ℓ1 = ℓ2 = ∞) in the form of (3.4) as:  I if ε = 1 k1 +k2 (F ) ℓ1 +1 ℓ2 +1 Ik1 (F ) •1−ε Ik2 (F ) ⊂ Ik1 (F ) •1−ε Ik2 (F ) ⊂ . [F , Ik +k −1 (F )] if ε = 0 1 2 Using the obvious inclusion Jkℓ+1 = Iℓk+1 (A, g) ⊂ gk+1 for all k, ℓ ≥ 0 and Lemma 3.3, we obtain  gk1 +k2 +2 if ε = 0 k2 +1 k1 +1 k2 +1 k1 +1 Jℓ 2 • ε Jℓ 1 ⊂ g •ε g ⊂ [g, gk1 +k2 +1 ] if ε = 1 for all k1 , k2 , ℓ1 , ℓ2 ≥ 0, ε ∈ {0, 1}. Therefore, we obtain the inclusion: (Ikℓ11 +1 •1−ε Ikℓ22 +1 ) · (Jℓk12 +1 •ε Jℓk11 +1 ) ⊂  I k1 +k2 +1 ] k1 +k2 (F ) · [g, g ⊂ k +k [F , Ik +k −1 (F )] · g 1 2 +2 1 2 if ε = 1 if ε = 0 ^ ⊂ (g, A)(F ) . ^ A)(F ) and finishes the proof of (c). This proves the inclusion (g, A)(F ) ⊂ (g, 20 A. BERENSTEIN and V. RETAKH Therefore, Theorem 3.2 is proved.  Now we will refine Theorem 3.2 by introducing a natural filtration on each involved Lie algebra and proving the ”filtered” version of the theorem. For any compatible pair (g, A), an object F of N , and each m ≥ 1 we define the ^ subspaces F · hgim , (g, A)m (F ), (g, A) (F ) and (g, A) (F ) of F · A by: m m F · hgim = X F · gk X (F · g)(k) 1≤k≤m (3.5) (g, A)m (F ) = 0≤k<m (3.6) ^ (g, A)m (F ) = F · g + X ≤m−k Ik≤m−k (F ) · [g, gk+1 ] + [F , Ik−1 (F )] · gk+1 , 1≤k<m where Ik≤ℓ (F ) is defined in (1.3) and X (3.7) (g, A)m (F ) = F · g + Ikℓ11 +1 Ikℓ22 +1 · [Jℓk11 +1 , Jℓk22 +1 ] + [Ikℓ11 +1 , Ikℓ22 +1 ] · Jℓk22 +1 Jℓk11 +1 , where the summation is over all k1 , k2 ≥ 0, ℓ1 , ℓ2 ≥ 0 such that k1 +k2 +ℓ1 +ℓ2 +2 ≤ m, and we abbreviated Ikℓ := Ikℓ (F ), Jkℓ := Ikℓ (A, g) in the notation (1.6). Recall that a Lie algebra h = (h1 ⊂ h2 ⊂ . . .) is called a filtered Lie algebra if [hk1 , hk2 ] ⊂ hk1 +k2 for all k1 , k2 ≥ 0. Taking into account that [gk1 +1 , gk2 +1 ] ⊂ gk1 +k2 +1 , we see that hm = F · hgim , m ≥ 0 defines an increasing filtration on the Lie algebra F · hgi (where hgi is as in (2.2). The following result is a filtered version of Theorem 3.2. Theorem 3.5. For any compatible pair (g, A) and an object F of N one has: ^ (a) (g, A)(F ) is a filtered Lie subalgebra of F · hgi. (b) (g, A)(F ) is a filtered Lie subalgebra of F · hgi. (c) A chain of inclusions of filtered Lie algebras: ^ (g, A)(F ) ⊆ (g, A)(F ) ⊆ (g, A)(F ) . The proof of Theorem 3.5 is almost identical to that of Theorem 3.2. Noncommutative loops over Lie algebras and Lie groups 21 4. Perfect pairs and achievable upper bounds Below we lay out some sufficient conditions on the compatible pair (g, A) which guarantee that the upper bounds are achievable. Definition 4.1. We say that a compatible pair (g, A) is perfect if (4.1) [g, gk ]g + (gk ∩ gk+1 ) = gk+1 for all k ≥ 2. Definition 4.2. We say that a Lie algebra g over an algebraically closed field k is strongly graded if there exists an element h0 ∈ g such that: (i) The operator ad h0 on g is diagonalizable, i.e., (4.2) g= M gc , c∈k where gc ⊂ g is an eigenspace of ad h0 with the eigenvalue c. (ii) The nulspace g0 of ad h0 is spanned by [gc , g−c ], c ∈ k \ {0}. The class of strongly graded Lie algebras is rather large; it includes all semisimple and Kac-Moddy Lie algebras, as well as the Virassoro algebra. Main Theorem 4.3. Let (g, A) be a compatible pair. Then (a) If (g, A) is perfect, then for any object F of N one has ^ (g, A)(F ) = (g, A)(F ) , ^ i.e., the N -loop Lie algebras (g, A), (g, A) : N → LieAlgN are equal. (b) If g = k ⊗ g is strongly graded, then (g, A) is perfect. (c) If g is semisimple over k, then for any object F of N one has X (4.3) (g, A)(F ) = F · g + Ik−1 (F ) · (gk )+ + [F , Ik−2 (F )] · Zk (g) , k≥2 where (gk )+ = [g, gk ] is the “centerless” part of gk , Zk (g) = Z(hgi) ∩ gk , and Z(hgi) P is the center of hgi = k≥1 gk . Proof. Prove (a). We need the following assertion regarding the lower bound for (g, A)(F ). 22 A. BERENSTEIN and V. RETAKH Proposition 4.4. Let (g, A) be a compatible pair and F be an object of N . (a) Assume that for some k ≥ 2 one has I · [g, gk ] ⊂ (g, A)(F ) where I is a left ideal in F . Then: [F , I] · [g, gk ]g ⊂ (g, A)(F ) . (4.4) (b) Assume that for some k ≥ 2 one has J · gk ⊂ (g, A)(F ) where J is a subset of F such that [F , J] ⊂ J. Then: (4.5) [F , J] · gk+1 + (F J + J) · [g, gk ] ⊂ (g, A)(F ) Proof. Prove (a). Indeed, [F · g, I · [g, gk ]] ≡ [F , I] · [g, gk ]g mod F I · [g, [g, gk ]] . Since F I ⊂ F and [g, [g, gk ]] ⊂ [g, gk ], and, therefore, F I · [g, [g, gk ]] ⊂ I · [g, gk ] ⊂ (g, A)(F ), the above congruence implies that [F , I] · [g, gk ]g also belongs to (g, A)(F ). This proves (a). Prove (b). For any g ∈ g we obtain: [F · g, J · g k ] = [F , J] · g k+1 k+1 ⊂ (g, A)(F ) (in the notation of Lemma 2.9). Using which implies that [F , J] · gg Lemma 2.9, we obtain k+1 ≡ [F , J] · gk+1 [F , J] · gg mod [F , J] · (gk ∩ gk+1 ) . Taking into account that [F , J] · (gk ∩ gk+1 ) ⊂ [F , J] · gk ⊂ (g, A)(F ), the above implies that [F , J] · gk+1 also belongs to (g, A)(F ). Furthermore, [F · g, J · gk ] ≡ F J · [g, gk ] mod [F , J] · gk+1 . Therefore, using the already proved inclusion [F , J] · gk+1 ⊂ (g, A)(F ), we see that F J · [g, gk ] also belongs to (g, A)(F ). Finally, using the fact that [g, gk ] ⊂ gk , we obtain J · [g, gk ] ⊂ J · gk ⊂ (g, A)(F ). This proves (b). Proposition 4.4 is proved.  Noncommutative loops over Lie algebras and Lie groups 23 Now we are ready to finish the proof of Theorem 4.3(a). In view of Theorem 3.2(c), ^ it suffices to prove that (g, A)(F ) ⊂ (g, A)(F ), that is, (4.6) Ik (F ) · [g, gk+1 ] ⊂ (g, A)(F ), [F , Ik (F )] · gk+2 ⊂ (g, A)(F ) for k ≥ 0. We will prove (4.6) by induction in k. First, verify the base of induction at k = 0. Obviously, I0 (F )·[g, g] ⊂ (g, A)(F ) = F ·[g, g] ⊂ (g, A)(F ). Furthermore, Proposition 4.4(b) taken with k = 1, J = F implies that [F , F ] · g2 ⊂ (g, A)(F ). Now assume that k > 0. Using a part of the inductive hypothesis in the form [F , Ik−1(F )]·gk+1 ⊂ (g, A)(F ) and applying Proposition 4.4(b) with J = [F , Ik−1(F )], we obtain (F J + J) · [g, gk+1 ] ⊂ (g, A)(F ). In its turn, Lemma 1.2(b) taken with ℓ = ∞ implies that F J + J = Ik (F ). Therefore, we obtain Ik (F ) · [g, gk+1 ] ⊂ (g, A)(F ) , which is the first inclusion of (4.6). To prove the second inclusion (4.6), we will use Proposition 4.4(a) with I = Ik (F ): (g, A)(F ) ⊃ [F , Ik (F )] · [g, gk+1 ]g . On the other hand, using the perfectness of the pair (g, A), we obtain: [F , Ik (F )] · [g, gk+1 ]g ≡ [F , Ik (F )] · gk+2 mod [F , Ik (F )] · gk+1 . But Lemma 1.2(a) taken with ℓ = ∞ implies that Ik (F ) ⊂ Ik−1 (F ), therefore, [F , Ik (F )] · (gk+1 ∩ gk+2 ) ⊂ [F , Ik (F )] · gk+1 ⊂ [F , Ik−1(F )] · gk+1 ⊂ (g, A)(F ) by the inductive hypothesis. This gives the second inclusion of (4.6). Therefore, Theorem 4.3(a) is proved. Prove (b) now. We will first show that each the pair (g, A) is perfect, whenever g = k ⊗ g is strongly graded. We need the following result. Lemma 4.5. Let (g, A) be a compatible pair. Assume that h0 ∈ g = k ⊗ g is such that ad h0 is diagonalizable, i.e., has a decomposition (4.2). Then: k+1 (a) For each k ≥ 1 and each c = (c1 , . . . , ck+1 ) ∈ k \ {0} the subspace gc1 · · · gck+1 k+1 k k k+1 of g belongs to [g, g ]g + (g ∩ g ). (b) If g = k ⊗ g is a strongly graded Lie algebra, then one has (in the notation of Definition 4.2): gk+1 ⊂ [g, gk ]g + (gk ∩ gk+1 ) , 0 24 A. BERENSTEIN and V. RETAKH Proof. Prove (a). Clearly, under the adjoint action of h0 on gk each vector of k x ∈ gc1 · · · gck satisfies [h0 , x] = (c1 + · · · + ck )x. Therefore, for any (c1 , . . . , ck ) ∈ k such that c1 + · · · + ck 6= 0 the subspace gc1 · · · gck belongs to [g, gk ]. Clearly, gc1 · · · gck+1 ≡ gcσ(1) · · · gcσ(k) gcσ(k+1) mod gk ∩ gk+1 for any permutation σ ∈ Sk+1 . k+1 It is also easy to see that for any c = (c1 , . . . , ck+1) ∈ k \ {0} there exists a permutation σ ∈ Sk+1 such that cσ(1) + · · · + cσ(k) 6= 0 and, therefore, gc1 · · · gck+1 ⊂ (gcσ(1) · · · gcσ(k) )gcσ(k+1) + gk ∩ gk+1 ∈ [g, gk ]g + (gk ∩ gk+1 ) . This proves (a). Prove (b) now. Let c ∈ k \ {0} be such that hc = [gc , g−c ] 6= 0. Then we obtain the following congruence: [gc , g0k−1 g−c ]g0 ≡ g0k−1 hc g0 mod [gc , g0k−1 ]g−c g0 Taking into the account that [gc , g0 ] ⊂ gc , we obtain: [gc , g0k−1 ]g−c g0 ⊂ k−1 X k−1−i gi−1 g−c g0 ⊂ [g, gk ]g + (gk ∩ gk+1 ) 0 gc g0 i=1 by the already proved part (a). Therefore, g0k−1 hc g0 ⊂ [g, gk ]g + (gk ∩ gk+1 ). Since g is strongly graded, the subspaces hc , c ∈ k \ {0} span g0 , and therefore, gk+1 ⊂ 0 k k k+1 [g, g ]g + (g ∩ g ). This proves (b). The lemma is proved.  Thus Lemma 4.5 guarantees that for any strongly graded Lie algebra g one has: [g, gk ]g + (gk ∩ gk+1 ) = gk+1 for all k ≥ 2, where g = k ⊗ g is the “complexification” of g. Since the “complexification” commutes with the multiplication and the commutator bracket in A, the restriction of the above equation to gk+1 ⊂ gk+1 becomes (4.1). This finishes the proof of Theorem 4.3(b). Prove Theorem 4.3(c) now. Since for each semisimple Lie algebra g the compatible pair (g, A) is perfect by the already proved Theorem 4.3(b), Theorem 4.3(a) implies ^ that (g, A)(F ) = (g, A)(F ). Therefore, in order to finish the proof of Theorem 4.3(c), Noncommutative loops over Lie algebras and Lie groups 25 it suffices to show that (4.7) ^ (g, A)(F ) = F · g + X Ik−1 (F ) · (gk )+ + [F , Ik−2(F )] · Zk (g) . k≥2 We need the following simple fact regarding compatible pairs (g, A) with g being semisimple. Lemma 4.6. Let g be a semisimple Lie algebra over k. Then for any associative k-algebra A containing g one has the following decomposition of the g-module gk , k ≥ 2: gk = [g, gk ] + Zk (g), [g, gk ] ∩ Zk (g) = {0} , P where Zk (g) = Z(hgi) ∩ gk , and Z(hgi) is the center of hgi = k≥0 gk . Proof. Clearly, gk is a semisimple finite-dimensional g-module (under the adjoint action). Therefore, it uniquely decomposes into isotypic components one of which, the component of invariants, is Zk (g). Denote the sum of all non-invariant isotypic components by (gk )+ . By definition, gk = (gk )+ + Zk (g) and (gk )+ ∩ Zk (g) = 0. It remains to prove that (gk )+ = [g, gk ]. Indeed, [g, gk ] ⊆ (gk )+ . On the other hand, each non-trivial irreducible g-submodule V ⊂ gk is faithful, i.e., [g, V ] = V (since [g, V ] is always a g-submodule of V ). Therefore, [g, gk ] contains all non-invariant isotypic components, i.e., [g, gk ] ⊂ (gk )+ . The obtained double inclusion implies that (gk )+ = [g, gk ]. The lemma is proved.  ^ Furthermore, using Lemma 4.6 and the definition (3.1) of (g, A)(F ), we obtain X ^ (g, A)(F ) = F · g + Ik (F ) · [g, gk+1 ] + [F , Ik−1(F )] · gk+1 k≥1 =F ·g+ X (Ik (F ) + [F , Ik−1 (F )]) · [g, gk+1 ] + [F , Ik−1(F )] · Zk+1(g) , k≥1 which, after taking into account that [F , Ik−1 (F )] ⊂ Ik (F ) (and shifting the index of summation), becomes the right hand side of (4.7). This finishes the proof of Theorem 4.3(b). Therefore, Theorem 4.3 is proved.  The following is a direct corollary of Theorem 4.3. 26 A. BERENSTEIN and V. RETAKH Corollary 4.7. Assume that a compatible pair (g, A) is pair and F is a unital kalgebra satisfying I1 (F ) = F (i.e., F F ′ = F ). Then (g, A)(F ) = F · g + F · [g, hgi] + [F , F ] · hgi (4.8) (where hgi = P k k≥1 g ). Proof. First, show by induction that Ik (F ) = F for all k ≥ 1. If k = 1, we have nothing to prove. Furthermore, using the inclusion Ik−1(F )F ′ ⊆ Ik (F ) and the inductive hypothesis Ik−1 (F ) = F , we obtain F = F F ′ = Ik−1 (F )F ′ ⊆ Ik (F ), therefore, Ik (F ) = F . This and (3.1) imply that X ^ (g, A)(F ) = F · g + F · [g, gk+1 ] + [F , F ] · gk+1 = F · g + F · [g, hgi] + [F , F ] · hgi . k≥1 This and Theorem 4.3 finish the proof.  Remark 4.8. The condition I1 (F ) = F holds for each noncommutative simple unital algebra F , e.g., for each noncommutative skew-field F containing k. Therefore, for all such algebras and any perfect pair (g, A), the Lie algebra (g, A)(F ) is given by the relatively simple formula (4.8), which also complements (2.6). The following result is a specialization of Theorem 4.3 to the case when g = sl2 (k). Theorem 4.9. Let A an object of Alg1 containing sl2 (k) as a Lie subalgebra. Then X (4.9) (sl2 (k), A)(F ) = F · sl2 (k) + [F · 1, Z1 (A, F )] + Zi (A, F ) · V2i , i≥1 where Zi (A, F ) = X Ii+2j−1 (F ) · ∆j , j≥0 ∆ = 2EF + 2F E + H 2 is the Casimir element, and V2i is the (2i + 1-dimensional) sl2 (k)-submodule of A generated by E i . In particular, if A = hgi is finite dimensional over k, then there exists m ≥ 1 such that E m+1 = 0, ∆ ∈ k · 1, and (4.10) (sl2 (k), A)(F ) = [F , F ] · 1 + m X k=1 Ik−1(F ) · V2k . Noncommutative loops over Lie algebras and Lie groups 27 Proof. Prove (4.9). Clearly, each gk is a finite-dimensional sl2 (k)-module generated by the highest weight vectors ∆j E i , i, j ≥ 0 , i + 2j ≤ k. That is, in notation of (4.3), one has X ∆j · V2i , (gk )+ = i>0,j≥0,i+2j≤k where the sum is direct (but some summands may be zero) and V2i = i X k · (ad F )i+r (E i ) r=−i is the corresponding simple sl2 (k)-module; and X Zk (g) = k · ∆j , 1≤j≤k/2 where the sum is direct. Therefore, taking into account that Ik (F ) ⊂ Ik−1 (F ), the equation (4.3) simplifies to X X (sl2 (k), A)(F ) = F · g + Ii+2j−1 (F ) · ∆j V2i + [F , I2j−2(F )] · ∆j i>0,j≥0 j≥1 = F · sl2 (k) + [F · 1, Z1(A, F )] + X Zi (A, F ) · V2i . i≥1 This finishes the proof of (4.9). Prove (4.10). Indeed, now (g, A) is of finite type, say, m, therefore using (3.3) and the fact that Zk (A, F ) = Ik−1 (F ) (because Ik (F ) ⊂ Ik−1 (F ) and ∆ is a scalar), we obtain from already proved (4.9): X (sl2 (k), A)(F ) = F · sl2 (k) + [F , Z1(A, F )] · 1 + Zi (A, F ) · V2i i≥1 = [F , F ] · 1 + X Ik−1 (F ) · V2k . 1≤k≤m Theorem 4.9 is proved.  5. N -groups Throughout the section we assume that each object of N is a unital k-algebra, i.e., N is a sub-category of Alg1 . 28 A. BERENSTEIN and V. RETAKH 5.1. From N -Lie algebras to N -groups and generalized K1 -theories. In this section will use F -algebras and the category LieAlgN defined in Section 2. Definition 5.1. An affine N -group is a triple (F , G, A), where F is an object of N , A is an F -algebra in Alg1 (i.e., ι : F → A respects the unit), and G is a subgroup of the group of units A× such that G contains the image ι(F × ) = ι(F )× . A morphism (F1 , G1 , A1 ) → (F2 , G2 , A2 ) in of affine N -groups is a pair (ϕ, ψ), where ϕ : F1 → F2 is a morphism in N and ψ : G1 → G2 is a group homomorphism such that ψ ◦ ι1 |F1× = ι2 |F1× ◦ ψ. Denote by GrN the category of affine N -groups. Note that if N has only one object F , then an N -Lie group is simply any object of GrN of the form (F , G, A). Next, we will construct a number of affine N -groups out of a given N -group or a given N -algebra as follows. Let LieAlg1N be the sub-category of LieAlgN whose objects are triples (F , L, A), where F is an object of Alg1 , A is an F -algebra in Alg1 , and L is a Lie subalgebra of A invariant under the adjoint action of ι(F ); morphisms in LieAlg1N are those morphisms in LieAlgN which respect the unit. Given an object (F , L, A) of LieAlg1N , define the triple (F , G, A) = Exp(F , L, A), where G is the subgroup of A× generated by ι(F )× and by the stabilizer {g ∈ A× : gLg −1 = L} of L in A× . Given an object (F , G, A) of GrN , define the triple (F , L′, A) = Lie(F , G, A), where L′ is the Lie subalgebra of A generated (over k) by the set {g · ι(f ) · g −1 : g ∈ G, f ∈ F }, that is, L′ is the smallest Lie subalgebra of A containing ι(F ) and invariant under conjugation by G (therefore, L′ is invariant under the adjoint action of ι(F )). The following result is obvious. Lemma 5.2. (a) The correspondence (F , L, A) 7→ Exp(F , L, A) defines a functor Exp : LieAlg1N → GrN . (b) The correspondence (F , G, A) 7→ Lie(F , G, A) defines a functor Lie : GrN → LieAlg1N . Remark 5.3. The functors Exp and Lie are analogues of the Lie correspondence (between Lie algebras and Lie groups). Noncommutative loops over Lie algebras and Lie groups 29 Composing these two functors with each other and the functors Lij , we obtain a large number of functors between LieAlg1N and GrN . By definition, one has a natural (forgetful) projection functor π : GrN → N by π(F , E, G) = F and π(ϕ, ψ) = ϕ|F1 . A noncommutative loop group (or simply N -group) is any functor G : N → GrN such that π ◦ G = IdN (i.e., G is a section of π). The above arguments allow for constructing a number of N -loop groups out of N -loop Lie algebras and vice versa. We will discuss the N -loop groups associated with compatible pairs (g, A) below in Section 5.2. More general N -loop groups will be considered elsewhere. Furthermore, similarly to Section 2, given an object F of Alg1 and a group G, we refer to a group homomorphism ι : F × → G in Alg as a F -group structure on G (we will also refer to G an F -group). Definition 5.4. A decorated group is a pair (F , G), where F is an object of Alg1 and G is an F -group. We denote by DecGr the category whose objects are decorated groups and morphisms are pairs (ϕ, ψ) : (F1 , G1 ) → (F2 , G2 ), where ϕ : F1 → F2 is a morphism in Alg1 and ψ : G1 → G2 is a group homomorphism such that ψ ◦ ι1 = ι2 ◦ ϕ|F1× . In particular, one has a natural (forgetful) projection functor π : DecGr → Alg1 by π(F , G) = F and π(ϕ, ψ) = ϕ. Note also that for any object (F , G, A) of GrN the pair (F , G) is a decorated group, therefore, the projection (F , G, A) 7→ (F , G) defines a (forgetful) functor GrN → DecGr. Definition 5.5. A generalized K1 -theory is a functor K : N → DecGr such that π ◦ K = π|N . In what follows we will construct a number of generalized K1 -theories as compositions of an N -group s : N → GrN with a certain functors κ from GrN to the category of decorated groups. Given an object (F , G, A) of GrN , we define κcom (F , G, A) := G/[G, G], where [G, G] is the (normal) commutator subgroup of G. Lemma 5.6. The correspondence (F , G, A) 7→ κcom (F , G, A) defines a functor κcom from GrN to the category of decorated abelian groups. In particular, for any N -Lie group G : N → GrN the composition κcom ◦ G is a generalized K1 -theory. 30 A. BERENSTEIN and V. RETAKH Note that each K1 -theory defined by Lemma 5.6 is still commutative. Below we will construct a number of non-commutative nilpotent K1 -theories in a similar manner. Definition 5.7. Let A be an F -algebra. Recall that a subset S of A is called a nilpotent if S n = 0 for some n. In particular, an element e ∈ A is nilpotent if {e} is nilpotent, i.e., en = 0. We denote by Anil the set of all nilpotents in A. Note that Anil is invariant under adjoint action of the group of units A× . We say that an element e ∈ A is an F -stable nilpotent if (F · e · F )n = 0 for some n > 0. Denote by × F× (F , A)nil the set of all F -stable nilpotents of A. Denote also by AF nil and (F , A)nil the centralizers of F × in Anil and (F , A)nil respectively. In particular, taking A = F , we see that f ∈ (F , F )nil if and only if the ideal F f F ⊂ F is nilpotent. Note that if F is commutative and the left F -action on A coincides with the right one, then (F , A)nil = Anil . Let (F , G, A) be an object of GrN and let S be a subset of Anil , denote by ES = ES (F , G, A) the subgroup of G generated by all 1 + xsx−1 , s ∈ S, x ∈ G. Clearly, ES is a normal subgroup of G. Then denote the quotient group G/ES by:   κnil (F , G, A) if S = Anil     κ if S = (F , A)nil stnil (F , G, A) (5.1) ×  κnil,inv (F , G, A) if S = AF  nil    κ F× stnil,inv (F , G, A) if S = (F , A)nil Lemma 5.8. Each of the four correspondences (F , L, A) 7→ κ(F , L, A) , where κ = κnil , κstnil , κnil,inv , κstnil,inv , defines a functor κ : GrN → DecGr . In particular, for any N -Lie group G : N → GrN the composition κ ◦ G is a generalized K1 -theory N → DecGr. Proof. Clearly, in each case of (5.1), the association A 7→ S = SA is functorial, i.e., commutes with morphisms of F -algebras. Therefore, the association A 7→ ES is also functorial in all four cases (5.1). This finishes the proof of the lemma.  We will elaborate examples of generalized (non-commutative) K1 -groups in a separate paper. Noncommutative loops over Lie algebras and Lie groups 31 5.2. N -loop groups for compatible pairs. Here we keep the notation of Section 2 with the additional assumption that A is unital (so that F ⊗ A = F · A is unital for any object F in N ). Note that for the N -Lie algebra s : F 7→ (F , (g, A)(F ), F ·A) the corresponding N group is of the form (F , Gg,A(F ), F · A), where Gg,A (F ) is the normalizer of (g, A)(F ) in (F · A)× (i.e., Gg,A (F ) = {g ∈ (F · A)× : g · (g, A)(F ) · g −1 = (g, A)(F )). The following facts are obvious. Lemma 5.9. Let (g, A) be a compatible pair and S ⊂ g be a generating set. Then for any object F of Alg1 an element g ∈ (F · A)× belongs to Gg,A (F ) if and only if: g(u · x)g −1 ⊂ (sl2 (k), An )(F ) (5.2) for all x ∈ S, u ∈ F . Lemma 5.10. For each compatible pair of the form (g, A) = (sln (k), Mn (k)) and an object F of N1 one has: Gg,A (F ) = GLn (F ) = (F · A)× . In what follows we will consider compatible pairs of the form (g, End(V )) where V is a simple finite-dimensional g-module. By choosing an appropriate basis in V , we identify A = End(V ) with Mn (k) so that Gg,A (F ) ⊂ GLn (F ). In all cases to be considered, we will, in fact, compute the “Cartan subgroup” (F × )n ∩ Gg,A (F ) of Gg,A (F ). Let Φ0 be a bilinear form on the k-vector space V = kn given by: Φ0 (x, y) = x1 yn + x2 yn−1 + · · · + xn y1 . Also define a bilinear form Φ1 on k2m by: Φ1 (x, y) = x1 y2m + x2 y2m−1 + · · · + xm ym − xm+1 ym−1 · · · − x2m y1 . Proposition 5.11. Let F be an object of Alg1 , Let A = Mn (k), and suppose that either g = o(Φ0 ) or g = o(Φ1 ) and n = 2m. Then an invertible diagonal matrix D = diag(f1, ..., fn ) ∈ GLn (F ) belongs to Gg,A (F ) if and only if fi fn−i+1 − f1 fn ∈ I1 (F ) = F [F , F ] for i = 1, . . . , n. Proof. We will prove the theorem for Lie algebras g defined by forms Φ0 . A proof for Φ1 goes in a similar way. It is easy to see that g is generated by all eij := Eij − En−j+1,n−i+1, i, j = 1, . . . , n, i 6= j. Since the set S = {eij ) generates g, 32 A. BERENSTEIN and V. RETAKH Lemma 5.9 guarantees that D = (f1 , . . . , fn ) ∈ (F × )n belongs to Gg,A (F ) if and only if D(u · eij )D −1 ⊂ (g, A)(F ) for all u ∈ F , i, j = 1, . . . , n, i 6= j. Note that −1 + δij (u)Ej ′,i′ , D(u · eij )D −1 = fi ufj−1 Eij − fj ′ ufi−1 ′ Ej ′ ,i′ = fi ufj and i′ = n + 1 − i, j ′ = n + 1 − j. Therefore, taking where δij (u) = fi ufj−1 − fj ′ ufi−1 ′ into account that (g, A)(F ) = [F , F ]·1+F ·g+I1 (F )·sln (k) by Corollary 2.14, we see that D(u·eij )D −1 ∈ (g, A)(F ) for i 6= j if and only if δij (u) ∈ I1 (F )·sln (k). Note that δij (u) ≡ uδij (1) mod I1 (F ). Since I1 (F ) is an ideal, uδij (1) ∈ I1 (F ) for all u ∈ F if mod I1 (F ). Taking into account that and only if δij (1) ∈ I1 (F ), i.e., fi fj−1 − fj ′ fi−1 ′ −1 −1 −1 −1 fi fj − fj ′ fi′ ≡ (fi fi′ − fj fj ′ )fj fi′ mod I1 (F ), we see that D ∈ Gg,A (F ) if and only if fi fn+1−i − fj fn+1−j ∈ I1 (F ) for all i, j. Clearly, it suffices to take j = 1. The proposition is proved.  To formulate the main result of this section we need the following notation. For any ℓ ≥ 0 and any m1 , . . . , mℓ+1 ∈ F denote   ℓ X (ℓ) k ℓ mk+1 (5.3) ∆ (m1 , . . . , mℓ+1 ) = (−1) k k=0 and refer to it as the ℓ-th difference derivative. Clearly, ∆(ℓ) (m1 , . . . , mℓ+1 ) = ∆(ℓ−1) (m1 , . . . , mℓ ) − ∆(ℓ−1) (m2 , . . . , mℓ+1 ) . Let An = Mn (k) = End(Vn−1 ), where Vn−1 is the n-dimensional irreducible sl2 (k)module. Then Gsl2 (k),An (F ) is naturally a subgroup of GLn (F ). Main Theorem 5.12. For any object F of Alg1 , the “Cartan subgroup” (F × )n ∩ Gsl2 (k),An (F ) consists of all D = (f1 , ..., fn ) ∈ (F × )n such that: (5.4) −1 ∆(k) (f1 f2−1 , . . . , fk+1 fk+2 ) ∈ Ik (F ) for k = 1, . . . , n − 2. Proof. We will start with the following characterization of (F × )n ∩ Gsl2 (k),An (F ). Proposition 5.13. A diagonal matrix D = (f1 , . . . , fn ) ∈ (F × )n belongs to the group Gsl2 (k),An (F ) if and only if: (5.5) −1 ) ∈ Ik (F ), ∆(k) (f1 uf2−1 , . . . , fk ufk+1 (5.6) −1 −1 ∆(k) (fn ufn−1 , . . . , fn−k ufn−1−k ) ∈ Ik (F ) . for k = 1, . . . , n − 2 and all u ∈ F . Noncommutative loops over Lie algebras and Lie groups 33 Proof. Denote by (An )k the set of all x ∈ An such that [H, x] = kx. Clearly, (An )k 6= 0 if and only if k even and −2(n − 1) ≤ k ≤ 2(n − 1). In fact, (An )2k is the span of all those Eij such that j − i = k. In particular, E ∈ (An )2 . Denote also (sl2 (k), An )(F )k := (sl2 (k), An )(F ) ∩ F · (An )k . Lemma 5.14. The components (sl2 (k), An )(F )j , j = −2, 2 are given by: (5.7) (sl2 (k), An )(F )2 = n−2 X Ik (F ) · E (k) , (sl2 (k), An )(F )−2 = k=0 where E (k) n−2 X Ik (F ) · F (k) , k=0   n−1  n−1  X X i−1 i−1 (k) En+1−i,n−i i Ei,i+1 , F = i = k k i=k+1 i=k+1 for k = 0, 1, . . . , n − 2 form a basis for (An )2 and (An )−2 respectively. Proof. Let us prove the formula for j = 2. Let p0 (x), . . . pn−2 (x) be any polynomials in k[x] such that deg pk (x) = k for all k. Then it is easy to see that (sl2 (k), An )(F )2 = n−2 X Ik (F ) · (pk (H) · E) . k=0
Pn Take pk (H) = k , where H ′ = 21 (n · 1 − H) = i=1 (i − 1)Eii . Then pk (H) = Pn i−1
Eii and i=1 k !  ! X n n  X i−1 Eii pk (H) · E = iEi,i+1 = E (k) . k i=1 i=1 H′ To prove the formula for j = −2, it suffices to conjugate that for j = 2 with the Pn matrix of the longest permutation w0 = i=1 Ei,n+1−i , i.e., apply the involution Eij 7→ En+1−i,n+1−j .  Lemma 5.15. For each diagonal matrix D = (f1 , f2 , . . . , fn ) ∈ (F × )n and u ∈ F one has n−2 X −1 −1 D(u · E)D = ∆(k) (f1 uf2−1 , . . . , fk ufk+1 ) · (−1)k+1 E (k) , k=0 D(u · F )D −1 = n−2 X −1 −1 ∆(k) (fn ufn−1 , . . . , fn−k ufn−1−k ) · (−1)k+1 F (k) , k=0 (k) where ∆ is the k-th divided difference as in (5.3). 34 A. BERENSTEIN and V. RETAKH Proof. It is easy to see that the elements E (k) , F (k) satisfy: (5.8)     n−2 n−2 X X k k (k) k+1−i k+1−i F (k) E , iEn+1−i,n−i = (−1) iEi,i+1 = (−1) i−1 i−1 k=i−1 k=i−1 for i = 1, . . . , n − 1. Furthermore, D(u · E)D −1 = n−1 X −1 fi fi+1 (−1)i k=0 i=1  −1 fi fi+1 · i=1 i=1 n−2 X n−1 X · iEi,i+1 = n−1 X n−2 X (−1) k+1−i k=0  k E (k) = i−1   n−2 X k −1 k+1 (k) fi fi+1 · (−1) E = ∆(k) (m1 , . . . , mk+1 ) · (−1)k+1 E (k) . i−1 k=0 The formula for D(u · F )D −1 follows. The lemma is proved.  Now we are ready to finish the proof of Proposition 5.13. Indeed, since the set S = {E, F } generates sl2 (k), Lemma 5.9 guarantees that D ∈ GLn (F ) belongs to Gsl2 (k),An (F ) if and only if D(u · E)D −1, D(u · F )D −1 ∈ (sl2 (k), An )(F ) for all u ∈ F . Using the obvious fact that D(u · E)D −1 ⊂ F · (An )2 for all D ∈ (F × )n , u ∈ F , we see that D(u · E)D −1 ∈ (sl2 (k), An )(F ) if and only if D(u · E)D −1 ∈ (sl2 (k), An )(F )2. In turn, using Lemmas 5.14 and 5.15, we see that this is equivalent to n−2 X D(u · E)D −1 = ∆(k) (m1 , . . . , mk+1 ) · (−1)k+1 E (k) ∈ k=0 n−2 X Ik (F ) · E (k) , k=0 (0) (n−2) which, provided that all E , . . . , E are linearly independent, is equivalent to (5.5). Applying the above argument to D(u · F )D −1 , we obtain D(u · E)D −1 = n−2 X (k) ∆ −1 −1 (fn ufn−1 , . . . , fn−k ufn−1−k ) k=0 which gives (5.6). The proposition is proved. · (−1) k+1 F (k) ∈ n−2 X Ik (F ) · F (k) , k=0  To finish the proof of Theorem 5.12, we need to establish some basic properties of inclusions (5.4). Lemma 5.16. Let m1 , m2 , . . . , mℓ be elements of F . Then the following are equivalent: (a) ∆(k) (m1 , . . . , mk+1 ) ∈ Ik (F ) for all 1 ≤ k ≤ ℓ − 2. (b) ∆(j−i) (mi , . . . , mj ) ∈ Ij−i (F ) for all 1 ≤ i ≤ j ≤ ℓ. Noncommutative loops over Lie algebras and Lie groups Proof. Denote (5.9) 35 The implication (b)=>(a) is obvious. Prove the implication (a)=>(b). mij := ∆(j−i) (mi , . . . , mj ) for all 1 ≤ i ≤ j ≤ ℓ. So that mii = mi and: (5.10) mij = mi,j−1 − mi+1,j for all 1 ≤ i ≤ j ≤ ℓ. Prove that the inclusions m1,k+1 ∈ Ik (F ) for 0 ≤ k ≤ ℓ imply inclusions mij ∈ Ij−i (F ) for all i ≤ j such that j − i ≤ ℓ. We will proceed by induction in i. The basis of the induction, when i = 1, is obvious. Assume that i > 1 and j ≤ ℓ − 1 − i. Then the inclusions (the inductive hypothesis) mi−1,j = mi−1,j−1 − mij ∈ Ij+1−i (F ) ⊂ Ij−i (F ) and mi−1,j−1 ∈ Ij−i (F ) imply that mij ∈ Ij−i (F ). This finishes the proof of the implication (a)=>(b). The lemma is proved.  Proposition 5.17. Let m1 , m2 , . . . , mℓ be invertible elements of F . Then the following are equivalent: (a) ∆(j−i) (mi , . . . , mj ) ∈ Ij−i (F ) for all i ≤ j ≤ ℓ. −1 (b) ∆(j−i) (m−1 i , . . . , mj ) ∈ Ij−i (F ) for all i ≤ j ≤ ℓ. Proof. Due to the symmetry, it suffices to prove only one implication, say, (a)=>(b). Indeed, similarly to (5.9), denote (5.11) −1 mij := ∆(j−i) (mi , . . . , mj ), m∗ij := ∆(j−i) (m−1 i , . . . , mj ) . −1 for all 1 ≤ i ≤ j ≤ ℓ. In particular, m∗ii = m−1 and m∗12 = m−1 1 (m1 − m2 )m2 = i −m∗11 m12 m∗22 . We need the following recursive formula for m∗ij . Lemma 5.18. In the above notation, we have for all 1 ≤ i < j ≤ ℓ: X j,j1 ,j2 ,j3 ∗ (5.12) m∗ij = ci,i mi1 ,j1 mi2 ,j2 m∗i3 ,j3 , 1 ,i2 ,i3 i≤i1 ≤j1 ≤j,i≤i2 <j2 ≤j,i≤i3 ≤j3 ≤j j1 −i1 +j2 −i2 +j3 −i3 =j−i where the coefficients are translation-invariant integers: j,j1 ,j2 ,j3 1 +1,j2 +1,j3 +1 cj+1,j i+1,i1 +1,i2 +1,i3 +1 = ci,i1 ,i2 ,i3 . 36 A. BERENSTEIN and V. RETAKH Proof. We will proceed by induction in j − i. If j = i + 1, we obtain: −1 ∗ ∗ ∗ ∗ mi,i+1 = m−1 i (mi − mi+1 )mi+1 = −mii mi,i+1 mi+1,i+1 = −mi+1,i+1 mi,i+1 mi,i . Furthermore, assume that j − i > 1. Then, using the translation invariance of the coefficients in (5.12) for m∗i,j−1 , we obtain X 1 ,j2 ,j3 j1 ,j2 ,j3 cj−1,j δi1 ,i2 ,i3 , m∗ij = m∗i,j−1 − m∗i+1,j = i,i1 ,i2 ,i3 i≤i1 ≤j1 ≤j−1,i≤i2 <j2 ≤j−1,i≤i3 ≤j3 ≤j−1 j1 −i1 +j2 −i2 +j3 −i3 =j−1−i where δij11,i,j22,i,j33 = m∗i1 ,j1 mi2 ,j2 m∗i3 ,j3 − m∗i1 +1,j1+1 mi2 +1,j2 +1 m∗i3 +1,j3 +1 . Furthermore, δij11,i,j22,i,j33 = m∗i1 ,j1+1 mi2 ,j2 m∗i3 ,j3 + m∗i1 +1,j1 +1 (mi2 ,j2 m∗i3 ,j3 − mi2 +1,j2 +1 m∗i3 +1,j3 +1 ) = m∗i1 ,j1 +1 mi2 ,j2 m∗i3 ,j3 + m∗i1 +1,j1 +1 mi2 ,j2 +1 m∗i3 ,j3 + m∗i1 +1,j1 +1 mi2 +1,j2 +1 m∗i3 ,j3+1 . This proves the formula (5.12) for m∗ij . The lemma is proved. The desired implication (a)=>(b) follows inductively from (5.11).   Note that, in the view of Lemma 5.16, the condition (a) (resp. the condition (b)) −1 of Proposition 5.17 for mi = fi fi+1 , i = 1, . . . , n−2, is a particular case of (5.5) (resp. of (5.6)) with u = 1. Therefore, to finish the proof of Theorem 5.12, we need to prove the converse. Since the conditions (a) and (b) of Proposition 5.17 are equivalent, all we have to do is to prove the following result. Theorem 5.19. The inclusions (5.4) imply the inclusions (5.5). Proof. We will start with a formalism of homogeneous maps F → F (relative to the ideals Ik (F ). Definition 5.20. We say that a k-linear map ∂ : F → F is homogeneous of degree ℓ if ∂(Ik (F )) ⊂ Ik+ℓ (F ) for all k ≥ 0; denote by End(ℓ) (F ) the set of all such maps. Lemma 1.4 guarantees that for each f1 ∈ Iℓ1 (F ), f2 ∈ Iℓ2 (F ), the map F → F given by u 7→ f1 uf2 is homogeneous of degree ℓ1 + ℓ2 . We construct a number of homogeneous maps of degree 1 as follows. For an invertible element m ∈ F × define ∂m : F → F by ∂m (u) = mum−1 − u = [m, um−1 ] . Clearly, ∂m : F → F is homogeneous of degree 1. Noncommutative loops over Lie algebras and Lie groups 37 Proposition 5.21. Let m1 , m2 , . . . , mℓ be invertible elements of F such that, in the notation (5.9), one has mij ∈ Ij−i (F ) for all 1 ≤ i ≤ j ≤ ℓ. Then ∆(j−i) (∂mi , . . . , ∂mj ) ∈ End(j+1−i) (F ) for all 1 ≤ i ≤ j ≤ ℓ. Proof. Similarly to (5.9), denote (5.13) ∂ij = ∆(j−i) (∂mi , . . . , ∂mj ) for 1 ≤ i ≤ j ≤ ℓ. By definition, ∂ii = ∂mi and ∂i,i+1 = ∂mi − ∂mi+1 . Lemma 5.22. For each u ∈ F and 1 ≤ i ≤ j ≤ ℓ one has: X ∗ 1 ,j2 cj,j (5.14) ∂ij (u) = i,i1 ,i2 [mi1 ,j1 , umi2 ,j2 ] i≤i1 ≤j1 ≤j,i≤i2 ≤j2 ≤j j1 −i1 +j2 −i2 =j−i in the notation (5.11), where the coefficients are translation-invariant integers: j+1,j1 +1,j2 +1 1 ,j2 ci+1,i = cj,j i,i1 ,i2 . 1 +1,i2 +1 Proof. We will proceed by induction in j − i. If j = i, we have ∂ii (u) = ∂mi (u) = [mi , um−1 i ]. −1 ∗ ∗ ∗ ∗ mi,i+1 = −m−1 i (mi − mi+1 )mi+1 = −mii mi,i+1 mi+1,i+1 = −mi+1,i+1 mi,i+1 mi,i . Furthermore, assume that j − i > 0. Then, using the translation-invariance of the coefficients in (5.14) for ∂i,j−1 (u), we obtain X j−1,j1 ,j2 j1 ,j2 ci,i δi1 ,i2 , ∂ij (u) = ∂i,j−1 (u) − ∂i+1,j (u) = 1 ,i2 i≤i1 ≤j1 ≤j−1,i≤i2 <j2 ≤j−1 j1 −i1 +j2 −i2 =j−1−i where δij11,i,j22 = [mi1 ,j1 , um∗i2 ,j2 ] − [mi1 +1,j1 +1 , um∗i2 +1,j2 +1 ]. Furthermore, δij11,i,j22 = [mi1 ,j1 +1 , um∗i2 ,j2 ] + [mi1 +1,j1 +1 , um∗i2 ,j2 ] − [mi1 +1,j1+1 , um∗i2 +1,j2 +1 ] = [mi1 ,j1 +1 , um∗i2 ,j2 ] + [mi1 +1,j1 +1 , um∗i2 +1,j2 +1 ] . This proves the formula (5.14) for ∂ij (u). The lemma is proved.  Since mi1 ,j1 ∈ Ij1 −i1 (F ), m∗i2 ,j2 ∈ Ij2 −i2 (F ), and [mi1 ,j1 , um∗i2 ,j2 ] belongs to the ideal Ik+j1−i1 +j2 −i2 +1 (F ) for all u ∈ Ik (F ), the formula (5.14) guarantees the inclusion ∂ij (Ik (F )) ⊂ Ik+j+1−i (F ) for all k ≥ 0. This proves Proposition 5.21.  38 A. BERENSTEIN and V. RETAKH Let mi , di : F → F , i = 1, . . . , ℓ be linear maps. Denote (5.15) ∂ (i,j) := ∆(j−i) (mi di+1 · · · dj , mi+1 di+2 · · · dj , . . . , mj−1 dj , mj ) for all 1 ≤ i ≤ j ≤ ℓ. For instance, ∂ (i,i) = mi , ∂ (i,i+1) = mi di+1 − mi+1 , and ∂ (i,i+2) = mi di+1 di+2 − 2mi+1 di+2 + mi+2 . −1 Lemma 5.23. Let D = (f1 , . . . , fn ) ∈ (F × )n and denote mi = fi fi+1 . Then for each u ∈ F one has (5.16) −1 −1 ∆(j−i) (fi ufi+1 , . . . , fj ufj+1 ) = ∂ (i,j) (u′ ) , where we abbreviated u′ = fj ufj−1 and di := ∂mi + 1. Proof. Indeed, for i ≤ k ≤ j one has −1 −1 fk ufk+1 = mk mk+1 · · · mj fj+1 ufj+1 (mk mk+1 · · · mj )−1 = mk (∂mk+1 ···mj + 1)(u′) = mk (∂mk + 1) · · · (∂mj + 1)(u′) = mk dk+1 · · · dj (u′) . −1 Substituting so computed fk ufk+1 into (5.3), we obtain (5.16).  Therefore, all we need to finish the proof of Theorem 5.19 is to prove the following result. Proposition 5.24. Let m1 , m2 , . . . , mℓ be elements of F and ℓ ≥ 0 such that, in the notation (5.9), one has mij ∈ Ij−i (F ) for all 1 ≤ i ≤ j ≤ ℓ. Then ∂ (i,j) ∈ End(j−i) (F ) for all 1 ≤ i ≤ j ≤ ℓ, where again we abbreviated ∂i := ∂mi . Proof. It suffices to prove the following result. Lemma 5.25. In the notation (5.15) one has X (5.17) ∂ (i,j) = cI MI , I where the summation is over all subsets I = {i0 < i1 < i2 · · · < ik } of {1, . . . , ℓ} such that i0 = i, ik = j, MI is a linear map F → F given by: MI = mi0 ,i1 ∂i1 +1,i2 ∂i2 +1,i3 · · · ∂ik−1 +1,ik , Noncommutative loops over Lie algebras and Lie groups 39 the coefficients cI ∈ Z are translation-invariants: cI+1 = cI , where for any subset I = {i0 , . . . , ik } ⊂ {1, . . . , ℓ}, we have abbreviated I + 1 = {i0 + 1, . . . , ik + 1} ⊂ {2, . . . , ℓ + 1}; and:  d ′ − 1 if i′ = j ′ ′ ′ i mi′ ,j ′ := ∆(j −i ) (mi′ , . . . , mj ′ ), ∂i′ ,j ′ = ∆(j ′ −i′ ) (di′ , . . . , dj ′ ) if i′ < j ′ for all 1 ≤ i′ ≤ j ′ ≤ ℓ; Proof. We will proceed by induction in j − i. The basis of the induction when j = i is obvious because ∂ (i,i) = MI = mi for all i, where I = {i}. Note that ∂ (i,j) = ∂ (i,j−1) dj − ∂ (i+1,j) = ∂ (i,j−1) ∂j,j + (∂ (i,j−1) − ∂ (i+1,j) ) for all i, j. Therefore, we have by the inductive hypothesis and the translationinvariance of the coefficients cI : X X ∂ (i,j) = cI MI⊔{j} + cI (MI − MI+1 ) , I I where the summations are over all subsets I = {i0 < i1 < · · · < ik } of {1, . . . , ℓ} such that i0 = i, ik = j − 1 (we have used the fact that MI ∂j,j = MI⊔{j} ). It is easy to see that for any I = {i0 < i1 < · · · < ik } one has MI − MI+1 = MI1 + MI2 · · · + MIk , where Ij = {i0 < i1 < i2 · · · < ij1 < ij + 1 < · · · < ik + 1} for j = 1, 2, . . . , k. Therefore, X X ∂ (i,j) = cI MI⊔{j} + cI MIj , I i.e., ∂ (i,j) I,j is of the form (5.17). The lemma is proved.  Now we can finish the proof of Proposition 5.21. Recall from (5.4) that mij = ∆ (mi , . . . , mj ) ∈ Ij−i (F ) (hence the map u 7→ mij u belongs to End(j−i) (F )) and from Proposition 5.21 that ∂ij = ∆(j−i) (∂mi , . . . , ∂mj ) ∈ End(j+1−i) (F ) for all 1 ≤ i ≤ j ≤ ℓ. This implies that for I = {i0 < i1 < i2 < · · · < ik } one has: (j−i) MI ∈ End(i1 −i0 ) (F ) ◦ End(i2 −i1 ) (F ) ◦ · · · ◦ End(ik −ik−1 ) (F ) ⊂ End(ik −i0 ) (F ) . Therefore, Lemma 5.25 guarantees that ∂ (i,j) ∈ End(j−i) (F ) for all 1 ≤ i ≤ j ≤ ℓ. Proposition 5.21 is proved.  40 A. BERENSTEIN and V. RETAKH Finally, note that Theorem 5.21 in conjunction with Lemma 5.16 and Proposition 5.17 guarantees that the inclusions (5.4) also imply the inclusions (5.6). This argument and Proposition 5.13 finish the proof of Theorem 5.19.  Therefore, Theorem 5.12 is proved.  We will finish the section with a natural (yet conjectural) generalization of Theorem 5.12. Conjecture 5.26. Let g = sl2 (k) and A = An be as in Theorem 5.12, and F be an object of Alg1 . Then a matrix g ∈ GLn (F ) belongs to Gg,A (F ) if and only if (5.18) g · g · g −1 ⊂ (g, A)(F ) . Remark 5.27. More generally, we would expects that for any perfect pair (g, A) an element g ∈ (F · A)× belongs to Gg,A (F ) if and only if (5.18) holds. References [1] E. Artin, Geometric algebra, Interscience Publishers, Inc., New York-London, 1957. [2] A. Berenstein, V Retakh, Noncommutative double Bruhat cells and their factorizations, Int. Math. Res. Notes (2005), no. 8, 477–516. [3] A. Berenstein, V Retakh, Noncommutative loops over Lie algebras, MPIM Preprint (2006), no. 131. [4] J. Cuntz, D. Quillen, Algebra extensions and nonsingularity, J. Amer. Math. Soc. 8 (1995), no. 2, 251–289. [5] J. Dieudonné, La géométrie des groupes classiques, 3rd edition, Springer-Verlag, Berlin Heidelberg New York, 1971. [6] B. Feigin, B. Shoikhet, On [A, A]/[A, [A, A]] and on a Wn -action on the consecutive commutators of free associative algebra, preprint math.QA/0610410. [7] A. J. Hahn, O. T. O’Meara, The classical groups and K-theory, Springer-Verlag, Berlin, 1989. [8] M. Kapranov, Noncommutative geometry based on commutator expansions, J. Reine Angew. Math. 505 (1998), 73–118. [9] M. Kontsevich, A. Rosenberg, Noncommutative smooth spaces, The Gelfand Mathematical Seminars 1996–1999, 85–108, Birkhuser Boston, Boston, MA, 2000. Department of Mathematics, University of Oregon, Eugene, OR 97403 E-mail address: arkadiy@math.uoregon.edu Department of Mathematics, Rutgers University, Piscataway, NJ 08854 E-mail address: vretakh@math.rutgers.edu