Remarks on the Second Homology Groups of Queer Lie Superalgebras
aa r X i v : . [ m a t h . R A ] F e b Remarks on the Second Homology Groups of Queer LieSuperalgebras
Yongjie Wang and Zhihua Chang ∗ February 3, 2021
Abstract
The aim of this note is to completely determine the second homology group of the specialqueer Lie superalgebra sq n ( R ) coordinatized by a unital associative superalgebra R , whichwill be achieved via an isomorphism between the special linear Lie superalgebra sl n ( R ⊗ Q )and the special queer Lie superalgebra sq n ( R ). MSC(2020):
Keywords: queer Lie superalgebra, root-graded Lie superalgebra, cyclic homology group, centralextension.
The universal central extension of a Lie (super)algebra plays an important role in Lie theory. Onone hand, an affine Kac-Moody Lie algebra over the field C can be realized as the universal centralof a twisted loop algebra based on a finite-dimensional Lie algebra (extended by a derivation). Therealizations of extended affine Lie algebras also involve the universal central extension of an infinite-dimensional Lie algebra such as a twisted multi-loop algebra or a matrix Lie algebra coordinatizedby quantum tori. On the other hand, an explicit description of the kernel of the universal centralextension provides us with the second homology group of this Lie (super)algebra with coefficientsin the trivial module [15]. From this point of view, C. Kassel and J. L. Loday studied in [11] theuniversal central extension of sl n ( A ) with A a unital associative algebra and established an elegantisomorphism between the second homology group of sl n ( A ) for n > ( A ). This establishes a connection between the Lie theoretical object sl n ( A ) and theK-theoretical object HC ( A ) in non-commutative geometry. This phenomenon also appears inunitary Lie algebras [6], where an isomorphism between the second homology group of a unitaryLie algebra and the first skew-dihedral homology group of the corresponding associative algebrawas established.Inspired by these works, the universal central extension of Lie superalgebras have been studiedextensively in recent decades. For a Lie superalgebra g of classical type over a field of characteristiczero, K. Iohara and Y. Koga determined in [8, 9] the second homology group of g ⊗ R with R asuper-commutative unital associative superalgebra. In the case where g is a Lie superalgebra with anon-degenerate invariant bilinear form and R a super-commutative unital associative superalgebra,K. H. Neeb and M. Yousofzadeh in [14] also obtained the universal central extension of g ⊗ R byexplicitly creating the corresponding 2-cocycle with K¨ahler differentials. ∗ Corresponding Author: Zhihua Chang, Email: [email protected] sl n ( R ) , n > sl m | n ( R ), m + n > Z / Z -graded version of the first dihedral and skew-dihedral homology group.This short note aims to determine the second homology group of a special queer Lie superalge-bra. It has been shown in [12] that every Q ( n − st n ( S ) for n >
4, where S is a unital associative superalgebrasuch that its odd part contains an element ν with ν = 1. We will show in Section 2 that such aLie superalgebra can be characterized by the special queer Lie superalgebra sq n ( R ) coordinatizedby an associative algebra R . Moreover, we observe that the special queer Lie superalgebra sq n ( R )is indeed isomorphic to the special linear Lie superalgebra sl n ( R ⊗ Q ), where Q is the Cliffordsuperalgebra of rank one (see (2.1) below).If R is a super-commuative superalgebra over k , where k is a commutative base ring with2 invertible, then sq n ( R ) is isomorphic to sq n ( k ) ⊗ R . Their universal central extensions havebeen determined in [9], [13] and [14] by using different methods. We deal with the general casewhere R is an arbitrary unital associative superalgebra. The isomorphism between sq n ( R ) and sl n ( R ⊗ Q ) allows us to identify the second homology group of sq n ( R ) with the first Z / Z -gradedcyclic homology group HC ( R ⊗ Q ). By further identifying HC ( R ⊗ Q ) with HC ( R ) (up to aparity change), we obtain the second homology group of sq n ( R ). We remark that one could alsointroduce the Steinberg queer Lie superalgebra and follow the same lines as C. Kassel and J. L.Loday’s argument to obtain this result, but it would be tedious.The remainder of this note is divided into three sections. In Section 2, we will define the queerLie superaglebra q n ( R ), the special queer Lie superalgebra sq n ( R ), and prove the isomorphismbetween sq n ( R ) and sl n ( R ⊗ Q ) for an arbitrary unital associative superalgebra R . Section 3 isdevoted to identifying HC ( R ⊗ Q ) with HC ( R ) up to a parity change. The second homologygroup of sq n ( R ) will be discussed in Section 4.Throughout this note, we always assume that k is a unital commutative associative base ringwith 2 invertible. All modules, associative (super)algebras and Lie (super)algebras are over k .We write Z / Z = { ¯0 , ¯1 } and the parity of an element x is denoted by | x | . If both A and B areassociative superalgebras, then A ⊗ B is understood as the associative superalgebra with gradedmultiplication( a ⊗ b )( a ⊗ b ) = ( − | a || b | a a ⊗ b b , for homogeneous elements a ∈ A, b ∈ B. It is shown in [12] that every Q ( n − st n +1 ( S ), where S is a unital associative superalgebra such that thereexists ν ∈ S ¯1 satisfying ν = 1. In fact, such an associative superalgebra is isomorphic to R ⊗ Q ,where R is a (non-super) associative algebra and Q is the associative superalgebra Q = (cid:26)(cid:18) a bb a (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) a, b ∈ k (cid:27) . (2.1)The associative superalgebra Q has one dimensional even part spanned by 1 = (cid:18) (cid:19) and onedimensional odd part spanned by ν = (cid:18) (cid:19) .Let R be a unital associative superalgebra over k and M m | n ( R ) be the associative superalgebraof all ( m + n ) × ( m + n )-matrices with entries in R , in which the parity of a matrix unit e ij ( a )2with a ∈ R at the ( i, j )-position and 0 elsewhere) is | e ij ( a ) | := | i | + | j | + | a | , a ∈ R, i, j m + n, and | i | = ( , if i m, , if i > m. The associative superalgebra M m | n ( R ) is naturally regraded as a Lie superalgebra under thestandard super-commutator. We denote this Lie superalgebra by gl m | n ( R ). Define the queer Liesuperalgebra coordinatized by R to be the Lie sub-superalgebra q n ( R ) = (cid:26)(cid:18) A Bρ ( B ) ρ ( A ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) A, B ∈ M n ( R ) (cid:27) ⊆ gl n | n ( R ) , where ρ ( A ) = ( ρ ( a ij )) n × n if A = ( a ij ) n × n and ρ ( a ) = ( − | a | ρ ( a ) for homogeneous element a ∈ R .The queer Lie superalgebra q n ( R ) is not perfect even when R = k is the base ring. Instead, wecall its derived sub-superalgebra the special queer Lie superalgebra : sq n ( R ) =: [ q n ( R ) , q n ( R )] . (2.2)If R is a non-super unital associative algebra, we will see that sq n ( R ) is indeed a Q ( n − st n ( S ) for some S . On theother hand, it is known from [3] that the Steinberg Lie superalgebra st n ( R ⊗ Q ) is a centralextension of sl n ( R ⊗ Q ) for n >
3. In fact, we can prove the following proposition which connects sq n ( R ) and sl n ( R ⊗ Q ). Proposition 2.1.
Let R be an arbitrary unital associative superalgebra. Then there is an isomor-phism of Lie superalgebras q n ( R ) ∼ = gl n ( R ⊗ Q ) . Consequently, sq n ( R ) ∼ = sl n ( R ⊗ Q ) as Lie superalgebras.Proof. We use e ij ( a ) to denote the n × n -matrix with a ∈ R at the ( i, j )-position and 0 elsewhere.Then q n ( R ) is spanned by u ij ( a ) = (cid:18) e ij ( a ) 00 ( − | a | e ij ( a ) (cid:19) , w ij ( a ) = (cid:18) e ij ( a )( − | a | e ij ( a ) 0 (cid:19) , for i, j = 1 , . . . , n and a ∈ R is homogeneous . They satisfy[ u ij ( a ) , u kl ( b )] = δ jk u il ( ab ) − ( − | a || b | δ il u kj ( ba ) , [ u ij ( a ) , w kl ( b )] = δ jk w il ( ab ) − ( − | a || b | δ il w kj ( ba ) , [ w ij ( a ) , w kl ( b )] = ( − | b | (cid:16) δ jk u il ( ab ) + ( − | a || b | δ il u kj ( ba ) (cid:17) . Hence, the k -linear map q n ( R ) → gl n ( R ⊗ Q ) , u ij ( a ) e ij ( a ⊗ , w ij ( a ) e ij ( a ⊗ ν ) , gives the desired isomorphism of Lie superalgebras. Corollary 2.2.
The Lie superalgebra sq n ( R ) is perfect for n > and it can be described as sq n ( R ) = (cid:26)(cid:18) A Bρ ( B ) ρ ( A ) (cid:19) | Tr( B ) ∈ [ R, R ] (cid:27) . (2.3)3 roof. Since sq n ( R ) ∼ = sl n ( R ⊗ Q ), the perfectness of sq n ( R ) follows from the perfectness of sl n ( R ⊗ Q ).Now, the Lie superalgebra sl n ( R ⊗ Q ) can be characterized as sl n ( R ⊗ Q ) = { X ∈ gl n ( R ⊗ Q ) | Tr( X ) ∈ [ R ⊗ Q , R ⊗ Q ] } Note that [1 ⊗ ν, ⊗ ν ] = 2(1 ⊗ , and [ a ⊗ , b ⊗ ν ] = [ a, b ] ⊗ ν. Since 2 is invertible, we deduce that [ R ⊗ Q , R ⊗ Q ] = ( R ⊗ ⊕ ([ R, R ] ⊗ ν ). Hence, for anelement X = P i,j e ij ( a ij ⊗ b ij ⊗ ν ) ∈ gl n ( R ⊗ Q ),Tr( X ) = X i ( a ii ⊗ b ii ⊗ ν ) ∈ [ R ⊗ Q , R ⊗ Q ]if and only if P i b ii ∈ [ R, R ]. Hence, the preimage of X in sq n ( R ) is the matrix (cid:18) A Bρ ( B ) ρ ( A ) (cid:19) where A = ( a ij ), B = ( b ij ) such that Tr( B ) ∈ [ R, R ]. Remark . (i) The Lie superalgebra sq ( R ) is not necessarily perfect. For instance, if R issuper-commutative, then q ( R ) := (cid:26)(cid:18) a bρ ( b ) ρ ( a ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) a, b ∈ R (cid:27) , and sq ( R ) = (cid:26)(cid:18) a ρ ( a ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) a ∈ R (cid:27) . The Lie superalgebra sq ( R ) is not perfect since [ sq ( R ) , sq ( R )] = 0.(ii) In [5], the author determined the second homology group of Lie algebra sl ( S ) whether S is an associative algebra or not. More concretely, the homology group H ( sl ( S ) , k ) is thegeneralization of cyclic homology group HC ( S ) denoted by hC ( S ) (more details could befound in [5, Theorem 2]) when S is an associtive algebra. For nonassociative algebra S ,the vector space sl ( S ) is a Lie algebra if and only if S satisfies certain conditions, that is sl -admissible. In this case, the homology group was also computed (see [5, Proposition 4]).Analogue to sl case, the definition of sq ( R ) for a (non)associative superalgebra R and itssecond homology group should be considered separately.(iii) As C. Martinez and E. I. Zelmanov pointed out in [12], a Q (2)-graded Lie superalgebrais centrally isogenous to st ( R ), where R is a unital alternative superalgebra that is notnecessarily associative. But one can check that the isomorphism in Proposition 2.1 is stillvalid if R is a unital alternative superalgebra.When R is specified to be a concrete unital associative superalgebra, we can find many wellacquainted examples in these queer Lie superalgebras. Example 2.4.
Let R be a unital super-commutative associative superalgebra, then q n ( R ) ∼ = q n ( k ) ⊗ k R, (2.4) where the bracket of q n ( k ) ⊗ k R is given by [ x ⊗ a, y ⊗ b ] = ( − | a || y | [ x, y ] ⊗ ab, x, y ∈ q n ( k ) , a, b ∈ R. In particular, this gives the untwisted loop queer Lie superalgebra when R is the algebra of Laurentpolynomials. Example 2.5.
Suppose that k contains √− . Let R be a unital associative superalgebra, thenthere is an isomorphism of Lie superalgebras q n ( R ⊗ Q ) ∼ = gl n | n ( R ) , iven by u ij ( r ⊗ e ij ( r ) + ( − | r | e n + i,n + j ( r ) , u ij ( r ⊗ ν ) e i,n + j ( r ) + ( − | r | e n + i,j ( r ) ,w ij ( r ⊗
7→ √− − e i,n + j ( r ) + ( − | r | e n + i,j ( r )) , w ij ( r ⊗ ν )
7→ √− e i,j ( r ) − ( − | r | e n + i,n + j ( r )) , where u ij , w ij have the same meaning as in Corollary 2.2. Consequently, sq n ( R ⊗ Q ) ∼ = sl n | n ( R ) as Lie superalgebras. Z / Z -graded cyclic homology We have seen from Section 2 that the queer Lie superalgebra sq n ( R ) is perfect for n > sl n ( R ⊗ Q ). It has been demonstrated in [3] that, if n >
5, the universal centralextension of sl n ( R ⊗ Q ) can be described by the so-called Steinberg Lie superalgebra st n ( R ⊗ Q ),and the kernel of the canonical homomorphism π : st n ( R ⊗ Q ) → sl n ( R ⊗ Q )is isomorphic to the first Z / Z -graded cyclic homology group HC ( R ⊗ Q ). We further identifyit with the first cyclic homology group HC ( R ) up to a parity change.We first recall the definition of HC ( R ) (we refer to [10] for more details about the Z / Z -gradedcyclic homology). Let I R be the k -submodule of R ⊗ R generated by a ⊗ b + ( − | a || b | b ⊗ a, and ( − | a || c | ab ⊗ c + ( − | b || a | bc ⊗ a + ( − | c || b | ca ⊗ b for homogeneous a, b, c ∈ R and h R, R i := ( R ⊗ R ) /I R . We denote by λ ( a, b ) the canonical imageof a ⊗ b in h R, R i . ThenHC ( R ) := (X i λ ( a i , b i ) ∈ h R, R i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i [ a i , b i ] = 0 ) . In order to distinguish HC ( R ) and HC ( R ⊗ Q ), we denote by h ( x, y ) the canonical image of x ⊗ y in h R ⊗ Q , R ⊗ Q i for x, y ∈ R ⊗ Q . Lemma 3.1.
The following relations hold in h R ⊗ Q , R ⊗ Q i : h ( a ⊗ , b ⊗ ν ) = − ( − | a || b | h ( b ⊗ , a ⊗ ν ) , for a, b ∈ R, (3.1) h ( a ⊗ , b ⊗
1) = h ( a ⊗ ν, b ⊗ ν ) = 0 , for a ∈ R ¯1 or b ∈ R ¯1 , (3.2) h ( a ⊗ , b ⊗
1) = 12 h ([ a, b ] ⊗ ν, ⊗ ν ) , for a, b ∈ R ¯0 , (3.3) h ( a ⊗ ν, b ⊗ ν ) = 12 h ( { a, b } ⊗ ν, ⊗ ν ) , for a, b ∈ R ¯0 , (3.4) where { , } is possion bracket, i.e., { a, b } = ab + ba. Proof.
It follows from the definition of h R ⊗ Q , R ⊗ Q i that h ( x, y ) satisfies h ( x, y ) + ( − | x || y | h ( y, x ) = 0 , (3.5)( − | x || z | h ( xy, z ) + ( − | y || x | h ( yz, x ) + ( − | z || y | h ( zx, y ) = 0 . (3.6)By setting y = z = 1 ⊗ h ( x, ⊗
1) = h (1 ⊗ , x ) = 0for all x ∈ R ⊗ Q . 5n order to show (3.1), we set x = a ⊗ ν, y = b ⊗ ν, z = 1 ⊗ ν in (3.6). Then h ( ab ⊗ , ⊗ ν ) + ( − | a || b | h ( b ⊗ , a ⊗ ν ) + h ( a ⊗ , b ⊗ ν ) = 0 . from which we observe that h ( a ⊗ , ⊗ ν ) = 0 for all a ∈ R . Hence, (3.1) follows.Similarly, set x = a ⊗ ν, y = b ⊗ ν, z = c ⊗ − | a || c | + | a | h ( ab ⊗ , c ⊗ − ( − | b || a | h ( bc ⊗ ν, a ⊗ ν ) + ( − | b || c | + | b | h ( ca ⊗ ν, b ⊗ ν ) = 0 . (3.7)Let c = 1 in (3.7), we have0 − ( − | a || b | h ( b ⊗ ν, a ⊗ ν ) + ( − | b | h ( a ⊗ ν, b ⊗ ν ) = 0 . Then it follows from (3.5) that (cid:16) − ( − | a | (cid:17) h ( a ⊗ ν, b ⊗ ν ) = 0 = (1 − ( − | a | ) h ( b ⊗ ν, a ⊗ ν ) , (3.8)which ensures that h ( a ⊗ ν, b ⊗ ν ) = 0 if a or b is odd.Considering the case of a = 1 and the case of b = 1 in (3.7) respectively, we deduce that h ( b ⊗ , c ⊗ − h ( bc ⊗ ν, ⊗ ν ) + ( − | b || c | + | b | h ( c ⊗ ν, b ⊗ ν ) = 0 , ( − | a || c | + | a | h ( a ⊗ , c ⊗ − h ( c ⊗ ν, a ⊗ ν ) + h ( ca ⊗ ν, ⊗ ν ) = 0 . Equivalently, for all a, b ∈ R ,( − | b | h ( a ⊗ , b ⊗
1) + h ( a ⊗ ν, b ⊗ ν ) − h ( ab ⊗ ν, ⊗ ν ) = 0 , (3.9) h ( a ⊗ , b ⊗
1) + ( − | b | h ( a ⊗ ν, b ⊗ ν ) − h ( ab ⊗ ν, ⊗ ν ) = 0 . (3.10)These imply that (cid:16) − ( − | b | (cid:17) h ( a ⊗ , b ⊗
1) = − (cid:16) − ( − | b | (cid:17) h ( a ⊗ ν, b ⊗ ν ) = 0 , and hence h ( a ⊗ , b ⊗
1) = 0 if a or b is odd.For (3.3) and (3.4), we assume that a, b are both even and exchanging a and b in (3.9), then h ( b ⊗ , a ⊗
1) + h ( b ⊗ ν, a ⊗ ν ) − h ( ba ⊗ ν, ⊗ ν ) = 0 . (3.11)Then (3.5), (3.9) and (3.11) yield (3.3) and (3.4). Theorem 3.2.
Let R be an arbitrary unital associative superalgebra over k . Then HC ( R ⊗ Q ) ∼ = HC ( R ) ⊗ k | . Proof.
The Z / Z -graded vector space k | has a zero even part and a one-dimensional odd part.Hence, HC ( R ) ⊗ k | is the Z / Z -graded space obtained by exchanging the even and odd parts ofHC ( R ). It suffices to show that there is an odd isomorphism between HC ( R ⊗ Q ) and HC ( R ).Note that every element in R ⊗ Q can be written as a ⊗ b ⊗ ν for a, b ∈ R . By Lemma 3.1,every element in h R ⊗ Q , R ⊗ Q i can be written as z = X j h ( c j ⊗ ν, ⊗ ν ) + X i h ( a i ⊗ , b i ⊗ ν )where c j ∈ R ¯0 , a i , b i ∈ R , and both summations run over some finite sets. Such an element z liesin HC ( R ⊗ Q ) if and only if0 = X j [ c j ⊗ ν, ⊗ ν ] + X i [ a i ⊗ , b i ⊗ ν ] = 2 X j c j ⊗ X i [ a i , b i ] ⊗ ν, P j c j = 0 and P i [ a i , b i ] = 0. Hence,HC ( R ⊗ Q ) = X j h ( a j ⊗ , b j ⊗ ν ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a j , b j ∈ R such that X j [ a j , b j ] = 0 . Now, we can define two odd k -linear maps: ϕ : HC ( R ⊗ Q ) → HC ( R ) , X i h ( a i ⊗ , b i ⊗ ν ) X i λ ( a i , b i ) ,ψ : HC ( R ) → HC ( R ⊗ Q ) , X i λ ( a i , b i ) X i h ( a i ⊗ , b i ⊗ ν ) . Lemma 3.1 imply that both ϕ and ψ are well-defined and they are inverse to each other. Hence, ϕ isan odd isomorphism. In other word, HC ( R ⊗ Q ) ∼ = HC ( R ) ⊗ k | as Z / Z -graded k -modules. Remark . If R = k , R ⊗ Q ∼ = Q is the Clifford superalgebra associated to the quadratic form q ( x ) = x . Its Z / Z -graded cyclic homology HC ∗ ( Q ) has been fully understood in [10]. For aunital associative superalgebra R , the higher degree Z / Z -graded cyclic homology HC n ( R ⊗ Q )with n > The second homology group of a Lie superalgebra is identified with the kernel of its universalcentral extension. Based on the interpretation of HC ( R ⊗ Q ) in Section 3, we obtain the secondhomology group of the queer Lie superalgebra sq n ( R ). Theorem 4.1.
Let R be a unital associative superalgebra and n > . Then the second homologyof the queer Lie superalgebra sq n ( R ) is H ( sq n ( R )) ∼ = HC ( R ) ⊗ k | . Proof.
Note that sq n ( R ) ∼ = sl n ( R ⊗ Q ), the proof can be done by applying the results obtainedin [3]. In fact, it is shown in [3] thatH ( sl n ( R ⊗ Q )) = HC ( R ⊗ Q ) , n > , HC ( R ⊗ Q ) ⊕ ( A ) ⊕ , n = 4 , HC ( R ⊗ Q ) ⊕ ( A ) ⊕ , n = 3 , (4.1)where A n = ( R ⊗ Q ) / I n and I is the two-sided ideal of R ⊗ Q generated by n ( R ⊗ Q ) and[ R ⊗ Q , R ⊗ Q ].Now, 2 is invertible in k , so [1 ⊗ ν, ⊗ ν ] = 2(1 ⊗ ∈ [ R ⊗ Q , R ⊗ Q ] is a unit in R ⊗ Q .i.e., A = A = 0. By Theorem 3.2,H ( sq n ( R )) ∼ = H ( sl n ( R ⊗ Q )) ∼ = HC ( R ⊗ Q ) ∼ = HC ( R ) ⊗ k | . Remark . The Steinberg Lie superalgebra st ( R ) and st ( R ) might fail to be centrally closedfor some unital associative superalgebra R . Nonetheless, st ( R ⊗ Q ) and st ( R ⊗ Q ) are alwayscentrally closed for arbitrary unital associative superalgebra R .Now, we apply the above results to a couple of examples.In the case where R is super commutative. sq n ( R ) ∼ = sq n ( k ) ⊗ R and HC ( R ) = Ω ( R ) / dΩ ( R )is the module of K¨ahler differentials modulo exact ones. Hence,H ( sq n ( k ) ⊗ R ) = Ω ( R ) / d( R ) ⊗ k | , for n > . sq n ( k ) for n > k I n | n , where I n | n is the identity matrix. The quotient psq n ( k ) = sq n ( k ) / k I n | n is then a simple Lie superalgebra if k is an algebraic closed field of characteristic zeroand n >
3. Similarly, if R is super-commutative, the scalar matrices with entries in R is centralin sq n ( R ) and psq n ( k ) ⊗ R = sq n ( R ) / ( RI n | n ). For n > ( psq n ( k ) ⊗ R ) = R ⊕ (HC ( R ) ⊗ k | ) , which is the result given in [9].Finally, we suppose that k contains √− R = S ⊗ Q , where S is an arbitraryunital associative superalgebra. We have already known from Example 2.5 that sq n ( R ) ∼ = sl n | n ( S ).Hence, we conclude thatH ( sl n | n ( S )) ∼ = H ( sq n ( S ⊗ Q )) ∼ = HC ( S ⊗ Q ) ⊗ k | ∼ = HC ( S ) ⊗ k | ⊗ k | ∼ = HC ( S ) , for n >
3. This is part of the result obtained in [4] and [7].
Acknowledgments
Zhihua Chang was supported by National Natural Science Foundation of China (No. 11771455and 12071150) and Guangdong Basic and Applied Basic Research Foundation 2020A1515011417.Yongjie Wang also thanks the support of National Natural Science Foundation of China (No.11901146 and 12071026).
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