Central Extensions of Filiform Zinbiel Algebras

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Lie a of structure the given be y soitv n omttv algebra. commutative and associative ) . ,cnrlextension. central n, c bo Khudoyberdiyev Abror & s)o l ope null-filiform complex all of ism) . n dmninlnull- -dimensional d 2 every Zinbiel algebra with the commutator multiplication gives a Tortkara algebra (also about Tortkara algebras, see, [24–26]), which have recently sprung up in unexpected areas of mathematics [18, 19]. Central extensions play an important role in quantum mechanics: one of the earlier encounters is by means of Wigner’s theorem which states that a symmetry of a quantum mechanical system determines an (anti-)unitary transformation of a Hilbert space. Another area of physics where one encounters central extensions is the quantum theory of conserved currents of a Lagrangian. These currents span an algebra which is closely related to so-called affine Kac-Moody algebras, which are universal central extensions of loop algebras. Central extensions are needed in physics, because the symmetry group of a quantized system usually is a central extension of the classical symmetry group, and in the same way the corresponding symmetry Lie algebra of the quantum system is, in general, a central extension of the classical symmetry algebra. Kac-Moody algebras have been conjectured to be symmetry groups of a unified superstring theory. The centrally extended Lie algebras play a dominant role in quantum field theory, particularly in conformal field theory, string theory and in M-theory. In the theory of Lie groups, Lie algebras and their representations, a Lie algebra extension is an enlargement of a given Lie algebra g by another Lie algebra h. Extensions arise in several ways. There is a trivial extension obtained by taking a direct sum of two Lie algebras. Other types are a split extension and a central extension. Extensions may arise naturally, for instance, when forming a Lie algebra from projective group representations. A central extension and an extension by a derivation of a polynomial loop algebra over a finite-dimensional simple Lie algebra gives a Lie algebra which is isomorphic to a non-twisted affine Kac-Moody algebra [6, Chap- ter 19]. Using the centrally extended loop algebra one may construct a current algebra in two spacetime dimensions. The Virasoro algebra is the universal central extension of the Witt algebra, the Heisenberg algebra is the central extension of a commutative Lie algebra [6, Chapter 18]. The algebraic study of central extensions of Lie and non-Lie algebras has a very long history [3, 28–30, 35, 39, 47, 48, 50]. For example, all central extensions of some filiform Leibniz algebras were classified in [3, 48] and all central extensions of filiform associative algebras were classified in [35]. Skjelbred and Sund used central extensions of Lie algebras for a classification of low dimensional nilpotent Lie algebras [47]. After that, the method introduced by Skjelbred and Sund was used to describe all non-Lie central extensions of all 4-dimensional Malcev algebras [30], all non-associative central extensions of 3-dimensional Jordan algebras [29], all anticommutative central extensions of 3-dimensional anticommutative algebras [8]. Note that the method of central extensions is an important tool in the classification of nilpotent algebras. It was used to describe all 4-dimensional nilpotent associative algebras [15], all 4-dimensional nilpotent assosymmetric algebras [32], all 4-dimensional nilpotent bicommutative algebras [40], all 4-dimensional nilpotent Novikov algebras [34], all 4-dimensional commutative algebras [23], all 5-dimensional nilpotent Jordan algebras [27], all 5-dimensional nilpotent restricted Lie algebras [14], all 5-dimensional anticommutative algebras [23], all 6-dimensional nilpotent Lie algebras [13,16], all 6-dimensional nilpotent Malcev algebras [31], all 6-dimensional nilpotent binary Lie algebras [1], all 6-dimensional nilpotent anticommutative CD-algebras [1], all 6-dimensional nilpotent Tortkara algebras [24, 26], and some others. 3

1. PRELIMINARIES All algebras and vector spaces in this paper are over C. 1.1. Filiform Zinbiel algebras. An algebra A is called Zinbiel algebra if for any x, y, z A it satisfies the identity ∈ (x y) z = x (y z)+ x (z y). ◦ ◦ ◦ ◦ ◦ ◦ For an algebra A, we consider the series i 1 i+1 k i+1 k A = A, A = A A − , i 1. ≥ Xk=1 We say that an algebra A is nilpotent if Ai = 0 for some i N. The smallest integer satisfying Ai =0 is called the nilpotency index of A. ∈ Definition 1. An n-dimensional algebra A is called null-filiform if dim Ai =(n+1) i, 1 i n+1. − ≤ ≤ It is easy to see that a Zinbiel algebra has a maximal nilpotency index if and only if it is null-filiform. For a nilpotent Zinbiel algebra, the condition of null-filiformity is equivalent to the condition that the algebra is one-generated. j All null-filiform Zinbiel algebras were described in [4]. Throughout the paper, Ci denotes the combi- i natorial numbers j . Theorem 2. [4] An arbitrary n-dimensional null-filiform Zinbiel algebra is isomorphic to the algebra 0 Fn : j ei ej = Ci+j 1ei+j, 2 i + j n, ◦ − ≤ ≤ where omitted products are equal to zero and e , e ,...,e is a basis of the algebra. { 1 2 n} As an easy corollary from the previous theorem we have the next result.

0 0 Theorem 3. Every non-split central extension of Fn is isomorphic to Fn+1. 0 Proof. It is easy to see, that every non-split central extension of Fn is a one-generated nilpotent algebra. It follows that every non-split central extension of a null-filiform Zinbiel algebra is a null-filiform Zinbiel algebra. Using the classification of null-filiform algebras (Theorem 2) we have the statement of the Theorem. Definition 4. An n-dimensional algebra is called filiform if dim(Ai)= n i, 2 i n. − ≤ ≤ All filiform Zinbiel algebras were classified in [4]. Theorem 5. An arbitary n-dimensional (n 5) filiform Zinbiel algebra is isomorphic to one of the following pairwise non-isomorphic algebras:≥

1 j Fn : ei ej = Ci+j 1ei+j, 2 i + j n 1; 2 ◦ j − ≤ ≤ − Fn : ei ej = Ci+j 1ei+j, 2 i + j n 1, en e1 = en 1; 3 ◦ j − ≤ ≤ − ◦ − Fn : ei ej = Ci+j 1ei+j, 2 i + j n 1, en en = en 1. ◦ − ≤ ≤ − ◦ − 4

1.2. Basic definitions and methods. Throughout this paper, we are using the notations and methods well written in [29,30] and adapted for the Zinbiel case with some modifications. From now, we will give only some important definitions. Let (A, ) be a Zinbiel algebra and V a vector space. Then the C-linear space Z2 (A, V) is defined as the set of all◦ bilinear maps θ : A A V, such that × −→ θ(x y, z)= θ(x, y z + z y). ◦ ◦ ◦ Its elements will be called cocycles. For a linear map f from A to V, if we write δf : A A V by δf(x, y) = f(x y), then δf Z2 (A, V). We define B2 (A, V) = θ = δf : f Hom(×A, V−→) . One can easily check◦ that B2(A, V)∈is a linear subspace of Z2 (A, V) whose{ elements are∈ called coboundaries} . We define the second cohomology space H2 (A, V) as the quotient space Z2 (A, V) B2 (A, V).

Let Aut (A) be the automorphism group of the Zinbiel algebra A and let φ Aut (A). For θ Z2 (A, V) deﬁne φθ(x, y)= θ (φ (x) ,φ (y)). Then φθ Z2 (A, V). So, Aut (A) acts∈ on Z2 (A, V). Itis∈ easy to verify that B2 (A, V) is invariant under the action∈ of Aut (A) and so we have that Aut (A) acts on H2 (A, V).

Let A be a Zinbiel algebra of dimension m < n, and V be a C-vector space of dimension n m. 2 − For any θ Z (A, V) deﬁne on the linear space Aθ := A V the bilinear product “ [ , ]A ” by ∈ ⊕ − − θ [x + x′,y + y′]A = x y+θ(x, y) for all x, y A, x′,y′ V. The algebra Aθ is a Zinbiel algebra which θ ◦ ∈ ∈ is called an (n m)-dimensional central extension of A by V. Indeed, we have, in a straightforward − 2 way, that Aθ is a Zinbiel algebra if and only if θ Z (A, V). We also call the set Ann (θ)= x A : θ (x, A∈ )+ θ (A, x)=0 the annihilator of θ. We recall that the annihilator of an algebra A is{ deﬁned∈ as the ideal Ann (A) =} x A : x A + A x =0 and observe that Ann (A ) = (Ann(θ) Ann (A)) V. { ∈ ◦ ◦ } θ ∩ ⊕ We have the next key result: Lemma 6. Let A be an n-dimensional Zinbiel algebra such that dim(Ann (A)) = m = 0. Then 6 there exists, up to isomorphism, a unique (n m)-dimensional Zinbiel algebra A′ and a bilinear map θ Z2(A, V) with Ann (A) Ann (θ)=0−, where V is a vector space of dimension m, such that A∈ A A A A∩ ∼= ′θ and / Ann ( ) ∼= ′.

However, in order to solve the isomorphism problem we need to study the action of Aut (A) on 2 2 H (A, V). To do that, let us ﬁx e1,...,es a basis of V, and θ Z (A, V). Then θ can be uniquely s ∈ written as θ (x, y)= θ (x, y) e , where θ Z2 (A, C). Moreover, Ann (θ) = Ann (θ ) Ann (θ ) i i i ∈ 1 ∩ 2 ∩ i=1 ... Ann (θ ). Further,Xθ B2 (A, V) if and only if all θ B2 (A, C). ∩ s ∈ i ∈ Deﬁnition 7. Let A be an algebra and I be a subspace of Ann(A). If A = A0 I then I is called an annihilator component of A. ⊕ 5

Deﬁnition 8. A central extension of an algebra A without annihilator component is called a non-split central extension.

It is not difﬁcult to prove (see [30, Lemma 13]), that given a Zinbiel algebra Aθ, if we write as above s θ (x, y)= θ (x, y) e Z2 (A, V) and we have Ann (θ) Ann (A)=0, then A has an annihilator i i ∈ ∩ θ i=1 X 2 component if and only if [θ1] , [θ2] ,..., [θs] are linearly dependent in H (A, C).

Let V be a ﬁnite-dimensional vector space. The Grassmannian Gk (V) is the set of all k-dimensional 2 linear subspaces of V. Let Gs (H (A, C)) be the Grassmannian of subspaces of dimension s in 2 2 H (A, C). There is a natural action of Aut (A) on Gs (H (A, C)). Let φ Aut (A). For 2 ∈ W = [θ1] , [θ2] ,..., [θs] Gs (H (A, C)) deﬁne φW = [φθ1] , [φθ2] ,..., [φθs] . Then φW 2 h i ∈ 2 h i ∈ Gs (H (A, C)). We denote the orbit of W Gs (H (A, C)) under the action of Aut (A) by Orb(W ). Since given ∈ W = [θ ] , [θ ] ,..., [θ ] , W = [ϑ ] , [ϑ ] ,..., [ϑ ] G H2 (A, C) , 1 h 1 2 s i 2 h 1 2 s i ∈ s s s we easily have that in case W1 = W2, then Ann (θi) Ann (A) = Ann (ϑi) Ann (A), and so i=1 ∩ i=1 ∩ we can introduce the set T T s T (A)= W = [θ ] , [θ ] ,..., [θ ] G H2 (A, C) : Ann (θ ) Ann (A)=0 , s h 1 2 s i ∈ s i ∩ ( i=1 ) \ which is stable under the action of Aut (A).

Now, let V be an s-dimensional linear space and let us denote by E (A, V) the set of all non-split s-dimensional central extensions of A by V. We can write s E (A, V)= A : θ (x, y)= θ (x, y) e and [θ ] , [θ ] ,..., [θ ] T (A) . θ i i h 1 2 s i ∈ s ( i=1 ) X We also have the next result, which can be proved as in [30, Lemma 17]. s Lemma 9. Let A , A E (A, V). Suppose that θ (x, y) = θ (x, y) e and ϑ (x, y) = θ ϑ ∈ i i i=1 s X ϑi (x, y) ei. Then the Zinbiel algebras Aθ and Aϑ are isomorphic if and only if i=1 X Orb [θ ] , [θ ] ,..., [θ ] = Orb [ϑ ] , [ϑ ] ,..., [ϑ ] . h 1 2 s i h 1 2 s i From here, there exists a one-to-one correspondence between the set of Aut (A)-orbits on Ts (A) and the set of isomorphism classes of E (A, V). Consequently we have a procedure that allows us, given the Zinbiel algebra A′ of dimension n s, to construct all non-split central extensions of A′. This procedure would be: − 6

Procedure 2 (1) For a given Zinbiel algebra A′ of dimension n s, determine H (A′, C), Ann (A′) and Aut (A′). − (2) Determine the set of Aut (A′)-orbits on Ts(A′). (3) For each orbit, construct the Zinbiel algebra corresponding to a representative of it.

Finally, let us introduce some of notation. Let A be a Zinbiel algebra with a basis e1, e2,...,en. Then by ∆i,j we will denote the bilinear form ∆i,j : A A C with ∆i,j (el, em) = δilδjm. Then the set ∆ :1 i, j n is a basis for the linear space× of the−→ bilinear forms on A. Then every θ Z2 (A, C) { i,j ≤ ≤ } ∈ can be uniquely written as θ = c ∆ , where c C. ij i,j ij ∈ 1 i,j n ≤X≤ 2. CENTRAL EXTENSION OF FILIFORM ZINBIEL ALGEBRAS

1 2 3 Proposition 10. Let Fn , Fn and Fn be n-dimensional filiform Zinbiel algebras defined in Theorem 5. Then: A basis of Z2(F k, C) is formed by the following cocycles • n s 1 2 1 C − i 1 Z (Fn , )= ∆1,1, ∆1,n, ∆n,1, ∆n,n, Cs−1∆i,s i; 3 s n , h i=1 − − ≤ ≤ i s 1 2 k C P− i 1 Z (Fn , )= ∆1,1, ∆1,n, ∆n,1, ∆n,n, Cs−1∆i,s i; 3 s n 1 ,k =2, 3. h i=1 − − ≤ ≤ − i A basis of B2(F k, C) is formed by the followingP coboundaries • n s 1 2 1 C − i 1 B (Fn , )= ∆1,1, Cs−1∆i,s i, 3 s n 1 , h i=1 − − ≤ ≤ − i s 1 n 2 2 2 C P− i 1 − i 1 B (Fn , )= ∆1,1, Cs−1∆i,s i, 3 s n 2, Cn− 2∆i,n 1 i + ∆n,1 , h i=1 − − ≤ ≤ − i=1 − − − i s 1 n 2 2 3 C P− i 1 P− i 1 B (Fn , )= ∆1,1, Cs−1∆i,s i, 3 s n 2, Cn− 2∆i,n 1 i + ∆n,n . h i=1 − − ≤ ≤ − i=1 − − − i A basis of H2(F k, C) is formedP by the following cocyclesP • n n 1 2 1 C − i 1 H (Fn , )= [∆1,n], [∆n,1], [∆n,n], [ Cn− 1∆i,n i] , h i=1 − − i 2 k C H (Fn , )= [∆1,n], [∆n,1], [∆n,n] ,P k =2, 3. h i Proof. The proof follows directly from the definition of a cocycle. Proposition 11. Let φn Aut (F k). Then k ∈ n 7

a1,1 0 0 ... 0 0 a1,1 0 0 ... 0 0 2 2 a2,1 a1,1 0 ... 0 0 a2,1 a1,1 0 ... 0 0 3 3  a a ... 0 0   a3,1 a ... 0 0  n 3,1 1,1 n 1,1 φ1 = . ∗. . . . . , φ2 = . ∗. . . . . ,  ......   ......   n 1   n 1   −  a a − a  an 1,1 a1,1 an 1,n  n 1,1 1,1 n 1,n  − ∗ ∗ −   − ∗ ∗ n− 2   an,1 0 0 ... 0 an,n   an,1 0 0 ... 0 a1,−1   a1,1 0 0 ...  0 0  2 a2,1 a1,1 0 ... 0 0  3  a3,1 a1,1 ... 0 0 n ∗ φ3 =  ......   ......   n 1  an 1,1 a1,−1 an 1,n   − ∗ ∗ (n−1)/2  an,1 0 0 ... 0 a −   1,1    1 2.1. Central extensions of Fn . Let us denote n 1 − j 1 1 = [∆1,n], 2 = [∆n,1], 3 = [∆n,n], 4 = [ Cn−1∆j,n j] ∇ ∇ ∇ ∇ − − j=1 X and x = a1,1,y = an,n, z = an 1,n,w = an,1. Since − 0 0 0 0 0 Cn−1α4 α1  ··· 1  0 0 0 C − α4 0 0 ··· n 1  .  0 ′ ′  . ··  ... Cn−1α4 α1  0 0 0 . 0 0   ∗ 1 ∗ ′   ·  ... Cn−1α4 0 0 . . . · . . ∗  . . . n−1−i · . .   . . . .   . . . C − α4 . .  . . . . = (φn)T  n 1 ·  φn,  ......  1  . . . .  1    . ·· . . .   n−2 ′   . 0 0 . . .   Cn−1 α4 ... 0 0 0     ′ ′   · . . . .   α2 ... 0 0 α3   n−3 . . . .   0 Cn−1 α4 0 . . . .   n−2   C − α4 0 0 0 0 0   n 1 ···   α2 0 0 0 0 α3  ··· for any θ = α1 1 + α2 2 + α3 3 + α4 4, we have the action of the automorphism group on the subspace θ as ∇ ∇ ∇ ∇ h i (α xy + α yw + α xz) +(α xy + α yw +(n 1)α xz) + α y2 + α xn . 1 3 4 ∇1 2 3 − 4 ∇2 3 ∇3 4 ∇4 D E 1 2.1.1. 1-dimensional central extensions of Fn . Let us consider the following cases:

(1) if α1 = 0,α2 = α3 = α4 = 0, then by choosing x = 1,y = 1/α1, we have the representative .6 h∇1i (2) if α2 = 0,α3 = α4 = 0, then by choosing x = 1,y = 1/α2,α = α1/α2, we have the family of representatives6 α + . h ∇1 ∇2i (3) if α1 = α2,α3 = 0,α4 = 0, then by choosing y = 1/√α3,w = α2/α3, x = 1, we have the representative 6 . − h∇3i 8

√α3 1 α2 (4) if α1 = α2,α3 =0,α4 =0, then by choosing x = ,y = ,w = , we have the 6 6 α1 α2 √α3 √α3(α2 α1) representative + . − − h∇1 ∇3i (5) if (n 1)α = α ,α =0,α =0, then by choosing x =1/√n α ,y =1, z = α /α , we have − 1 2 3 4 6 4 − 1 4 the representative 4 . h∇ i √n α4 (6) if (n 1)α = α ,α = 0,α = 0, then by choosing x = 1/ n α ,y = , z = 1 2 3 4 √ 4 α2 (n 1)α1 −n 6 6 − − √α4 , we have the representative + . α2 (n 1)α1 2 4 − − − h∇ ∇ i n α1 α2 α2 (n 1)α1 (7) if α3 =0,α4 =0, then by choosing x =1/√α4,y =1/√α3, z = − ,w = − n− , 6 6 (n 2)√α3α4 (n 2) √α4α3 we have the representative + . − − h∇3 ∇4i It is easy to verify that all previous orbits are different, and so we obtain

1 T1(Fn ) = Orb 1 Orb α 1 + 2 Orb 3 Orb 1 + 3 Orbh∇ i ∪ Orbh ∇+ ∇ i ∪Orb h∇+i ∪ . h∇ ∇ i∪ h∇4i ∪ h∇2 ∇4i ∪ h∇3 ∇4i

1 2.1.2. 2-dimensional central extensions of Fn . We may assume that a 2-dimensional subspace is generated by

θ = α + α + α + α , 1 1∇1 2∇2 3∇3 4∇4 θ = β + β + β . 2 1∇1 2∇2 3∇3 Then we have the six following cases:

(1) if α4 =0, β3 =0, then we can suppose that α3 =0. Now 6 6 1/(n 2) (α2 (n 1)α1)(β2 β1) − β2 β1 (a) for (n 1)α1 = α2, β1 = β2, by choosing x = − − − ,y = − x, − 6 6 α4 β3 α1(β1 β2) z = − x, w = β x/β , we have the representative + , + . α4β3 1 3 2 3 2 4 − h∇ 1/∇(n 1)∇ ∇ i α2 (n 1)α1 − (b) for (n 1)α1 = α2, β1 = β2, by choosing x = − − , y = 1/√β3, z = − 6 α4√β3 α1 ,w = β1x/β3, we have the representative 3, 2 + 4 . − α4√β3 − h∇ ∇ ∇ i n β2 β1 α1(β1 β2) (c) for (n 1)α1 = α2, β1 = β2, by choosing x = 1/√α4, y = − x, z = − x, − 6 β3 α4β3 w = β1x/β3, we have the representative 2 + 3, 4 . − h∇ ∇ ∇n i (d) for (n 1)α1 = α2, β1 = β2, by choosing x = 1/√α4, y = 1/√β3, z = α1y/α4, w = −β x/β , we have the representative , . − − 1 3 h∇3 ∇4i (2) if α4 =0, β3 =0, β2 =0, then we can suppose that α2 =0. Now 6 6 n α1 (n 1)α1 (a) for α3 = 0, by choosing x = 1/√α4,y = 1/√α3, z = ,w = −n , and 6 (n 2)√α3α4 − (n 2) √α4α3 α = β /β we have the family of representatives α + − , + . − 1 2 h ∇1 ∇2 ∇3 ∇4i (b) for α =0, (n 1)β = β , then by choosing x =1/ n α ,y =1, z = α1β2 and 3 1 2 √ 4 α4((n 1)β1 β2) − 6 − − − α = β1/β2, we have the family of representatives α 1 + 2, 4 α= 1 . h ∇ ∇ ∇ i 6 n−1 n (c) for α3 = 0, (n 1)β1 = β2 and α1 = 0, by choosing x = 1/√α4,y = 1, z = 0, we have − 1 the representative n 1 1 + 2, 4 . h − ∇ ∇ ∇ i 9

n α4 (d) for α = 0, (n 1)β = β and α = 0, by choosing x = 1/ n α ,y = √ , z = 3 1 2 1 √ 4 (n 1)α1 n − 6 − − √α4 , we have the representative 1 + , + . (n 1)α4 n 1 1 2 2 4 − h − ∇ ∇ ∇ ∇ i (3) if α4 =0, β3 = β2 =0, β1 =0, then 6 6 n α1 α2 α2 (n 1)α1 (a) for α3 = 0, by choosing x = 1/√α4,y = 1/√α3, z = − ,w = − n− , we 6 (n 2)√α3α4 (n 2) √α4α3 have the representative , + . − − h∇1 ∇3 ∇4i (b) for α3 =0, after a linear combination of θ1 and θ2 we can suppose that (n 1)α1 = α2, by n − choosing x =1/√α4,y =1, z = α1/α4, we have the representative 1, 4 . − h∇ ∇ i (4) if α4 =0,α3 =0, β2 =0, then (a) for β = 6β , after6 a linear combination of θ and θ we can suppose that α = α , by choos- 1 6 2 1 2 1 2 ing y =1/√α ,w = α /α , x =1 and α = β /β we have the family of representatives 3 − 2 3 1 2 α 1 + 2, 3 α=1. h ∇ ∇ ∇ i 6 (b) for β1 = β2,α1 = α2, after a linear combination of θ1 and θ2 we have the representative + , . h∇1 ∇2 ∇3i (c) for β = β ,α = α , by choosing x = √α3 ,y = 1/ α ,w = α2 , we have the 1 2 1 2 α1 α2 √ 3 √α3(α2 α1) 6 − − representative 1 + 2, 1 + 3 . (5) if α = 0,α = 0, βh∇= 0,∇ β ∇= 0, then∇ i after a linear combination of θ and θ we can suppose 4 3 6 2 1 6 1 2 that α1 = α2, by choosing y = 1/√α3,w = α2/α3, x = 1 and α = β1/β2 we have the − representative 1, 3 . (6) if α = α =0h∇, β =0∇ i, then we have the representative , . 3 4 3 h∇1 ∇2i It is easy to verify that all previous orbits are different, and so we obtain

1 T2(Fn ) = Orb 1, 2 Orb 1, 3 Orb 1, 3 + 4 Orb 1, 4 h∇1 ∇ i ∪ h∇ ∇ i ∪ h∇ ∇ ∇ i ∪ h∇ ∇ i∪ Orb n 1 1 + 2, 2 + 4 Orb 1 + 2, 1 + 3 Orb α 1 + 2, 3 h − ∇ ∇ ∇ ∇ i ∪ h∇ ∇ ∇ ∇ i ∪ h ∇ ∇ ∇ i∪ Orb α 1 + 2, 3 + 4 Orb α 1 + 2, 4 Orb 2 + 3, 2 + 4 Orbh ∇+ ∇, ∇ Orb∇ i ∪ , h+∇ ∇Orb∇ i ∪, .h∇ ∇ ∇ ∇ i∪ h∇2 ∇3 ∇4i ∪ h∇3 ∇2 ∇4i ∪ h∇3 ∇4i 1 2.1.3. 3-dimensional central extensions of Fn . We may assume that a 3-dimensional subspace is generated by θ = α + α + α + α , 1 1∇1 2∇2 3∇3 4∇4 θ = β + β + β , 2 1∇1 2∇2 3∇3 θ = γ + γ . 3 1∇1 2∇2 Then we have the following cases:

(1) if α4 =0, β3 =0,γ2 =0, then we can suppose that α2 =0,α3 =0, β2 =0 and 6 6 6 n α1γ2y (a) for γ1 = γ2, (n 1)γ1 = γ2, then by choosing x =1/√α4,y =1/√β3, z = , 6 − 6 α4((n 1)γ1 γ2) β1γ2x − − w = , we have the family of representatives α + , , 1 . α4(γ1 γ2) 1 2 3 4 α 1, − − h ∇ ∇ ∇ ∇ i 6∈{ n 1 } (b) for γ1 = γ2, then n β1x α1y (i) for β1 = 0, by choosing x = 1/√α4, y = , z = , w = 0, we have the 6 β3 (n 2)α4 representative + , + , . − h∇1 ∇2 ∇1 ∇3 ∇4i 10

(ii) for β = 0, by choosing x = 1/ n α ,y = 1/√β , z = α1y ,w = 0, we have the 1 √ 4 3 (n 2)α4 representative + , , . − h∇1 ∇2 ∇3 ∇4i (c) for (n 1)γ1 = γ2, then − α1y n−1 (n 1)β1x (i) for α = 0, by choosing y = 1/√β , z = , x = (n 1)z, w = − , 1 3 α4 (n 2)β3 6 − − − − we have the representative 1 + , , + . n 1 1 2 3 2 4 p h − ∇ ∇ ∇ ∇ ∇ i (n 1)β1x (ii) for α = 0, by choosing x = 1/ n α ,y = 1/√β , z = 0,w = − , we have 1 √ 4 3 (n 2)β3 1 − − the representative n 1 1 + 2, 3, 4 . h − ∇ ∇ ∇ ∇ i (2) if α4 =0, β3 =0,γ2 =0,γ1 =0, then we can suppose that α3 =0 and after a linear combination 6 6 6 n of θ1, θ2, θ3 we can suppose that (n 1)α1 = α2, β1 = β2. By choosing x =1/√α4,y =1/√β3, − z = α1y/α4,w = β1x/β3, we have the representative 1, 3, 4 . (3) if α −= 0, β = 0, β−= 0,γ = 0,γ = 0, then and afterh∇ a linear∇ ∇ combinationi of θ , θ , θ we 4 6 3 2 6 2 1 6 1 2 3 can suppose that α1 = α2 = β1 =0. Now n (a) for α3 =0, by choosing y =1/√α3, x =1/√α4 we have the representative 1, 2, 3 + . 6 h∇ ∇ ∇ ∇4i (b) for α3 =0, we have the representative 1, 2, 4 . (4) if α =0, β =0,γ =0, then we have theh∇ representative∇ ∇ i , , . 4 3 2 h∇1 ∇2 ∇3i It is easy to verify that all previous orbits are different, and so we obtain

1 T3(Fn ) = Orb 1, 2, 3 Orb 1, 2, 3 + 4 Orb 1, 2, 4 Orbh∇ +∇ ∇, i ∪+ h∇, ∇ Orb∇ 1∇ i ∪+ ,h∇ , ∇ +∇ i∪ h∇1 ∇2 ∇1 ∇3 ∇4i ∪ h n 1 ∇1 ∇2 ∇3 ∇2 ∇4i∪ Orb α + , , Orb , −, . h ∇1 ∇2 ∇3 ∇4i ∪ h∇1 ∇3 ∇4i 1 2.1.4. 4-dimensional central extensions of Fn . There is only one 4-dimensional non-split central extension of the algebra F 1. It is deﬁned by , , , . n h∇1 ∇2 ∇3 ∇4i 1 2.1.5. Non-split central extensions of Fn . So we have the next theorem 1 Theorem 12. An arbitrary non-split central extension of the algebra Fn is isomorphic to one of the following pairwise non-isomorphic algebras one-dimensional central extensions: • n+1 n+1 n+1 n+1 1 2 3 µ1 , µ2 (α), µ3 , µ4 , Fn+1, Fn+1, Fn+1 two-dimensional central extensions: • n+2 n+2 n+2 n+2 n+2 n+2 n+2 n+2 n+2 n+2 n+2 n+2 n+2 µ5 , µ6 , µ7 , µ1 , µ8 , µ9 , µ10 (α), µ11 (α), µ2 (α), µ12 , µ4 , µ13 , µ3 three-dimensional central extensions: • n+3 n+3 n+3 n+3 n+3 n+3 n+3 µ14 , µ15 , µ5 , µ9 , µ16 , µ10 (α), µ6 four-dimensional central extensions: • n+4 µ14 with α C. ∈ 11

2 2.2. Central extensions of Fn . Let us denote = [∆ ], = [∆ ], = [∆ ] ∇1 1,n ∇2 n,1 ∇3 n,n and x = a ,w = a . Let θ = α + α + α . Then by 1,1 n,1 1∇1 2∇2 3∇3 ... 0 0 α1′ 0 ... 0 0 α1 ∗0 ... 00 0 0 ... 00 0  . . . .  n T  . . . .  n ...... =(φ2 ) ...... φ2 ,      0 ... 00 0   0 ... 00 0   α ... 0 0 α   α ... 0 0 α   2′ 3′   2 3  we have the action of the automorphism group on the subspace θ as  h i n 2 n 2 2n 4 x − (xα + wα ) + x − (xα + wα ) + x − α . 1 3 ∇1 2 3 ∇2 3∇3 D 2 E 2.2.1. 1-dimensional central extensions of Fn . Let us consider the following cases: (1) if α3 =0, then (a) for α2 =0,α1 =0, we have the representative 1 . 6 1/(n 1) h∇ i (b) for α2 = 0, by choosing x = α2− − and α = α1/α2, we have the family of representatives α6 + . h ∇1 ∇2i (2) if α3 =0, then 6 α2 α1 1/(n 3) xα1 (a) for α1 = α2, by choosing x =( − ) − ,w = we have the representative 2 + 6 α3 − α3 h∇ 3 . ∇ i xα1 (b) for α1 = α2, by choosing w = we have the representative 3 . − α3 h∇ i It is easy to verify that all previous orbits are different, and so we obtain T (F 2) = Orb Orb α + Orb + Orb . 1 n h∇1i ∪ h ∇1 ∇2i ∪ h∇2 ∇3i ∪ h∇3i 2 2.2.2. 2-dimensional central extensions of Fn . We may assume that a 2-dimensional subspace is generated by θ = α + α + α , 1 1∇1 2∇2 3∇3 θ = β + β . 2 1∇1 2∇2 We consider the following cases:

(1) if α3 =0 and β1 = β2, then after a linear combination of θ1 and θ2 we can suppose that α1 = α2. Now,6 6 1/(n 1) xα1 (a) for β2 = 0, by choosing x = β− − ,w = and α = β1/β2 we have the family of 6 2 − α3 respresentatives α 1 + 2, 3 α=1. 6 h ∇ ∇ ∇ 1i/(n 1) xα1 (b) for β2 =0, by choosing x = β− − ,w = , we have the respresentative 1, 3 . 1 − α3 h∇ ∇ i (2) if α3 =0 and β1 = β2, then 6 α1 α2 1/(n 1) xα2 (a) for α1 = α2, by choosing x =( − ) − ,w = we have the representative 1 + 6 α3 − α3 h∇ , + . ∇2 ∇1 ∇3i 12

(b) for α = α , after a linear combination of θ and θ we have the representative + 1 2 1 2 h∇1 2, 3 . (3) if α ∇=0∇, theni we have the representative , . 3 h∇1 ∇2i It is easy to verify that all previous orbits are different, and so we obtain T (F 2) = Orb , Orb , Orb + , + Orb α + , . 2 n h∇1 ∇2i ∪ h∇1 ∇3i ∪ h∇1 ∇2 ∇1 ∇3i ∪ h ∇1 ∇2 ∇3i

2 2.2.3. 3-dimensional central extensions of Fn . There is only one 3-dimensional non-split central extension of the algebra F 2. It is deﬁned by , , . n h∇1 ∇2 ∇3i 2 2.2.4. Non-split central extensions of Fn . So we have the next result. 2 Theorem 13. An arbitrary non-split central extension of the algebra Fn is isomorphic to one of the following pairwise non-isomorphic algebras one-dimensional central extensions: • 1 µn+1, µn+1(α) with α = , µn+1, µn+1, µn+1 1 2 6 n 3 8 12 16 − two-dimensional central extensions: • 1 µn+2, µn+2, µn+2, µn+2(α) with α = , µn+2 5 6 9 10 6 n 4 16 − three-dimensional central extensions: • n+3 µ14 with α C. ∈ 3 2.3. Central extensions of Fn . Let us denote = [∆ ], = [∆ ], = [∆ ] ∇1 1,n ∇2 n,1 ∇3 n,n and x = a ,w = a . Let θ = α + α + α . Then by 1,1 n,1 1∇1 2∇2 3∇3

... 0 0 α1′ 0 ... 0 0 α1 ∗0 ... 00 0 0 ... 00 0  . . . .  n T  . . . .  n ...... =(φ3 ) ...... φ3 ,      0 ... 00 0   0 ... 00 0   α ... 0 0 α   α ... 0 0 α   2′ 3′   2 3      we have the action of the automorphism group on the subspace θ as h i (n 1)/2 (n 1)/2 n 1 x − (xα + wα ) + x − (xα + wα ) + x − α . 1 3 ∇1 2 3 ∇2 3∇3 D E 13

3 2.3.1. 1-dimensional central extensions of Fn . Let us consider the following cases: (1) if α3 =0, then 2/(n+1) (a) for α2 =0,α1 =0, by choosing x = α1− , we have the representative 1 . 6 2/(n+1) h∇ i (b) for α2 =0, by choosing x = α2− and α = α1/α2 we have the family of representatives α +6 . h ∇1 ∇2i (2) if α3 =0, then 6 α2 α1 2/(n 3) xα1 (a) for α2 = α1, by choosing x =( − ) − ,w = we have the representative 2 + 6 α3 − α3 h∇ 3 . ∇ i xα1 (b) for α2 = α1, by choosing w = we have the representative 3 . − α3 h∇ i It is easy to verify that all previous orbits are different, and so we obtain T (F 3) = Orb Orb α + Orb + Orb . 1 n h∇1i ∪ h ∇1 ∇2i ∪ h∇2 ∇3i ∪ h∇3i 3 2.3.2. 2-dimensional central extensions of Fn . We may assume that a 2-dimensional subspace is generated by θ = α + α + α , 1 1∇1 2∇2 3∇3 θ = β + β . 2 1∇1 2∇2 We consider the following cases:

(1) if α3 =0 and β1 = β2, then after a linear combination of θ1 and θ2 we can suppose that α1 = α2. Now,6 6 2/(n+1) xα1 (a) for β2 = 0, by choosing x = β− ,w = and α = β1/β2 we have the family of 6 2 − α3 respresentatives α 1 + 2, 3 α=1. 6 h ∇ ∇ ∇ 2i/(n+1) xα1 (b) for β2 =0, by choosing x = β− ,w = , we have the respresentative 1, 3 . 1 − α3 h∇ ∇ i (2) if α3 =0 and β1 = β2, then 6 α1 α2 2/(n 3) xα2 (a) for α1 = α2, by choosing x =( − ) − ,w = we have the representative 1 + 6 α3 − α3 h∇ 2, 1 + 3 . (b) for∇ α∇ = ∇α i, after a linear combination of θ and θ we have the representative + 1 2 1 2 h∇1 2, 3 . (3) if α ∇=0∇, theni we have the representative , . 3 h∇1 ∇2i It is easy to verify that all previous orbits are different, and so we obtain T (F 3) = Orb , Orb , Orb + , + Orb α + , . 2 n h∇1 ∇2i ∪ h∇1 ∇3i ∪ h∇1 ∇2 ∇1 ∇3i ∪ h ∇1 ∇2 ∇3 i 3 2.3.3. 3-dimensional central extensions of Fn . There is only one 3-dimensional non-split central extension of the algebra F 3. It is deﬁned by , , . n h∇1 ∇2 ∇3i 3 2.3.4. Non-split central extensions of Fn . So we have the next theorem. 3 Theorem 14. An arbitrary non-split central extension of the algebra Fn is isomorphic to one of the following pairwise non-isomorphic algebras 14

one-dimensional central extensions: • n+1 n+1 n+1 n+1 µ7 , µ11 (α), µ12 , µ3 two-dimensional central extensions: • n+2 n+2 n+2 n+2 µ15 , µ6 , µ9 , µ10 (α) three-dimensional central extensions: • n+3 µ14 with α C. ∈ 3. APPENDIX: THE LIST OF THE ALGEBRAS

n j µ : ei ej = C , 2 i + j n 2, e1 en = en−1, 1 ◦ i+j−1 ≤ ≤ − ◦ n j µ (α): ei ej = C , 2 i + j n 2, e1 en = αen−1, en e1 = en−1, 2 ◦ i+j−1 ≤ ≤ − ◦ ◦ n j µ : ei ej = C , 2 i + j n 2, en en = en−1, 3 ◦ i+j−1 ≤ ≤ − ◦ n j µ : ei ej = C , 2 i + j n 2, e1 en = en−1, en en = en−1, 4 ◦ i+j−1 ≤ ≤ − ◦ ◦ n j µ : ei ej = C , 2 i + j n 3, e1 en = en−1, en e1 = en−2, 5 ◦ i+j−1 ≤ ≤ − ◦ ◦ n j µ : ei ej = C , 2 i + j n 3, e1 en = en−1, en en = en−2, 6 ◦ i+j−1 ≤ ≤ − ◦ ◦ n j µ : ei ej = C , 2 i + j n 2, e1 en = en−1, en en = en−2, 7 ◦ i+j−1 ≤ ≤ − ◦ ◦ n j 1 µ : ei ej = C , 2 i + j n 2, e1 en = en−1, en e1 = en−2 + en−1, 8 ◦ i+j−1 ≤ ≤ − ◦ n−3 ◦ n j µ : ei ej = C , 2 i + j n 3, e1 en = en−2 + en−1, en e1 = en−1, en en = en−2 9 ◦ i+j−1 ≤ ≤ − ◦ ◦ ◦ n j µ (α): ei ej = C , 2 i + j n 3, e1 en = αen−1, en e1 = en−1, en en = en−2, 10 ◦ i+j−1 ≤ ≤ − ◦ ◦ ◦ n j µ (α): ei ej = C , 2 i + j n 2, e1 en = αen−1, en e1 = en−1, en en = en−2, 11 ◦ i+j−1 ≤ ≤ − ◦ ◦ ◦ n j µ : ei ej = C , 2 i + j n 2, en e1 = en−2 + en−1, en en = en−1, 12 ◦ i+j−1 ≤ ≤ − ◦ ◦ n j µ : ei ej = C , 2 i + j n 2, en e1 = en−2, en en = en−1, 13 ◦ i+j−1 ≤ ≤ − ◦ ◦ n j µ : ei ej = C , 2 i + j n 4, e1 en = en−2, en e1 = en−1, en en = en−3, 14 ◦ i+j−1 ≤ ≤ − ◦ ◦ ◦ n j µ : ei ej = C , 2 i + j n 3, e1 en = en−2, en e1 = en−1, en en = en−3, 15 ◦ i+j−1 ≤ ≤ − ◦ ◦ ◦ n j 1 µ : ei ej = C , 2 i + j n 3, e1 en = en−1, en e1 = en−3 + en−1, en en = en−2. 16 ◦ i+j−1 ≤ ≤ − ◦ n−4 ◦ ◦

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