THE CLASSIFICATION OF ALGEBRAS OF LEVEL ONE A.KH.KHUDOYBERDIYEV,B.A.OMIROV Abstract. Inthepresentpaperweobtainthelistofalgebras,uptoisomorphism,suchthatclosure ofanycomplexfinite-dimensionalalgebracontains oneofthealgebraofthegivenlist. 3 1 0 Mathematics Subject Classification 2010: 14D06; 14L30. 2 Key Words and Phrases: closure of orbit, degeneration, level of algebra. n a J 1. Introduction 4 Itis knownthat any n-dimensionalalgebraovera field F may be consideredas anelement λof the 2 affine variety Hom(V ⊗V,V) via the bilinear mapping λ:V ⊗V →V on a vector space V. ] Since the space Hom(V ⊗V,V) formann3-dimensionalaffine space B(V) overF we shallconsider A the Zariskitopologyonthis spaceandthe linearreductivegroupGL (F) actsonthe spaceasfollows: R n h. (g∗λ)(x,y)=g(λ(g−1(x),g−1(y))). t a The orbits (Orb(−)) under this action are the isomorphic classes of algebras. Note that algebras m which satisfy identities (like, commutative, antisymmetric, nilpotency etc.) of the same dimension [ form also an invariant subvariety of the variety of algebras under the mentioned action. 1 In the study of a variety of algebrasplay the crucialrole closuresof orbitof algebras. Since closure v 7 of open set forms irreducible component of a variety, algebras whose orbits are open (so-called rigid 7 algebras) give the description of the variety of algebras. One of the tools of finding rigid algebras are 7 5 degenerations. The description of a variety of algebras by means of degenerations can be interpreted . 1 via down directed graph with the highest vertexes rigid algebras. Since any n-dimensional algebra 0 degenerates to the abelian (denoted by a ), any edge ends with the algebra a . For some examples of n n 3 1 descriptions of varieties by means of degeneration graphs we refer to papers [1, 3, 4] and others. : InthepaperofV.V.Gorbatsevich[2]thenearest-neighboralgebrasindegenerationgraph(algebras v i of levelone) to the algebra a are investigated. Namely, such algebrasin the varieties of commutative X n (respectively, antisymmetric) algebras are indicated. r a Forthe casea groundfieldis algebraicclosedfrom[3]it is knownthatclosuresoforbits ofalgebras (denoted by Orb(−)) in Zariski and Euclidean topologies are coincide. That is λ ∈ Orb(µ) can be realized by the following: ∃g ∈GL (C(t)) such that limg ∗λ=µ, t n t t→0 where C(t) is the field of fractions of the polynomial ring C[t]. In this workwe show that the paper [2] has some incorrectness andwe describe allalgebrasof level one in the variety of all complex finite-dimensional algebras. Let λ and µ are complex algebras of the same dimension. Definition 1.1. An algebra λ is said to degenerate to algebra µ, if Orb(µ) lies in Zariski closure of Orb(λ). We denote this by λ→µ. 1 2 A.KH.KHUDOYBERDIYEV,B.A.OMIROV Thedegenerationλ→µiscalledadirectdegenerationifthereisnochainofnon-trivialdegenerations of the form: λ→ν →µ. Definition 1.2. Level of an algebra λ is the maximum length of a chain of direct degeneration. We denote the level of an algebra λ by lev (λ). n Consider the following algebras: ± p : e e =e , e e =±e , i≥2, n 1 i i i 1 i ± n : e e =e , e e =±e . 3 1 2 3 2 1 3 Theorem 1.3. [2] Let λ be an n-dimensional algebra. Then 1. if the algebra λ is skew-commutative, then lev (λ) = 1 if and only if it is isomorphic to p− or n n − (with n≥3) to the algebra n3 ⊕an−3. In particular, the algebra λ is a Lie algebra. 2. if the algebra λ is commutative, then lev (λ) = 1 if and only if it is isomorphic to p+ or (for n n n≥3) to the algebra n+3 ⊕an−3. In particular, the algebra λ is an Jordan algebra. 2. Main result In this section we describe all complex finite dimensional algebras of level one. Consider following algebras λ : e e =e , 2 1 1 2 ν (α): e e =e , e e =αe , e e =(1−α)e , 2≤i≤n. n 1 1 1 1 i i i 1 i In the following proposition we prove that algebras p+n and n+3 ⊕an−3 are not of level one. Proposition 2.1. p+n →λ2⊕an−2 and n+3 ⊕an−3 →λ2⊕an−2. Proof. The first degeneration is given by the family of transformations g : t t−2 t−2 g (e )=t−1e − e , g (e )= e , g (e )=t−2e , 3≤i≤n. t 1 1 2 t 2 2 t i i 2 2 The second one is realized by the family f : t t−2 f (e )=t−1e −t−2e , f (e )=t−2e , f (e )= e , f (e )=e , 4≤i≤n. t 1 1 3 t 2 3 t 3 2 t i i 2 (cid:3) The above Proposition shows that the second assertion of result of Theorem 1.3 is not correct. In order to prove the main theorem we need the following interim result. Proposition 2.2. Any n-dimensional (n≥3) non-abelian algebra degenerates to one of the following algebras p−n, n−3 ⊕an−3, λ2⊕an−2, νn(α), α∈C. Proof. Let A be an n-dimensional non-abelian algebra. Firstly we consider the case when A is antisymmetric algebra. Clearly, xx=0 for any x of A. If there exist elements x,y ∈A such that xy ∈/<x,y >, then we can consider a basis of A: e =x, e =y, e =xy, e , ..., e . 1 2 3 4 n THE CLASSIFICATION OF ALGEBRAS OF LEVEL ONE 3 − Itis easyto check thatalgebraA degeneratesto the algebran3 ⊕an−3 by the use ofthe family gt : g (e )=t−1e , g (e )=t−1e , g (e )=t−2e , 3≤i≤n. t 1 1 t 2 2 t i i Consider now the contrary case, i.e. xy ∈<x,y > for all x,y ∈L. Then the table of multiplication of the algebra A have the form e e =γi e +γj e , 1≤i,j ≤n. i j i,j i i,j j Since the algebra A is non-abelian, without lost of generality, we can assume γ2 6=0. 1,2 Taking the change e′ = 1 e , e′ =e + γ11,2e , we can suppose e e =e . 1 γ12,2 1 2 2 γ12,2 1 1 2 2 Consider the product e (e +e )=γ1 e +(e +e )+(γi −1)e . 1 2 i 1,i 1 2 i 1,i i Taking into account e (e + e ) ∈< e ,e + e > we deduce γi = 1, 3 ≤ i ≤ n. Setting e′ = 1 2 i 1 2 i 1,i i e +γ1 e , 3≤i≤n, we obtain e e =e , 2≤i≤n. i 1,i 1 1 i i Putting g as follows: t g (e )=e , g (e )=t−1e , 2≤i≤n, t 1 1 t i i we get limt→0gt∗A=p−n. Now we assume that the algebraA is not antisymmetric. Then there exists an element x of A such that xx6=0. Case 1. Let there exists x of the algebra A such that xx ∈/< x > . Then we can chose a basis e1 =x, e2 =xx,...,en. The degeneration A→λ2⊕an−2 is realized by the family gt : g (e )=t−1e , g (e )=t−2e , 2≤i≤n. t 1 1 t i i Case 2 Let xx ∈< x > for all x ∈ A. Then for any x,y ∈ A we have (x+y)(x+y) = xx+xy+ yx+yy ∈<x+y >. Therefore, xy+yx∈<x,y >. If there exist elements x and y such that xy ∈/<x,y >, then we chose a basis {e =x,e =y,e = 1 2 3 e e ,...,e } of the algebra A. The following family 1 2 n g :g (e )=t−1e , g (e )=t−1e , g (e )=t−2e , 3≤i≤n t t 1 1 t 2 2 t i i − derives the degeneration A→n3 ⊕an−3. Now we consider the case when xy ∈< x,y > for all x,y ∈ A. Then for a basis {e ,e ,e ,...,e } 1 2 3 n ofA we have e e =α e , 1≤i≤n. Taking into accountthat algebraA is non-antisymmetric,we can i i i i supposeα 6=0.Withoutlossofgeneralitywecanassumeα 6=0, 1≤i≤kandα =0, k+1≤i≤n. 1 i i By scaling of basis elements we get e e =e , 1≤i≤k, α =0, k+1≤i≤n. i i i i The includings of the following products (e ±e )(e ±e )=e ±e e ±e e +e ∈<e ±e >, 1 i 1 i 1 1 i i 1 i 1 i imply e e +e e =e +e , 1≤i≤k. 1 i i 1 1 i Similarly, we obtain e e +e e =e , k+1≤i≤n. 1 i i 1 i Making the the change of basis ′ ′ e =e −e , 2≤i≤k e =e , k+1≤i≤n, i i 1 i i 4 A.KH.KHUDOYBERDIYEV,B.A.OMIROV we get the following products e e =e , e e =0, 2≤i≤n, e e =α e +β e , e e =(1−α )e −β e , 2≤i≤n, 1 1 1 i i 1 i i i i 1 i 1 i i i 1 for some α ,β ∈C. i i The product e (e +e )=(α −α )e +α (e +e )+(β +β )e 1 i j j i j i i j i j 1 and e (e +e )∈<e ,e +e > imply α =α, 2≤i≤n. 1 i j 1 i j i The degeneration A→ν (α) which is realized by using the family n g : g (e )=e , g (e )=t−1e , 2≤i≤n, t t 1 1 t i i complete the proof of proposition. (cid:3) Theorem 2.3. Let A be an n-dimensional (n ≥ 3) algebra of level one, then it is isomorphic to one of the following algebras: p−n, n−3 ⊕an−3, λ2⊕an−2, νn(α), α∈C. Proof. Due to Proposition 2.2 it is sufficient to prove that these four algebras do not degenerate to each other. Since p−n and n−3 ⊕an−3 are antisymmetric algebras, but λ2 ⊕ an−2 is commutative, we obtain − − Orb(pn)∩Orb(λ2⊕an−2)={an} and Orb(n3 ⊕an−3)∩Orb(λ2⊕an−2)={an}. Moreover,algebras n−3 ⊕an−3 and λ2⊕an−2 are nilpotent, but p−n and ν(α) are not nilpotent. Therefore, n−3 ⊕an−3 and λ2⊕an−2 do not degenerate to algebras p−n and ν(α). − − Let us show that Orb(p ) = {p ,a }. Consider a family of basis transformation g of the algebra n n n t p−. Then we have n n n eiej =tl→im0gt(gt−1(ei)gt−1(ej))=tl→im0gt(Xβi,k(t)ekXβj,k(t)ek)= k=1 k=1 n n n limgt(βi,1(t)Xβj,k(t)ek−βj,1(t)Xβi,k(t)ek)= limgt(βi,1(t)Xβj,k(t)ek− t→0 t→0 k=2 k=2 k=1 n βj,1(t)Xβi,k(t)ek)=tl→im0gt(βi,1(t)gt−1(ej)−βj,1(t)gt−1(ei))=tl→im0(βi,1(t)ej −βj,1(t)ei). k=1 If limt→0βi,1(t)=0 for any i, then we get the algebra an. If there exist i0 (1≤i0 ≤n) such that limt→0βi0,1(t)=βi0 6=0, then without lost of generality we can suppose i =1. 0 Taking the change of basis e′1 = β11e1, e′i =ei− ββ1ie1 in the algebra limt→0gt∗p−n we have ′ ′ ′ ′ ′ e e =−e e =e . 1 i i 1 i Thus, we obtain limt→0gt∗p−n =p−n. In a similar way we show that Orb(ν(α) ={ν(α),a }. n Consider n n eiei =tl→im0gt(gt−1(ei)gt−1(ei))=tl→im0gt(Xβi,k(t)ekXβi,k(t)ek)= k=1 k=1 n n limgt(βi,1(t)2e1+βi,1(t)Xβi,k(t)ek)= limgt(βi,1(t)Xβi,k(t)ek)= limβi,1(t)ei, t→0 t→0 t→0 k=2 k=1 THE CLASSIFICATION OF ALGEBRAS OF LEVEL ONE 5 n n eiej =tl→im0gt(gt−1(ei),gt−1(ej))=tl→im0gt(Xβi,k(t)ekXβj,k(t)ek)= k=1 k=1 n n limgt(βi,1(t)βj,1(t)e1+αβi,1(t)Xβj,k(t)ek+(1−α)βj,1(t)Xβi,k(t)ek)= t→0 k=2 k=2 n n limgt(αβi,1(t)Xβj,k(t)ek+(1−α)βj,1(t)Xβi,k(t)ek)= lim(αβi,1(t)ej +(1−α)βj,1(t)ei). t→0 t→0 k=1 k=1 If limt→0βi,1(t)=0 for all i, (1≤i≤n) then we have the algebra an. If there exist i0 (1 ≤ i0 ≤ n) such that limt→0βi0,1(t) = βi0 6= 0, then, without lost of generality, we can assume that β 6=0 for 1≤i≤k and β =0 for k+1≤i≤n. i i Taking the change 1 1 1 ′ ′ ′ e = e , e = e − e , 1≤i≤k, e =e , k+1≤i≤n 1 β 1 i β i β 1 i i 1 i 1 in the algebra limt→0gt∗νn(α), we derive the table of multiplication: ′ ′ ′ ′ ′ ′ ′ ′ e e =e , e e =αe , 2≤i≤n, e e =(1−α)e , 2≤i≤n. 1 1 1 1 i i i 1 i (cid:3) Remark that two-dimensional algebras of level one are the following − p , λ , ν (α). 2 2 2 References [1] BurdeD.Degenerationof 7-dimensional nilpotent Lie algebras, Comm.Algebra,vol.33(4), 2005,p.1259–1277. [2] GorbatsevichV.V.Oncompressions and degenerations of finite dimensional algebras,IzvestiyaVuzovMatem.,vol. 10,1991,p.19–27. [3] GrunewaldF.,O’HalloranJ.VarietiesofnilpotentLiealgebrasofdimensionlessthansix,J.Algebra,vol.112,1988, p.315–325. [4] SeeleyC.,Degenerationof 6-dimensional nilpotent LiealgebrasoverC,Comm.Algebra,vol.18(10), 1990,p.3493– 3505. [A.Kh.KhudoyberdiyevandB.A.Omirov]InstituteofMathematics,NationalUniversityofUzbekistan, Tashkent, 100125,Uzbekistan. E-mail address: [email protected], [email protected]