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Research Article Open Access A Class of Nonassociative Algebras Including Flexible and Alternative Algebras, Operads and Deformations Remm E* and Goze M Associate Professor, Université de Haute Alsace, 18 Rue des Frères Lumière, 68093 Mulhouse Cedex, France
Abstract There exists two types of nonassociative algebras whose associator satisfies a symmetric relation associated ∑ with a 1-dimensional invariant vector space with respect to the natural action of the symmetric group 3. The first one corresponds to the Lie-admissible algebras and this class has been studied in a previous paper of Remm and Goze. Here we are interested by the second one corresponding to the third power associative algebras.
Nonassociative algebras; Alternative algebras; Third Keywords: is an irreducible invariant subspace of [Σ3] with respect to the right power associative algebras; Operads action of Σ3 on [Σ3]. Introduction Recall that there exists only two one-dimensional invariant subspaces of [Σ ], the second being generated by the vector Recently, we have classified for binary algebras, Cf. [1], relations 3 ∑σ ∈Σ ε()σσwhere ε(σ) is the sign of σ. As we have precised in the of nonassociativity which are invariant with respect to an action of the 3 introduction, this case has been studied in literature of Remm [1]. symmetric group on three elements Σ3 on the associator. In particular we have investigated two classes of nonassociative algebras, the first 3 Definition 2.A Gi-p -associative algebra is a -algebra (,µ) whose one corresponds to algebras whose associator Aµ satisfies associator 2 Aµ ( Id −τττ − − ++c c ) = 0, (1) 12 23 13 Aµ =(µµ ⊗−⊗ Id Id µ ) and the second satisfies
2 Aµ ( Id +τττ12 + 23 + 13 ++c c ) = 0, (2) Aµ Φv = 0, G6 where τij denotes the transposition exchanging i and j, c is the 3-cycle ⊗⊗33 (1,2,3). where Φ→v : is the linear map Gi These relations are in correspondence with the only two irreducible Φ(xxxxxx ⊗⊗ ) = ( −−−⊗ ⊗ ). vG 123111∑ σσσ(1) (2) (3) i σ∈ one-dimensional subspaces of [Σ3] with respect to the action of Σ3, Gi where [Σ3] is the group algebra of Σ3. In studies of Remm [1], we have Let ()vG be the orbit of v with respect to the right action studied the operadic and deformations aspects of the first one: the class i Gi of Lie-admissible algebras. We will now investigate the second class and Σ×33[] Σ → [] Σ3 in particular nonassociative algebras satisfying (2) with nonassociative −1 (,σ∑∑aaii σ ) i σσ i relations in correspondence with the subgroups of Σ3. ii
Then putting FKv = ( (vG )) we have Convention: We consider algebras over a field of characteristic Gii zero. dimFv = 6, G 3 1 Gi -p -associative Algebras dimFFFvvv = dim = dim = 3, GGG Definition 234 dimFv = 2, Let Σ3 be the symmetric group of degree 3 and a field of G 5 characteristic zero. We denote by [Σ3] the corresponding group algebra, that is the set of formal sums ∑σ∈Σ aσσ , a endowed with dimFv = 1. 3 σ G the natural addition and the multiplication induced by multiplication 6 ∈ in Σ3, and linearity. Let {Gi}i = 1, ,6 be the subgroups of Σ3. To fix notations we define *Corresponding author: Remm E, Associate Professor, Université de Haute ⋅⋅⋅ Alsace, 18 Rue des Frères Lumière, 68093 Mulhouse Cedex, France, Tel:
G1 ={} Id , G2 =<τττ 12 >, G 3 =< 23 >, G 4 =< 13 >, G 5 =< c >, G6 =Σ 3 , +3338933652; E-mail: [email protected] where < σ > is the cyclic group subgroup generated by σ. To each Received October 15, 2015; Accepted November 10, 2015; Published November 17, 2015 subgroup Gi we associate the vector vG of [Σ3]: i Citation: Remm E, Goze M (2015) A Class of Nonassociative Algebras Including v =.σ Flexible and Alternative Algebras, Operads and Deformations. J Generalized Lie Gi ∑ σ∈ Theory Appl 9: 235. doi:10.4172/1736-4337.1000235 Gi Copyright: © 2015 Remm E, et al. This is an open-access article distributed under Lemma 1. The one-dimensional subspace {}vΣ of [Σ ] generated by 3 3 the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and vvG ==Σ ∑ σ 6 3 σ∈Σ source are credited. 3
J Generalized Lie Theory Appl Volume 9 • Issue 2 • 1000235 ISSN: 1736-4337 GLTA, an open access journal Citation: Remm E, Goze M (2015) A Class of Nonassociative Algebras Including Flexible and Alternative Algebras, Operads and Deformations. J Generalized Lie Theory Appl 9: 235. doi:10.4172/1736-4337.1000235
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Proposition 3. Every G -p3-associative algebra is third power i Remark: Poisson algebras. A -Poisson algebra is a vector space associative. provided with two multiplications, an assocative commutative one x • y and a Lie bracket [x, y], which satisfy the Leibniz identity Recall that a third power associative algebra is an algebra (,µ) [x• yz ,] −• x [,] yz − [,] xz • y =0 whose associator satisfies Aµ (x, x, x) = 0. Linearizing this relation, we obtain In studies of Remm [3], it is shown that these conditions are A Φ = 0. µ vΣ equivalent to provide with a nonassociative multiplication x y 3 satisfying Since each of the invariant spaces FG contains the vector vΣ , we i 3 1 ⋅ deduce the proposition. x⋅()=() yz ⋅ xy ⋅ ⋅− z{ () xz ⋅ ⋅+ y () yz ⋅ ⋅− x () yx ⋅ ⋅− z () zx ⋅ ⋅ y} 3 R L ⋅⋅ Remark. An important class of third power associative algebras If we denote by A⋅ ( xyz , , )= x⋅⋅ ( y z ) and A⋅ ( xyz , , )=( x y ) z then is the class of power associative algebras, that is, algebras such that any the previous identity is equivalent to element generates an associative subalgebra. RL AA⋅⋅Φ+ww Φ = 0 12 3 What are Gi-p -associative algebras? w =3−Id +ττ +−− c c . where w1 = 3Id and 2 23 1 2 12 In fact the class of (1) If i =1, since v =,Id the class of G -p3-associative algebras is Poisson algebras is a subclass of a family of nonassociative algebras G1 1 the full class of associative algebras. defined by conditions on the associator. The product satisfies 3 A(,,)(,,)(,,)=0 xyz++ A yzx A zxy (2) If i =2, the associator of a G2-p -associative algebra satisfies ⋅⋅⋅
Aµ (x1, x2, x3) + Aµ (x2, x1, x3) = 0 and
and this is equivalent to A⋅⋅(,,) xyz+ A (,,)=0, zyx
Aµ (x, x, y), so it is a subclass of the class of algebras which are flexible and G -p3-associative [1]. for all x, y . 5 3 3 The Operads Gi-p ss and their Dual (3) If i = 3,∈ the associator of a G3-p -associative algebra satisfies For each i {1, ,6}, the operad for G -p3-associative algebras will be Aµ (x1, x2, x3) + Aµ (x1, x3, x2) = 0, i 3 3 denoted by G -p ss. The operads {Gi − p ss} are binary quadratic that is, i ∈ ⋅⋅⋅ i=1, ,6 operads, that is, operads of the form = Γ(E)/(R), where Γ(E) denotes Aµ (x, y, y), the free operad generated by a Σ2-module E placed in arity 2 and (R)
for all x, y . is the operadic ideal generated by a Σ3-invariant subspace R of Γ(E)(3). ! ! Γ ∨⊥ 3 Then the dual operad is the quadratic operad := (ER ) / ( ) , where Sometimes∈ G2-p -associative algebras are called left-alternative ⊥∨ 3 RE⊂Γ( )(3) is the annihilator of R Γ(E)(3) in the pairing algebras, G3-p -associative algebras are right-alternative algebras. 3 An alternative algebra is an algebra which satisfies theG and G -p - ⋅⋅ ⋅⋅ ≠′′′ 2 3 <(xxxxxxij ) k ,( i′′ j ) k ′⊂ >=0,if(, ijk , ) ( ijk , , ), associativity. ⋅⋅ ⋅⋅ −εσ() <(xxij ) x k ,( xx ij ) x k >=( 1) , (4) If i = 4, we have A (x, y, x) for all x, y , and the class of G -p3- 123 µ 3 withσ = , associative algebras is the class of flexible algebras. i jR ∈
(b) a skew-symmetric multiplication [,] satisfying the identity Let A2 be the two-dimensional associative algebra given in a basis {e1, e2} by e1e1 = e2, e1e2 = e2e1 = e2e2 = 0. Recall that the Hochschild homology [x yz ,][++ y zx ,][ z xy ,]=0 of an associative algebra is given by the complex (Cdnn ( , ), ) where ⊗ ⊗n dC: (,)→ C (,) for any x, y . Cn ( , )= and the differentials nn n−1 are given by Then a G - p3-associative algebra can be defined as ([ , ], )-admissible 6 ∈ algebra.
J Generalized Lie Theory Appl Volume 9 • Issue 2 • 1000235 ISSN: 1736-4337 GLTA, an open access journal Citation: Remm E, Goze M (2015) A Class of Nonassociative Algebras Including Flexible and Alternative Algebras, Operads and Deformations. J Generalized Lie Theory Appl 9: 235. doi:10.4172/1736-4337.1000235
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xxxx1234=−− xxxx 2134 = x 2 ( xx 13 )= x 4 xxxx 1324 =− xxxx 1234 daan(,,,)=(01 an aaa012 ,,,) an +−∑ (1)(,,,aa01 aaii 1 ,,) a n and (G -p3ss)! (4) = {0}. Let be (G -p3ss). The generating function +−( 1) (aann01 , a , , a 1 ). 2 2 ! 3 ! of = (G2-p ss) is Concerning the algebra A , we have 3 2 1 3!a 2x g !2(x ) =∑ dim(G− p ss ) ( a ) x = x++ x . a≥1 a!2 d1( ei , e j ) = ee i j− ee ji = 0,
3 for any i, j = 1,2. Similarly we have But the generating function of = (G2-p ss) is 35 d(,,)=2(,) eee ee− (, ee ), g (x )= x++ x234 x + x + Ox() 5 2111 21 12 22 d2112(,,)= eee d 2121 (,,)= eee− d 2211 (,,)=(,) eee ee 22 3 and if (G2-p ss) is Koszul, then the generating functions should be and 0 in all the other cases. Then dim Im d2 = 2 and dim Ker d1 = 4. related by the functional equation Then H (A , A ) is isomorphic to A . We have also 1 2 2 2 gg(−− (xx )) = ! d(,,,) eeee =(,−+ eee ,) (,, eee ), 31111 121 112 and it is not the case so both (G -p3ss) and (G -p3ss)! are not Koszul. − 2 2 d31112(,,,) eeee =(,,)(,,), eee 212 eee 122 By definition, a quadratic operad is Koszul if any free -algebra d31121(,,,) eeee =−− d 22111 (,,,)=(,,)(,,), eeee eee 221 eee 212 on a vector space V is a Koszul algebra. Let us describe the free algebra deeee31211(,,,)=(,,)(,,), eee 122− eee 221 d(,,,)= eeee− d (,,, eeee)=−d (,,,)=(,,,) eeee d eeee ()V 3112 2 312 21 3 2112 3 2 211 − 3! when dim V =1 and 2. ()G2 p ss =(eee222 , , ) 3 ! A (G2-p ss) -algebra is an associative algebra satisfying and d3 = 0 in all the other cases. Then dim Im d3 = 4 and dim Ker d2 = 6.
Thus H2(A2, A2) is non trivial and A2 is not a Koszul algebra. xyz = −yxz, 3 Now we will study all the operads Gi-p ss. for any x, y, z . This implies xyzt = 0 for any x, y, z . In particular we have 3 The operad (G1-p ss) ∈ ∈ x3 = 0, 3 Since G -p ss=ss, where ss denotes the operad for associative 2 1 xy= 0, algebras, and since the operad ss is selfdual, we have ()V 3! ! 3 for any x, y . If dim V =1, − 3! is of dimension 2 and (G−− p ss )= ss = G p ss . ()G2 p ss 11 given by We also have ∈ ee=, e 11 2 3 ee= ee = ee = 0. G1− p ss= ss =, ss 12 21 22 where is the maximal current operad of defined in [7,8]. ()V e3 = 0. In fact if V = {e1} thus in − 3! we have 1 We deduce ()G2 p ss 3 The operad (G -p ss) (VA )= ()V 2 that − 3! 2 and − 3! is not Koszul. It is easy to ()G2 p ss ()G2 p ss The operad G -p3ss is the operad for left-alternative algebras. It is generalize this construction. If dim V = n, then dim (V ) = 2 ()G− p3! ss the quadratic operad = Γ(E) / (R), where the Σ -invariant subspace R 2 2 3 nn(++ n 2) 2 and if {e1, ,en} is a basis of V then {ei,, e i ee i j , ee lmp e } for i, j of Γ (E)(3) is generated by the vectors 2 = 1, ,n and l, m, p = 1, ,n with m > l, is a basis of (V ). For ()xx⋅ ⋅−⋅ x x ()() xx ⋅ + xx ⋅ ⋅−⋅ x x (). xx ⋅ ⋅⋅⋅ ()G− p3! ss 12 3 1 23 21 3 2 13 example, if n = 2, the basis of ()V is 2 ()G− p3! ss The annihilatorR of R with respect to the pairing (3) is generated ⋅⋅⋅ ⋅⋅⋅ 2 ⊥ 22 2 by the vectors {vv112231425126217121812=,=,=,=,=e e v e v e v ee ,= v ee ,= v eee ,= v ee }
(xx12⋅⋅−⋅⋅ ) x 3 x 1 ( xx 23 ), (4) and the multiplication table is ()().xx12⋅ ⋅+ x 3 xx 21 ⋅ ⋅ x 3 We deduce from direct calculations that dim R = 9 and v1 v2 v3 v4 v5 v6 v7 v8 ⊥ v v v 0 v 0 v 0 0 Proposition 5. The (G -p3ss)!-algebras are associative algebras 1 3 5 8 7 2 v v v v 0 v 0 0 0 satisfying 2 6 4 − 7 − 8 v3 0 0 0 0 0 0 0 0 abc = −bac. v4 0 0 0 0 0 0 0 0 ! Recall that (G2ss) -algebras are associative algebras satisfying v5 v7 v8 0 0 0 0 0 0 v v v 0 0 0 0 0 0 abc = bac. 6 − 7 − 8 v7 0 0 0 0 0 0 0 0 and this operad is classically denoted erm. v8 0 0 0 0 0 0 0 0 3 ! Theorem 6. The operad (G2-p ss) is not Koszul [9]. 3 ! For this algebra we have Proof. It is easy to describe (G2-p ss) (n) for any n. In fact 3 ! (G2-p ss) (4) corresponds to associative elements satisfying
J Generalized Lie Theory Appl Volume 9 • Issue 2 • 1000235 ISSN: 1736-4337 GLTA, an open access journal Citation: Remm E, Goze M (2015) A Class of Nonassociative Algebras Including Flexible and Alternative Algebras, Operads and Deformations. J Generalized Lie Theory Appl 9: 235. doi:10.4172/1736-4337.1000235
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xx34 fx( )= x+++ x2 . dv( , v )= v− v , 2 12 11 2 56 ()V 3 ! Let ()G− p3! ss be the free (G4-p ss) -algebra based on the 1 4 − dvv11( , 6 )= v 7 = dvv 11 ( , 5 )= dv 12 ( , v 3 ), vector space V. In this algebra we have the relations 2 a3 = 0, 1 dv( , v )=−− v = dv ( , v )= dvv ( , ), aba = 0, 12 5 8 16 2 11 4 2 for any a, b V. Assume that dim V =1. If {e1} is a basis of V, then 3 ()=VV (). ()V and Ker d is of dim 64. The space Im d doesn’t contain in particular the e1 =0and −−3! 3! We deduce that − 3! 1 2 ()G42∈ p ss ()G p ss ()G4 p ss v v v vectors ( i , i) for i = 1, 2 because these vectors i are not in the derived is not a Koszul algebra. subalgebra. Since these vectors are in Ker d1 we deduce that the second space of homology is not trivial. Proposition 10. The operad for flexible algebra is not Koszul.
Proposition 7. The current operad of G -p3ss is Let us note that, if dim V=2 and {e1,e2} is a basis of V, then 2 22 22 22 2222 ()V {eeeeeeeeeeeeeeeeeeee1,,,, 2 1 2 12 , 21 , 12 , 1 2 , 21 , 21 , 1 2 , 21} ()G− p3! ss is generated by and G− p3 ss=. erm 4 2 is of dimension 12. This is directly deduced of the definition of the current operad [7]. Proposition 11. We have 3 The operad (G3-p ss) 3! G44−− p ss=( G ss ). It is defined by the module of relations generated by the vector 3 This means that a G4− p ss is an associative algebra satisfying ()xx x−+− x ()() xx xx x x (), xx 12 3 1 23 13 2 1 32 abc = cba, for any a, b, c . ⊥ and R is the linear span of 3 The operad (G5-p ss∈)
(xx12 ) x 3− x 1 ( xx 23 ), It coincides with (G5-ss) and this last has been studied by Remm (xx ) x+ ( xx ). x 12 3 13 2 [2]. Proposition 8. A (G -p3ss)!-algebra is an associative algebra 3 3 The operad (G6-p ss) satisfying 3 A (G6-p ss)-algebra (, µ) satisfies the relation abc = −acb, Axxxµµµ(,,)123++ Axxx (,,) 213 Axxx (,,) 321 for any a, b, c . ++Axxxµµµ(,,)132 Axxx (,,) 231 + Axxx (,,)=0. 312 Since (G -p3ss)! is basically isomorphic to (G -p3ss)! we deduce 3 ∈ 3 3 ! 3 The dual operad (G6-p ss) is generated by the relations that (G3-p ss) is not Koszul.
3 (xx12 ) x 3 = x 1 ( xx 23 ), The operad (G -p ss) εσ 4 − () σ ∈Σ (xx1 2 ) x 3 = ( 1) ( xσσ(1) x (2) ) x σ (3) , for all3 . 3 Remark that a (G4-p ss)-algebra is generally called flexible algebra. The relation We deduce
3 ! Axxxµµ(,,)123+ Axxx (,,)=0 321 Proposition 12. A (G6-p ss) -algebra is an associative algebra which satisfies is equivalent to Aµ (x, y, x) = 0 and this denotes the flexibility of (, µ). abc=====,−−− bac cba acb bca cab 3 ! for any a, b, c . In particular Proposition 9. A (G4-p ss) -algebra is an associative algebra satisfying a3 = 0, ∈ abc = −cba. aba= aab = baa = 0. 3 ! 3 ! 3 ! Lemma 13. The operad (G6 - p ss) satisfies (G6 - p ss) (4) = {0}. This implies that dim (G4-p ss) (3) = 3. We have also xxxx= (− 1)εσ() xxxx for any σ Σ . This gives 3 ! σσσσ(1) (2) (3) (4) 1 2 3 4 4 Proof. We have in (G6-p ss) (4) that 3 ! dim (G4-p ss) (4) = 1. Similarly ∈ x1()=()=( xx 23 x 4 x 2 xx 34 x 1−− x 1 xxx 342 )= xxxx 1324 = xxxx 1234
xx12()=()=(())=() xxx 345−− x 3 xxx 452 x 1 x 1 xx 45 xx 23 xx 12 xxx 354 so x1x2x3x4 = 0. We deduce that the generating function of 3 ! = (xx12 ( xx 45 )) x 3 =− ( xx 45 )( xx 21 ) x 3 = ( xxx 321 ) xx 45 (G6-p ss) is − 3 = xxxxx12345 x fx!2( )= x++ x . (the algebra is associative so we put some parenthesis just to explain 6 3 ! If this operad is Koszul the generating function of the operad how we pass from one expression to another). We deduce (G4-p ss) 3 ! (G -p3ss) should be of the form (5) = {0} and more generally (G4-p ss) (a) = {0} for a ≥ 5. 6
3 ! 2311 25 4 127 5 The generating function of (G4-p ss) is fx( )= x++ x x + x + x + 6 6 12
J Generalized Lie Theory Appl Volume 9 • Issue 2 • 1000235 ISSN: 1736-4337 GLTA, an open access journal Citation: Remm E, Goze M (2015) A Class of Nonassociative Algebras Including Flexible and Alternative Algebras, Operads and Deformations. J Generalized Lie Theory Appl 9: 235. doi:10.4172/1736-4337.1000235
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But if we look the free algebra generated by V with dimV =1, it p H ( , )st = 0 for p ≥ 4, 3 (V ). 3 satisfiesa = 0 and coincides with − 3! Then (G -p ss) is not ()G2 p ss 6 Koszul. because the cochain complex is a short sequence
δδ12 1st 2st 30 Proposition 14. We have (,) st → (,) st → (,) st → 0.
3! G6 − p ss= ieAdm The coboundary operator are given by 1 that is the binary quadratic operad whose corresponding algebras δ fab( , ) =fab ( )+− afb ( ) fab ( ), are associative and satisfying δϕ2 (,,)a b c =ϕ ( ab ,) c+−− ϕ ( ba ,) c ϕ (, a bc ) ϕ (, b ac ) ϕϕ(,)abc+−− (, bac ) a ϕϕ (,) bc b (,). ac abc = acb = bac. 3 The deformations cohomology: The minimal model of (G2-p ss) Cohomology and Deformations is a homology isomorphism
3 ρ Let ( ,µ) be a -algebra defined by quadratic relations. It is (G− p ss ,0) → ( Γ (E ), ∂ ) attached to a quadratic linear operad . By deformation of (,µ), we 2 mean [10] of dg-operads such that the image of ∂ consists of decomposable 3 * elements of the free operad Γ(E). Since (G2-p ss)(1) = , this • A non archimedian extension field of , with a valuation v minimal model exists and it is unique. The deformations cohomology such that, if A is the ring of valuation and the unique ideal of A, then * H (,)defo of is the cohomology of the complex the residual field A / is isomorphic to . 123 123(,) →δδ (,) → (,) → δ • The A / vector space is -isomorphic to . defo defo defo where • For any a, b we have that 1 (,) defo =Hom (,), µµ(,)ab− (,) ab kq⊗ ∈ (,) =Hom (⊕⊗≥− E () q Σ ,). defo q22 k q belongs to the -module (isomorphic to ). The Euler characteristic of E(q) can be read off from the inverse of The most important example concerns the case where A is [[t]], 3 ⊗ the generating function of the operad (G2-p ss) =ati ,, a∈ K * the ring of formal series. In this case {∑ i≥1 ii } = ((t)) 2343 5 53 5 g ()=t tt++ t + t + t − 3 the field of rational fractions. This case corresponds to the classical G2 p ss 2 2 12 Gerstenhaber deformations. Since A is a local ring, all the notions of which is valued deformations coincides [11]. t3 13 g(t ) = t−+ t2 + t 56 + Ot( ). We know that there exists always a cohomology which 23 parametrizes deformations. If the operad is Koszul, this cohomology We obtain in particular is the "standard”-cohomology called the operadic cohomology. If the χ(E (4)) = 0. operad is not Koszul, the cohomology which governs deformations (Ep ( )) is based on the minimal model of and the operadic cohomology and Each one of the modules E(p) is a graded module * and deformations cohomology differ [12]. χ(())=dim()Ep E012 p−++ dim() E p dim() E p In this section we are interested by the case of left-alternative We deduce 3 algebras, that is, by the operad (G2-p ss) and also by the classical µ ⋅→ alternative algebras. • E(2) is generated by two degree 0 bilinear operations 2 :VV V, ⊗ Deformations and cohomology of left-alternative algebras • E(3) is generated by three degree 1 trilinear operations VV→ , 3 • E(4) = 0. A -left-alternative algebra (,µ) is a -(G2-p ss)-algebra. Then µ satisfies Considering the action of Σn on E(n) we deduce that E(2) is generated by a binary operation of degree 0 whose differential satisfies Aµµ(,,) xxx123+ A (,,)=0. xxx 213 ∂(µ ) = 0, A valued deformation can be viewed as a [[t]]-algebra (A 2 [[t]], µt) whose product µt is given by E(3) is generated by a trilinear operation of degree one such that ⊗ µµ=.+ ti ϕ ti∑ ∂(µ3 )= µµµµµµτµµτ 2 1 2 − 2 2 2 + 2 1()(). 2 ⋅− 12 2 2 2 ⋅ 12 i≥1
The operadic cohomology:It is the standard cohomology (we have (µ2 2 µτ 2 )⋅ 12 (abc , , )= bac ( )). H * (,) 3 − 3 st of the (G -p ss)-algebra (,µ). It is associated to ()G2 p ss 2 Since E(4) = 0 we deduce the cochains complex * Proposition 15. The cohomology H (,)defo which governs δδδ1 23 1st 23st st (,) st → (,) st → (,) st → deformations of right-alternative algebras is associated to the complex
3 where = (G -p ss) and 1234δδ123 δ 2 (,) defo → (,) defo → (,) defo → (,) defo → pp! ⊗ (,)= Hom (() p ⊗Σ ,). st p with 3 ! Since (G2-p ss) (4) = 0, we deduce that
J Generalized Lie Theory Appl Volume 9 • Issue 2 • 1000235 ISSN: 1736-4337 GLTA, an open access journal Citation: Remm E, Goze M (2015) A Class of Nonassociative Algebras Including Flexible and Alternative Algebras, Operads and Deformations. J Generalized Lie Theory Appl 9: 235. doi:10.4172/1736-4337.1000235
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11⊗ ( , ) =Hom ( V , V ), 23455 1 1 11 6 7 defo x++ x x + x + x − x + Ox( ). 22⊗ ( , )defo =Hom ( V , V ), 6 2 8 72 33⊗ ( , )defo =Hom ( V , V ), Because of the minus sign it can not be the generating function of 4⊗⊗ 55 ! ! (, )defo =Hom ( V ,) V⊕⊕ Hom ( V ,), V the operad =lt .So this implies also that both operad are not Koszul. But it gives also some information on the deformation cohomology. In In particular any 4-cochains consists of 5-linear maps. fact if Γ(E) is the free operad associated to the minimal model, then Alternative algebras dimχ (E (2)) =− 2, Recall that an alternative algebra is given by the relation dimχ (E (3)) =− 5, Axxx(,,)=− Axxx (,,)= Axxx (,,). µ123 µµ213 231 dimχ (E (4)) =− 12, Theorem 16. An algebra (,µ) is alternative if and only if the dimχ (E (5)) =− 15, associator satisfies dimχ (E (6)) =+ 110. Aµ Φv = 0, v =2Id ++++τττ c χ − i with 12 13 23 1 . Since (EE (6)) =∑ i ( 1) dimi (6) , the graded space E(6) is not +τ concentred in degree even. Then the 6-cochains of the deformation Proof. The associator satisfies AAµµΦΦ= with v1 = Id 12 vv12 cohomology are 6-linear maps of odd degree. v =.Id +τ and 2 23 The invariant subspace of [Σ3] generated by v1 and σ References v is of dimension 5 and contains the vector ∑σ∈Σ . From literature of 2 3 v 1. Goze M, Remm E (2007) A class of nonassociative algebras. Algebra Remm [1], the space is generated by the orbit of the vector . Colloquium 14: 313-326.
Proposition 17. Let lt be the operad for alternative algebras. Its 2. Goze M, Remm E (2004) Lie-admissible algebras and operads. Journal of dual is the operad for associative algebras satisfying Algebra 273: 129-152.
abc−−−++ bac cba acb bca cab = 0. 3. Markl M, Remm E (2006) Algebras with one operation including Poisson and other Lie-admissible algebras. Journal of Algebra 299: 171-189. Remark. The current operad lt is the operad for associative 4. Ginzburg V, Kapranov M (1994) Koszul duality for operads. Duke Math Journal algebras satisfying abc==== bac cba acb bca , that is, 3-commutative 76: 203-272. algebras so 5. Markl M, Shnider S, Stasheff J (2002) Operads in algebra, topology and physics. American Mathematical Society, Providence, RI. lt=. ie dm! 6. Markl M, Remm E (2015) (Non-) Koszulness of operads for n-ary algebras, In literature of Dzhumadil’daev and Zusmanovich [9], one gives galgalim and other curiosities. Journal of Homotopy and Related Structures the generating functions of =lt and ! =lt ! 10: 939-969. 2 7 32 175 180 7. Goze M, Remm E (2011) On algebras obtained by tensor product. Journal of g (x ) = x+++ x23 x x 4 + x 5 + x 6 + Ox( 7), 2! 3! 4! 5! 6! Algebra 327: 13-30. 2 5 12 15 8. Goze M, Remm E (2014) Rigid current Lie algebras. Algebra, geometry and g (xx )= +++ x23 x x 4 + x 5. ! 2! 3! 4! 5! mathematical physics, Springer Proc Math Stat 85: 247-258. and conclude to the non-Koszulness of lt. 9. Dzhumadil’daev A, Zusmanovich P (2011) The alternative operad is not Koszul. Experimental Mathematics Formerly 20: 138-144. The operadic cohomology is the cohomology associated to the 10. Goze M, Remm E (2004) Valued deformations. Journal of Algebra and complex Applications 3: 345-365. pp! ⊗ ( ( , ) = (Hom ( lt ( p )⊗Σ , ),δ ). lt st p st 11. Fialowsky A, Schlichenmaier M (2003) Global deformations of the Witt algebra of Krichever-Novikov type. Communications in Contemporary Mathematics 5: ! Since lt (p) = 0 for p ≥ 6 we deduce the short sequence 921-945.
δ1 12. Markl M (2004) Homotopy algebras are homotopy algebras. Forum Math 16: 12 →st → → 5→ lt (,) st lt (,) st lt (,) st 0. 129-160.
But if we compute the formal inverse of the function −−glt ()x we obtain
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