One-parameter deformations of the diassociative and dendriform operads
aa r X i v : . [ m a t h . QA ] F e b ONE-PARAMETER DEFORMATIONS OF THE DIASSOCIATIVEAND DENDRIFORM OPERADS
MURRAY R. BREMNER
Abstract.
Livernet and Loday constructed a polarization of the nonsym-metric associative operad A with one operation into a symmetric operad SA with two operations (the Lie bracket and Jordan product), and defined a one-parameter deformation of SA which includes Poisson algebras. We combinethis with the dendriform splitting of an associative operation into the sum oftwo nonassociative operations, and use Koszul duality for quadratic operads,to construct one-parameter deformations of the nonsymmetric dendriform anddiassociative operads into the category of symmetric operads. The dendriform and diassociative operads
We write O for the free nonsymmetric operad generated by one binary operation( a, b ) ab , and O for the free nonsymmetric operad generated by left and rightbinary operations ( a, b ) a ≺ b and ( a, b ) a ≻ b . Definition 1.1.
The operad morphism split : O → O , which is injective but notsurjective, is determined by its value on the generator: split ( ab ) = a ≺ b + a ≻ b. Definition 1.2.
The (nonsymmetric) associative operad A is the quotient of O by the ideal generated by the associator α = ( ab ) c − a ( bc ). Since α = 0 is aquadratic relation (every term contains two operations), A is a quadratic operad. Lemma 1.3.
The image of the associator under the splitting morphism is: split ( α ) = ( a ≺ b ) ≺ c + ( a ≺ b ) ≻ c + ( a ≻ b ) ≺ c + ( a ≻ b ) ≻ c − a ≺ ( b ≺ c ) − a ≺ ( b ≻ c ) − a ≻ ( b ≺ c ) − a ≻ ( b ≻ c ) . Definition 1.4.
We rearrange split ( α ) into the dendriform splitting of α :[ ( a ≺ b ) ≺ c − a ≺ ( b ≺ c ) − a ≺ ( b ≻ c ) ]+ [ ( a ≺ b ) ≻ c + ( a ≻ b ) ≻ c − a ≻ ( b ≻ c ) ]+ [ ( a ≻ b ) ≺ c − a ≻ ( b ≺ c ) ] . Mathematics Subject Classification.
Primary 17A30. Secondary 15A21, 15A54, 16S80,17A50, 17B63, 18D50.
Key words and phrases.
Diassociative algebras, dendriform algebras, Hermite normal form,LLL algorithm, polarization, deformation, Poisson algebras, algebraic operads, Koszul duality.The author was supported by a Discovery Grant from NSERC, the Natural Sciences andEngineering Research Council of Canada.
Definition 1.5.
The three relations in brackets in Definition 1.4 define the (qua-dratic nonsymmetric) dendriform operad
Dend ; see [1, 9, 10]:( a ≺ b ) ≺ c − a ≺ ( b ≺ c ) − a ≺ ( b ≻ c ) = 0 , ( a ≺ b ) ≻ c + ( a ≻ b ) ≻ c − a ≻ ( b ≻ c ) = 0 , ( a ≻ b ) ≺ c − a ≻ ( b ≺ c ) = 0 . The (quadratic nonsymmetric) diassociative operad
Dias is the Koszul dual of
Dend ; see [8].
Lemma 1.6. [8, 9]
The diassociative operad
Dias is defined by these relations: ( a ≺ b ) ≺ c = a ≺ ( b ≺ c ) = a ≺ ( b ≻ c ) , ( a ≺ b ) ≻ c = ( a ≻ b ) ≻ c = a ≻ ( b ≻ c ) , ( a ≻ b ) ≺ c = a ≻ ( b ≺ c ) . Proof.
In the matrix of the dendriform relations in Definition 1.5, the ( i, j ) entry isthe coefficient of the j -th basis monomial in the i -th relation; the basis monomialsare ordered as in the expansion of Lemma 1.3: " − − −
10 0 1 0 0 0 − To compute the matrix of relations for the Koszul dual operad, we first right-multiply by the block diagonal matrix diag( I , − I ), changing the signs of columns5–8 corresponding to the second placement of parentheses; see [11, Ch. 7]: A = " From this we compute a basis for the null space consisting of short vectors. To finda short basis for the null space of an m × n integer matrix A of rank r we use integerGaussian elimination to construct an invertible n × n integer matrix U for which U A t is the HNF (Hermite normal form) of the transpose A t , extract the submatrix N consisting of the last n − r rows of U , and use the LLL (Lenstra-Lenstra-Lov´asz)algorithm for lattice basis reduction on the rows of N ; see [2, 4]. We obtain: N = − − − −
10 0 1 0 0 0 − This is the coefficient matrix for the diassociative relations. (cid:3)
Definition 1.7.
In Lemma 1.6, the first (second) pair of equations are left (right)associativity and the left (right) bar identity ; the last is inner associativity .General references for algebraic operads are [11] for theory and [3] for algorithms.2.
Polarization of the associative operad
We take a different approach from [7, 12] which emphasizes the method by whichthe shortest defining relations for the polarization of an operation may be discoveredusing the HNF and LLL algorithms.
EFORMATIONS OF THE DIASSOCIATIVE AND DENDRIFORM OPERADS 3
Definition 2.1.
As in Definition 1.1 we write an operation ab as the sum of twoother operations, but now these two operations are defined in terms of ab ; they arethe (scaled) Lie bracket [ a, b ] and the Jordan product a ◦ b :[ a, b ] = ( ab − ba ) , a ◦ b = ( ab + ba ) , ab = [ a, b ] + a ◦ b. The Lie bracket is anti-commutative, [ b, a ] = − [ a, b ], and the Jordan product iscommutative, b ◦ a = a ◦ b ; hence the use of the term polarization : these two newoperations are eigenvectors for the transposition of the arguments in ab . Notation 2.2.
We have passed from the nonsymmetric operad O to its sym-metrization SO : we have introduced permutations of the arguments. The sub-space O (3) of arity 3 in O has dimension 2 and basis { ( ab ) c, ab ( c ) } ; applying all6 permutations to each monomial we see that SO (3) has dimension 12. Definition 2.3.
Using the polarized operations of Definition 2.1, we obtain a differ-ent ordered basis of SO (3): (skew-)symmetry allows us to write every monomialin the form ( a σ ◦ b σ ) ◦ c σ for σ ∈ S where a σ precedes b σ in lex order. The polarized basis for SO (3) is:[[ a, b ] , c ] , [[ a, c ] , b ] , [[ b, c ] , a ] , [ a ◦ b, c ] , [ a ◦ c, b ] , [ b ◦ c, a ] , [ a, b ] ◦ c, [ a, c ] ◦ b, [ b, c ] ◦ a, ( a ◦ b ) ◦ c, ( a ◦ c ) ◦ b, ( b ◦ c ) ◦ a. Lemma 2.4.
The expansion of the associator in terms of the polarized basis is: α ± = (cid:0) [[ a, b ] , c ] + [[ b, c ] , a ] (cid:1) + (cid:0) [ a ◦ b, c ] + [ b ◦ c, a ] (cid:1) + (cid:0) [ a, b ] ◦ c − [ b, c ] ◦ a (cid:1) + (cid:0) ( a ◦ b ) ◦ c − ( b ◦ c ) ◦ a (cid:1) . Proof.
We expand the associator into the polarized basis using ab = [ a, b ] + a ◦ b ,and apply (anti-)commutativity to the last four terms:[[ a, b ] , c ] + [ a ◦ b, c ] + [ a, b ] ◦ c + ( a ◦ b ) ◦ c − [ a, [ b, c ]] − [ a, b ◦ c ] − a ◦ [ b, c ] − a ◦ ( b ◦ c )= [[ a, b ] , c ] + [ a ◦ b, c ] + [ a, b ] ◦ c + ( a ◦ b ) ◦ c + [[ b, c ] , a ] + [ b ◦ c, a ] − [ b, c ] ◦ a − ( b ◦ c ) ◦ a. We then collect terms with the same pattern of polarized operations. (cid:3)
Definition 2.5.
We call α ± the polarized associativity relation . Proposition 2.6.
The S -submodule of SO (3) generated by the polarized associa-tivity relation is also generated by these two relations with only three terms each: [[ a, c ] , b ] + ( a ◦ b ) ◦ c − ( b ◦ c ) ◦ a, [ a ◦ b, c ] − [ a, c ] ◦ b − [ b, c ] ◦ a. Proof.
Our goal is to find the shortest integer vectors in the submodule generatedby α ± , and then from these, find a minimal set of generators for the submodule.We apply the permutations in S to the arguments of α ± and store the results inthe 6 ×
12 matrix P whose ( i, j ) entry is the coefficient of the j -th polarized basismonomial in the application of the i -th permutation (in lex order). We obtain thismatrix whose rows form a basis of the S -submodule of SO (3) generated by α ± : P = − −
10 1 − − − − − − − − − − − − − − − − We want to find the shortest integer vectors in the row space of P . This requirestwo iterations of the method of the previous section to find a short integer basis of MURRAY R. BREMNER the null space: the row space is the null space of the null space. We first computean invertible 12 ×
12 integer matrix U for which U P t is the HNF of P t , and let N be the 6 ×
12 matrix consisting of the bottom half of U . We then compute aninvertible 12 ×
12 integer matrix U for which U N t is the HNF of N t , and let N be the 6 ×
12 matrix consisting of the bottom half of U : N = − − − − − − − − − − − − The squared Euclidean lengths of the rows of N are 3, 4, 3, 3, 3, 3. If we applythe LLL algorithm for lattice basis reduction with higher reduction parameter 9/10(instead of the usual 3/4) then we obtain a lattice basis N for the integer rowspace of N in which every vector has square-length 3; we have also put these basisvectors into upper triangular form: N = − −
10 0 1 0 0 0 0 0 0 1 − − − − We write out the relations corresponding to the rows of N , but recalling that theseare basis vectors, where all we need is a set of module generators:(1) [[ a, b ] , c ] − [[ a, c ] , b ] + [[ b, c ] , a ] = 0 , [[ a, c ] , b ] + ( a ◦ b ) ◦ c − ( b ◦ c ) ◦ a = 0 , [[ b, c ] , a ] + ( a ◦ b ) ◦ c − ( a ◦ c ) ◦ b = 0 , [ a ◦ b, c ] + [ a ◦ c, b ] + [ b ◦ c, a ] = 0 , [ a ◦ b, c ] − [ a, c ] ◦ b − [ b, c ] ◦ a = 0 , [ a ◦ c, b ] − [ a, b ] ◦ c + [ b, c ] ◦ a = 0 . We extract a set of module generators in two stages. First, we retain row i if andonly if the relation it represents does not belong to the submodule generated by theprevious rows: this leaves us with rows 1, 2, 4, 5. Second, we remove each row fromthe set of four generators and compute the submodule generated by the remainingthree relations, retaining a row if and only if the three relations generate a propersubmodule; this leaves us with rows 2 and 5. (cid:3) Definition 2.7.
The relations of Proposition 2.6 are the associator relation andthe derivation relation , since they may be written as follows:( a ◦ b ) ◦ c − a ◦ ( b ◦ c ) = [[ c, a ] , b ] , [ a ◦ b, c ] = [ a, c ] ◦ b + a ◦ [ b, c ] . The first expresses the Jordan associator in terms of the Lie triple product, and thesecond states that the Lie bracket is a derivation of the Jordan product.
EFORMATIONS OF THE DIASSOCIATIVE AND DENDRIFORM OPERADS 5 Deforming the polarization: the Poisson operad
Definition 3.1.
The deformation of the polarization of the symmetrizedassociative operad [7, 12] is defined by these three relations satisfied by an anti-commutative product [ a, b ] and a commutative product a ◦ b :[[ a, b ] , c ] − [[ a, c ] , b ] + [[ b, c ] , a ] = 0 ,q [[ a, c ] , b ] + ( a ◦ b ) ◦ c − ( b ◦ c ) ◦ a = 0 , [ a ◦ b, c ] − [ a, c ] ◦ b − [ b, c ] ◦ a = 0 . This symmetric operad SA q is a module over the polynomial ring F [ q ]. Remark 3.2.
In Definition 3.1, the deformation parameter q appears only in thecoefficient of the first term of the second relation. Otherwise, these relations areidentical to 1, 2, 5 of (1). The first relation is the Jacobi identity for Lie algebras.If q = 0 then the first relation is the alternating sum of the second, but it must beincluded for q = 0. If q = 1 then these relations are equivalent to the associatorand derivation relations of Proposition 2.6. Lemma 3.3.
The HNF of the matrix whose row module over F [ q ] is the S -modulegenerated by the second and third relations in Definition 3.1 is: q − q −
10 0 q − − − − If q = 0 , this matrix has rank 6, but if q = 0 , its rank is only 5. Definition 3.4.
If we set q = 0 in Definition 3.1 then we obtain the relationsdefining a Poisson algebra , which has a commutative associative operation a ◦ b and a Lie bracket [ a, b ] (anticommutative operation satisfying the Jacobi identity),where the Lie bracket acts as derivations of the commutative associative product:[[ a, b ] , c ] + [[ b, c ] , a ] + [[ c, a ] , b ] = 0 , ( a ◦ b ) ◦ c = ( b ◦ c ) ◦ a, [ a ◦ b, c ] = [ a, c ] ◦ b + a ◦ [ b, c ] . Splitting the polarization: deformation of dendriform
Algorithm 4.1.
First, we expand the three relations of Definition 3.1, using thedefinitions of the Lie bracket and Jordan product, into the free symmetric operad SO with one binary operation (not the symmetrized associative operad, since weneed to keep track of the placements of parentheses).Second, we replace the binary operation in SO by the sum of left and right op-erations in SO using the splitting morphism of Definition 1.1, and apply Definition1.4 to decompose each relation into three corresponding dendriform relations.The results belong to SO (3), the 48-dimensional space of arity 3 relations inthe free symmetric operad generated by two binary operations ≺ , ≻ . Notation 4.2.
An ordered basis for SO (3) consists of 8 groups of 6 elements; eachgroup consists of the permutations of a, b, c in lex order applied to the arguments MURRAY R. BREMNER (indicated by dashes) of the following ordered association types:( − ≺ − ) ≺ − , ( − ≺ − ) ≻ − , ( − ≻ − ) ≺ − , ( − ≻ − ) ≻ − , − ≺ ( − ≺ − ) , − ≺ ( − ≻ − ) , − ≻ ( − ≺ − ) , − ≻ ( − ≻ − ) . We write vectors with 48 components as 4 ×
12 matrices in which the ( i, j ) entry isthe k -th entry of the vector for k = 12( i −
1) + j . Lemma 4.3.
After applying part 1 of Algorithm 4.1 to the relations of Definition3.1 we obtain the following elements of SO (3) : − − − − − − − − − − − − − − − − − − − − − − − − q − − q − q − − q − q − − q − q − − q − − − − q q − − − q q − − − q q − − − q q − − − − − − − − − − − − − − − − − − − − − − − − Definition 4.4.
We construct the 18 ×
48 matrix X in which rows 1–6, 7–12, 13–18respectively are obtained by applying all permutations of the arguments a, b, c tothe relations of Lemma 4.3. To apply part 2 of Algorithm 4.1, we partition X into18 × X , . . . , X corresponding to the association types of Notation 4.2,and rearrange them following the dendriform splitting into the 54 ×
48 matrix Y : Y = (cid:2) X X X (cid:3) (cid:2) X X X (cid:3)
00 0 (cid:2) X X (cid:3) The diagonal blocks in Y have sizes 18 ×
18, 18 ×
18, 18 ×
12 but each has rank 6.We compute the HNFs of the diagonal blocks of Y ; after removing zero rows (forthis we write HNF), they have sizes 6 ×
18, 6 ×
18, 6 ×
12; see Figure 1. We writeHNF( Y ) = HNF (cid:0) (cid:2) X X X (cid:3) (cid:1) (cid:0) (cid:2) X X X (cid:3) (cid:1)
00 0 HNF (cid:0) (cid:2) X X (cid:3) (cid:1) We sort the rows of HNF( Y ) so that all the rows containing q come first, and apartfrom this, rows with fewer nonzero entries come first. We retain only those rowswhich do not belong to the S -submodule generated by the previous rows, andobtain the following relations. Theorem 4.5.
The following three relations define a one-parameter deformation ofthe nonsymmetric dendriform operad
Dend into the category of symmetric operads: ( q + 3) (cid:2) (( a ≻ b ) ≺ c ) − ( a ≻ ( b ≺ c )) (cid:3) + ( q − (cid:2) ( a ≻ ( c ≺ b )) − ( b ≻ ( a ≺ c )) + ( b ≻ ( c ≺ a )) − ( c ≻ ( a ≺ b )) (cid:3) = 0 , ( q + 3) (cid:2) (( a ≺ b ) ≻ c ) + (( a ≻ b ) ≻ c ) − ( a ≻ ( b ≻ c )) (cid:3) + ( q − (cid:2) ( a ≻ ( c ≻ b )) − ( b ≻ ( a ≻ c )) + ( b ≻ ( c ≻ a )) − ( c ≻ ( a ≻ b )) (cid:3) = 0 , EFORMATIONS OF THE DIASSOCIATIVE AND DENDRIFORM OPERADS 7 . . . . − . . . . − − . . . . − . . . − . − . . − . − . . − . . . . . . − . − . . . − . − .. . . − . . . − − . . . − − . . . . q +3 . q − − q +1 . − q +1 − q − q − q − − q +1 . − q +1 − q − q − . . . . . q +3 . − q +1 q − − q +1 q − − q − . − q +1 q − − q +1 q − − q − . . . . . . . . − . . . . − . . . − . . . − . − . . − . . . . . . . . . . − . − .. . . − . . . − . . . − − . . . . q +3 . . . . . q +3 . q − − q +1 . − q +1 − q − q − . . . . . q +3 . . . . . q +3 . − q +1 q − − q +1 q − − q − . . . . − . . . . − . . . − . − . . − . . . . . . − . − .. . . − . . . − − . . . . q +3 . q − − q +1 . − q +1 − q − q − . . . . . q +3 . − q +1 q − − q +1 q − − q − Figure 1.
Hermite normal forms of the diagonal blocks of Y ( q + 3) (cid:2) (( a ≺ b ) ≺ c ) − ( a ≺ ( b ≺ c )) − ( a ≺ ( b ≻ c )) (cid:3) + ( q − (cid:2) ( a ≺ ( c ≺ b )) − ( b ≺ ( a ≺ c )) + ( b ≺ ( c ≺ a )) − ( c ≺ ( a ≺ b )) (cid:3) + ( q − (cid:2) ( a ≺ ( c ≻ b )) − ( b ≺ ( a ≻ c )) + ( b ≺ ( c ≻ a )) − ( c ≺ ( a ≻ b )) (cid:3) = 0 . For q = 1 we obtain (4 times) the nonsymmetric dendriform relations. Dualizing the split polarization: deformation of diassociative
The original references on Koszul duality for operads are [5, 6]. To preparefor computing the Koszul dual of the deformed dendriform operad, we multiplyeach column of X by the sign of the corresponding permutation of a, b, c and thenmultiply columns 25–48 (which have the second placement of parentheses) by − X ′ .As before, we partition X ′ into 18 × X ′ , . . . , X ′ corresponding to theassociation types, and rearrange them following the dendriform splitting into the54 ×
48 block diagonal matrix Y ′ : Y ′ = (cid:2) X ′ X ′ X ′ (cid:3) (cid:2) X ′ X ′ X ′ (cid:3)
00 0 (cid:2) X ′ X ′ (cid:3) This implies a certain permutation ξ ∈ S of the basis monomials of SO (3) fromtheir original order, of which we must keep track in order to undo it later.We compute an invertible 48 ×
48 integer matrix U for which U ( Y ′ ) t is theHNF of ( Y ′ ) t over F [ q ], and recalling that Y ′ has rank 18, we define N to be thematrix consisting of the bottom 30 rows of U . This is the only point at whichthe computations become difficult; the entries of N are polynomials in q of degrees0–8 with distribution 40 , , , , , , , ,
16. (We used the Maple packageLinearAlgebra for this.) For example, one of the entries of degree 8 is − (cid:0) q +550 q +2386 q +5230 q +5184 q − q − q +5018 q +13361 (cid:1) . MURRAY R. BREMNER
We compute the HNF of N ; its nonzero entries are ± ± ( q − ± ( q +3): a remark-able improvement. Moreover, the number of nonzero entries in each row is 2, 4 or6. We sort the rows as before (rows containing q first, then by increasing numberof nonzero entries), extract the rows which do not belong to the S -submodulegenerated by the previous rows, and obtain the following relations. Theorem 5.1.
The following five relations define a one-parameter deformation ofthe nonsymmetric diassociative operad
Dias into the category of symmetric operads: ( q + 3) (cid:2) (( a ≻ b ) ≺ c ) − ( a ≻ ( b ≺ c )) (cid:3) + ( q − (cid:2) ( a ≻ ( c ≺ b )) − ( b ≻ ( a ≺ c )) + ( b ≻ ( c ≺ a )) − ( c ≻ ( a ≺ b )) (cid:3) = 0 , ( q + 3) (cid:2) (( a ≻ b ) ≻ c ) − ( a ≻ ( b ≻ c )) (cid:3) + ( q − (cid:2) ( a ≻ ( c ≻ b )) − ( b ≻ ( a ≻ c )) + ( b ≻ ( c ≻ a )) − ( c ≻ ( a ≻ b )) (cid:3) = 0 , ( q + 3) (cid:2) (( a ≺ b ) ≺ c ) − ( a ≺ ( b ≻ c )) (cid:3) + ( q − (cid:2) ( a ≺ ( c ≻ b )) − ( b ≺ ( a ≻ c )) + ( b ≺ ( c ≻ a )) − ( c ≺ ( a ≻ b )) (cid:3) = 0 , ( a ≺ b ) ≻ c − ( a ≻ b ) ≻ c = 0 ,a ≺ ( b ≺ c ) − a ≺ ( b ≻ c ) = 0 . For q = 1 we obtain (4 times) the nonsymmetric diassociative relations. Note thatthe associativities deform but the bar relations do not. References [1]
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Department of Mathematics and Statistics, University of Saskatchewan, Canada
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