On special identities for dialgebras

0.19 MB·English

by  P. S. Kolesnikov

ON SPECIAL IDENTITIES FOR DIALGEBRAS
P.S.KOLESNIKOVANDV.YU.VORONIN
2
1
0
Abstract. For every variety of algebras over a field, there is a natural defi-
2
nitionofacorrespondingvarietyofdialgebras(Loday-type algebras). Inpar-

n ticular,LiedialgebrasareequivalenttoLeibnizalgebras. Weuseanapproach
a based on the notion of an operad to study the problem of finding special
J identities for dialgebras. It is proved that all polylinear special identities for
5 dialgebras can be obtained from special identities for corresponding algebras
2 by means ofasimpleprocedure. A particularcase ofthis resultconfirms the
conjecturebyM.Bremner,R.Felipe,andJ.Sanchez-Ortega,arXiv:1108.0586.

]
A
Q
. 1. Introduction
h
t The notion of a Leibniz algebra appeared first in [2] and later independently
a
m in [14] gave rise to a series of research devoted to the theory of dialgebras. By
definition, a (left) Leibniz algebra is a linear space with a bilinear operation [·,·]
[
which satisfies the Jacobi identity in the form [x,[y,z]]=[[x,y],z]+[y,[x,z]], i.e.,

1 the operator of left multiplication [x,·] is a derivation. This is one of the most
v
studied noncommutative analogues of Lie algebras.
9
Various classes of dialgebras appeared in the literature since they are related to
7
1 Leibniz algebras in the same way as the corresponding classes of ordinary algebras
5 are related to Lie algebras. Associative dialgebras were introduced in [16] as ana-
1. loguesofassociativeenvelopingalgebrasforLeibnizalgebras,alternativedialgebras
0 appeared in [13] in the study of universal central extensions for Leibniz algebras,
2 Jordandialgebras(firstunderthenameofquasi-Jordanalgebras)wereproposedin
1
[19], see also [3] and [11]. All dialgebras of these classes are linear spaces equipped
:
v by two bilinear operations ⊢ and ⊣ such that
i
X (1) (x⊣y)⊢z =(x⊢y)⊢z, x⊣(y ⊢z)=x⊣(y ⊣z).
r
a These identities are common for associative, alternative, Jordan dialgebras men-
tioned above, and they also hold for Leibniz algebras provided that a ⊢ b = [a,b],
a⊣b=−[b,a]. Otherdefiningidentities ofthesevarietiesinitiallyappearedfroma
posteriori considerations motivated by relations with Leibniz algebras. For exam-
ple, a dialgebra is associative if in addition to (1) the following identities hold
x⊢(y ⊢z)=(x⊢y)⊢z, x⊢(y ⊣z)=(x⊢y)⊣z,
x⊣(y ⊣z)=(x⊣y)⊣z.
Then the same space with respect to the new operation [a,b] = a ⊢ b−b ⊣ a is
a Leibniz algebra. A systematical study of this relation between Leibniz algebras
and associative dialgebras may be found in [15].
Anideaofamoreconceptualapproachtothedefinitionofwhatshouldbecalled
a dialgebra associatedwith a givenvariety P of ordinaryalgebras was proposedin
[7] in the case of associative algebras: It was shown that the operad governing the
1
2 P.S.KOLESNIKOVANDV.YU.VORONIN
variety of associative dialgebras in the sense of [16] coincides with the Hadamard
product As⊗Perm, where As is the operad governing the variety of associative
algebras and Perm is the operad governing associative algebras satisfying the left
commutativity relation (xy)z−(yx)z =0.
For an arbitraryvariety P of ordinary algebras with one binary operation (gov-
ernedbyanoperadP),thealgorithmproposedin[11]and[17]allowstodeducethe
defining identities for the class of (di-)algebras governed by the operad P ⊗Perm
starting with the defining identities of P. In [4], this algorithm was generalized
to the case of arbitrary varieties of algebras of any type (i.e., linear spaces with a
family of polylinear operations of arbitrary arity). In this note, we will show that
thisgeneralizedalgorithmalsoleadstotheclassofP⊗Perm-algebras. Thisiswhy
we denote by P ⊗Perm by diP.
Thisfactallowstoconsideraseriesofquestionsdevotedtoelementaryproperties
and relations between various classes of dialgebrasfrom a unified point of view. In
particular, a morphism of operads ω : P → R always gives rise to a functor from
the category of R-algebras to the category of P-algebras. So are the well-known
functors:
R P ω
Associative Lie x x 7→x x −x x
1 2 1 2 2 1
Associative Jordan x x 7→x x +x x
1 2 1 2 2 1
Alternative Jordan x x 7→x x +x x
1 2 1 2 2 1
Alternative Mal’cev x x 7→x x −x x
1 2 1 2 2 1
Associative Jordan triple hx ,x ,x i7→x x x +x x x
1 2 3 1 2 3 3 2 1
system
Jordan Jordan triple hx ,x ,x i7→(x x )x −(x x )x
1 2 3 1 2 3 1 3 2
system +x (x x )
1 2 3
For each triple (P,R,ω) as above the following speciality problem makes sense:
Whether the variety generated by all those P-algebras obtained from R-algebras
coincides with the class of all P-algebras? If no, what are the identities separating
these classes (special identities)? The same question is actual for dialgebras: The
corresponding varieties of diR- and diP-algebras are related by a functor raising
from the morphism ω⊗id of operads.
The purpose of this note is to show that the speciality problem for dialgebras
raising from the triple (diP,diR,ω ⊗id) can always be solved modulo the same
problem for ordinary algebras.
2. The BSO algorithm
Let us start with the construction from [4], assuming the base field k is of zero
characteristic.
Let A be an associative algebra over k equipped by new n-ary operation
(2) ω(x ,...,x )= α x ...x ,
1 n X σ σ(1) σ(n)
σ∈Sn
where α ∈k.
σ
Choose an index i∈{1,...,n}. One may rewrite (2) as
n
ω(x ,...,x )= α x ...x x x ...x ,
1 n XX σ σ(1) σ(j−1) i σ(j+1) σ(n)
j=1 Snji
ON SPECIAL IDENTITIES FOR DIALGEBRAS 3
where Sji — is a set of permutations such that σ(j)=i.
n
Denote byId(ω)thesetofallpolylinearidentities satisfiedforalln-aryalgebras
obtained in this way from associative ones.
Starting from the identities Id(ω), one may canonically construct a set of iden-
tities Id(ω)(2) of type {ω ,...,ω }, where each ω is an n-ary operation. The
1 n i
algorithmofsuchaconstructionwasdescribedin[4](asaKPalgorithm),andalso
in Section 6.
Onthe otherhand,considerthefollowingoperationsonanassociativedialgebra
D:
n
ω (x ,...,x )= α x ⊢···⊢x ⊢x ⊣x ⊣···⊣x ,
i 1 n XX σ σ(1) σ(j−1) i σ(j+1) σ(n)
j=1 Snji
i = 1,...,n (the bracketing is not essential here). The family of n-ary operations
ω ,...,ω obtained are denoted by BSO(ω). Let Id(BSO(ω)) stand for the set
1 n
of all polylinear identities satisfied for all algebras with n-ary operations BSO(ω)
obtained in this way from associative dialgebras.
Problem 1 ([4]). Let chark = 0. Prove that for every choice of ω we have
Id(BSO(ω))=Id(ω)(2).
Next, suppose chark = p > 0. Then the relation in Problem 1 is not valid in
general, but it is reasonable to state
Problem 2 ([4]). For chark = p > 0 and d < p, prove that for every choice of ω
we have Id (BSO(ω)) = Id (ω)(2), where Id (·) stands for the subset of identities
d d d
of degree d in Id(·).
In this paper, we solve these problems.
3. Preliminaries in operads
In this section, we state the necessary notions of the operad theory following
mainly [8], with a particular accent on the operads governing varieties of algebras.
A language Ω is a set of functional symbols {f | i ∈ I} equipped by an arity
i
function ν : f 7→ n ≡ ν(f ) ∈ N. An Ω-algebra is a linear space A over a base
i i i
field k endowed with linear maps fA : A⊗ni → A, i ∈ I [12]. Below, we will use
i
the term algebra of type Ω for an Ω-algebra to avoid confusion. In this paper, we
assume n ≥2.
i
Denote by F hXi the free algebra of type Ω generated by the countable set
Ω
X = {x ,x ,...}. The linear basis of this algebra consists of all terms of type Ω
1 2
in variables from X. Let us call such terms monomials, their linear combinations
(elements of the free algebra) are called polynomials.
Foreveryn∈Nconsiderthe spaceF (n)ofallpolylinearpolynomialsofdegree
Ω
n in x ,...,x . The composition
1 n
γ :F (n)⊗F (m )⊗···⊗F (m )→F (m +···+m )
m1,...,mn Ω Ω 1 Ω n Ω 1 n
of such maps is naturally defined by the rule
γ (f;g ,...,g )=f(g (x ,...,x ),g (x ,...,x ),...),
m1,...,mn 1 n 1 1 m1 2 m1+1 m1+m2
where f(x ,...,x ) ∈ F (n), g (x ,...,x ) ∈ F (m ), i = 1,...,n; the result
1 n Ω i 1 mi Ω i
belongs to F (m + ··· + m ). The simplest term x ∈ F (1) behaves as an
Ω 1 n 1 Ω
4 P.S.KOLESNIKOVANDV.YU.VORONIN
identity with respect to this composition. Symmetric groups S act on F (n) by
n Ω
permutations of variables.
Thecollectionofspaces{FΩ(n)}n∈Ntogetherwithabove-mentionedcomposition
rule, identity element, and S -action is a particular case of an operad which is
n
natural to call the free operad F generated by Ω.
Ω
GiventwooperadsP andR,amorphismα:P →RisjustafamilyofS -linear
n
maps {α(n)}n∈N,
α(n):P(n)→R(n)
preserving the composition and the identity element. The kernel of α is the collec-
tion of subspaces (even S -submodules) Kerα(n) ⊆ P(n), n ∈ N, which is closed
n
with respect to compositions in the obvious sense. Such a family of subspaces is
called an operad ideal in P.
To define a morphism π from F to an operad P it is enough to determine
Ω
π(n )(f ), f = f (x ,...,x ) ∈ F (n ), where f range through the language Ω,
i i i i 1 ni Ω i i
n =ν(f ). Moreover,everyfamily g ∈P(n ), i∈I, defines a unique morphism of
i i i i
operads π :F →P such that π(n )(f )=g .
Ω i i i
EverylinearspaceAgivesrisetoanoperadE(A), theoperadofendomorphisms
of A. Namely, E(A)(n) = Hom(A⊗n,A), n ∈ N, compositions and S -actions are
n
defined in the ordinary way.
A structure ofanalgebraof type Ω ona linear spaceA may be identified with a
morphism of operads α : F →E(A) such that α(n )(f ) =fA, i ∈I. Conversely,
Ω i i i
every morphism α:F →E(A) defines a structure of an algebra on A.
Ω
SupposePisavarietyofalgebrasoftypeΩdefinedbypolylinearidentities(this
is a generic case if chark=0). Then the following consideration makes sense.
Let T(P) be the ideal of identities (T-ideal) in F hXi corresponding to the
Ω
variety P. Denote P(n) = F (n)/(T(P)∩F (n)), n ∈ N. The composition rule
Ω Ω
and Sn-actions are well-defined on the family {P(n)}n∈N, so this collection is also
an operad. Such an operad is said to be the governing operad for the variety P.
There exists a natural quotient morphism π : F → P. If S is a defining family
Ω
of polylinear identities of the variety P then the kernel of π is exactly the operad
ideal generated by S in F .
Ω
EveryalgebraAfromthe varietyP isdeterminedbyacompositionπ◦α¯,where
α¯ is a morphism from P to E(A). Thus, A is defined by a morphism of operads
P → E(A). Conversely, every morphism of this kind defines an algebra structure
on A, and the algebra obtained belongs to P.
In general, given an operad P, a P-algebra is a pair (A,α) of a linear space A
and a morphism of operads α:P →E(A).
4. Conformal algebras
The notion of a conformal algebra was introduced in [10] as tool of vertex op-
erator algebra study. In a more general context, a conformal algebra is a pseudo-
algebra over the polynomial algebra k[T] in one variable [1]. Here we consider the
last approach for arbitrary set of operations Ω.
As we have already mentioned, every linear space A gives rise to the operad
E(A). A similar construction exists for left unital modules over a cocommutative
bialgebra H. Suppose M is such a module, then denote
E∗(M)(n)=Hom (M⊗n,H⊗n⊗ M).
H⊗n H
ON SPECIAL IDENTITIES FOR DIALGEBRAS 5
Hereinafter, the symbol ⊗ without a subscript stands for the tensor product of
spaces over the base field. The space H⊗n is considered as the outer product of
regular right H-modules, i.e.,
(h ⊗···⊗h )·h= h h ⊗···⊗h h ,
1 n X 1 (1) n (n)
(h)
where h ⊗···⊗h is the value of n-iterated coproduct on h. Compositions
(1) (n)
P
(h)
γ ofsuchmapsandtheactionofS onE∗(M)(n)weredefinedin[1],seealso
m1,...,mn n
[11] (one needs cocommutativity of H to ensure the action of S is well-defined).
n
A conformal algebra over H is a pair (C,α), where C is an H-module as above
and α: F →E∗(C) is a morphism of operads. If α splits into F →π P →α¯ E∗(C)
Ω Ω
then C is said to be a P-conformal algebra.
A simple but important example of a conformal algebra may be constructed as
follows. Let (A,α) be a P-algebra. Consider the free H-module C = H ⊗A and
define β =Curα:P →E∗(C) by the rule
β(n)(f):(h ⊗a )⊗···⊗(h ⊗a )7→(h ⊗···⊗h )⊗ α(f)(a ⊗···⊗a ),
1 1 n n 1 n H 1 n
f ∈ P(n), h ∈ H, a ∈ A. This is a morphism of operads, and the P-conformal
k k
algebra (C,β) obtained is denoted (CurA,Curα), the current conformal algebra
over A.
The correspondence A7→CurA is a functor from the category of P-algebras to
the category of P-conformal algebras: Every morphism ϕ between P-algebras can
be continued by H-linearity to the morphism Curϕ of the corresponding current
algebras.
5. The operad Perm
The operad Perm introduced in [7] is given by a family of spaces Perm(n)=kn
with natural composition rule
γ :e(n)⊗e(m1)⊗···⊗e(mn) =e(m1+···+mn) ,
m1,...,mn k j1 jn m1+···+mk−1+jk
where e(n), k =1,...,n, is the standard basis of kn, n∈N. Symmetric groups S
k n
act on Perm(n) by permutations of coordinates.
LetP beanoperad. DenotebydiP theHadamardproductP⊗Perm: diP(n)=
P(n)⊗Perm(n),compositionsandS -actionaredefinedinthecomponentwiseway.
n
Let us fix a cocommutative bialgebra H, and let ε stand for its counit. A left
unital H-module C is in particular a linear space over the base field k. For every
n∈N consider k-linear maps µk, k =1,...,n, from H⊗n⊗ C to C defined by
n H
k
µk :(h ⊗···⊗h )⊗ c7→ε(h .ˆ..h )h c.
n 1 n H 1 n k
Lemma1. If(C,α)isaP-conformalalgebrathenthefamilyofmaps{α(0)(n)}n∈N,
α(0)(n):diP(n)→E(C)(n), defined by
(3) α(0)(n)(f ⊗e(n))=α(n)(f)◦µk,
k n
f ∈P(n), k =1,...,n, a ∈C, defines a morphism α(0) of operads.
j
6 P.S.KOLESNIKOVANDV.YU.VORONIN
Proof. First, note that α(0)(n) is S -linear. Indeed,
n
α(0)(n):(f ⊗e(n))σ =fσ⊗e(n) 7→α(n)(f)σ ◦µσ(k) =(α(n)(f)◦µk)σ
k σ(k) n n
since the action of σ on E∗(C) permutes the arguments of α(n)(f) together with
tensor factors in H⊗n⊗ C, see [11].
H
Next, this is obvious that α(0)(1) preserves the identity.
Finally, consider a composition γ (f;g ,...,g ) in P. By abuse of nota-
m1,...,mn 1 n
tions, assume
α(ml)(gl):a(1l)⊗···⊗a(ml)l 7→F(l)⊗H b(l), F(l) ∈H⊗ml,
l=1,...,n, and
α(n)(f):b(1)⊗···⊗b(n) 7→G⊗ c, G∈H⊗n,
H
a(l),b(l),c∈C. Then by definition
j
(4) α(m +···+m )(γ (f;g ,...,g )):
1 n m1,...,mn 1 n
a(1)⊗···⊗a(1) ⊗···⊗a(n)⊗···⊗a(n)
1 m1 1 mn
7→((F(1)⊗···⊗F(n))⊗H 1)(∆[m1]⊗···⊗∆[mn])G⊗H c,
where ∆[m](h)= h ⊗···⊗h , h∈H.
(1) (n)
P
(h)
Let us fix some k ∈ {1,...,n}, j ∈ {1,...,m }, l = 1,...,n. Suppose F(l) =
l l
h(l) ⊗···⊗h(l), G = h ⊗···⊗h . Then by the properties of the counit ε the
1 ml 1 n
image of the right-hand side of (4) under µm1+···+mk−1+jk is equal to
m1+···+mn
k
(5) ε(F(1)).ˆ..ε(F(n))ε(F(k))ε(G )h(k)h c,
jk k jk k
k
where G =h .ˆ..h and F(k) is defined similarly.
k 1 n jk
On the other hand, let us compute the composition
γ (α(0)(n)(f ⊗e(n));α(0)(m )(g ⊗e(m1)),...,α(0)(m )(g ⊗e(mn)))
m1,...,mn k 1 1 j1 n n jn
in E(C). By (3), we have
α(0)(m )(g ⊗e(ml)):a(l)⊗···⊗a(l) 7→ε(F(l))h(l)b(l).
l l jl 1 ml jl jl
The H⊗n-linearity of α(n)(f) implies
(6) α(n)(f):ε(F(1))h(1)b(1)⊗···⊗ε(F(n))h(n)b(n)
j1 j1 jn jn
7→ε(F(1))...ε(F(n))(h(1)⊗···⊗h(n))G⊗ c.
j1 jn j1 jn H
Thisisnowobviousthatthe imageofthe right-handsideof (6)underµk coincides
n
with (5). (cid:3)
Hence, every P-conformal algebra (C,α) gives rise to a diP-algebra C(0) =
(C,α(0)). The correspondence C 7→ C(0) is obviously a functor from the category
of P-conformal algebras to the category of diP-algebras.
ON SPECIAL IDENTITIES FOR DIALGEBRAS 7
6. Dialgebras
Suppose P is a quotient operad of F , π is the corresponding morphism. This
Ω
is easy to see that diP is a quotient of diF = F ⊗Perm with respect to the
Ω Ω
morphism π⊗id. In general, this is hard to determine the generatorsand defining
relations of the Hadamard product of two operads, but due to the nice properties
of Perm this is easy to do for diP.
Consider the free operad F , where the language Ω(2) is constructed in the
Ω(2)
following way. If Ω = {f | i ∈ I}, n = ν(f ), then Ω(2) = {fk | i ∈ I, k =
i i i i
1,...,n }, ν(fk)=n .
i i i
Define the morphism ζ : F → diF in the following way: ζ (n) maps
Ω Ω(2) Ω Ω
fk(x ,...,x ) ∈ F (n ) to f (x ,...,x )⊗e(ni). The composition of ζ with
i 1 ni Ω(2) i i 1 ni k Ω
π⊗id provides a morphism π(2) :F →diP.
Ω(2)
Lemma 2. For every n∈N the linear maps ζ (n) and π(2)(n) are surjective.
Ω
Proof. To prove the surjectivity of ζ (and hence of π(2)) it is enough to show
Ω
that diF is generated by f ⊗e(ni), i ∈ I, k = 1,...,n . In the binary case it
Ω i k i
was actually done in [18] and [11], the general case can be processed analogously.
It is enough to construct a section ρ(n) : (F ⊗Perm)(n) → F (n) such that
Ω Ω(2)
ρ(n)◦ζ (n)=id. It was done in [4], let us recall here the construction in terms of
Ω
planartrees. Everymonomialf ∈F (n)canbeidentifiedwithaplanartreewithn
Ω
leaves(variables)labeledbynumbers1,...,nandverticeslabeledbysymbolsfrom
Ω, the degree (number of out-coming branches) of a vertex labeled by f ∈ Ω is
i
equalton . Thenf⊗e(n) maybeconsideredasatreewithkthemphasizedvertex.
i k
Togetρ(n)(f⊗e(n))weshouldaddsuperscriptstothelabelsofverticesinthetree
k
correspondingtof inthefollowingway. Ifakth(countingfromthe left-handside)
out-coming branchof a vertex labeled by f ∈Ω contains the emphasized leaf then
i
the label is replaced with fk. If neither of the out-coming branches in this vertex
i
contain the emphasized leaf then the label is replaced with f1. (cid:3)
i
SupposeS isasetofpolylinearpolynomialssuchthatthekernelofπisgenerated
by S (e.g., if P is a governingoperadfor a variety P then S consists ofits defining
identities). Consider the operad ideal J(S) in F generated by
Ω(2)
(7) fk(x1,...,xj−1,gl(xj,...,xj+m−1),xj+m,...,xn+m−1)
−fk(x1,...,xj−1,gp(xj,...,xj+m−1),xj+m,...,xn+m−1),
f,g ∈Ω, n=ν(f), m=ν(g), k,j =1,...,n, k 6=j, l,p=1,...,m,
and
(8) sk(x ,...,x ), s∈S∩F (n), n∈N, k=1,...,n,
1 n Ω
wheresk =ρ(n)(s⊗e(n)). DenotebyP(2) thequotientoperadofF withrespect
k Ω(2)
to J(S), and let πˆ(2) be the corresponding morphism from F to P(2).
Ω(2)
This is easy to see that J(S) is contained in the kernel of π(2). Thus we have
the following commutative diagram:
F −−−ζΩ−→ F ⊗Perm −−−p−r→ F
Ω(2) Ω Ω
πˆ(2) π⊗id π
  
Py(2) −−−−→ P ⊗yPerm −−−p−r→ Py
8 P.S.KOLESNIKOVANDV.YU.VORONIN
where pr stands for the natural projection.
Our aim is to show that the kernels of πˆ(2) and π(2) are equal.
Suppose (A,α) is a P(2)-algebra. By abuse of notations, let us identify fk ∈
F (n) and their images under πˆ(2)(n) in P(2)(n).
Ω(2)
Let A be the k-linear span of all
0
α(n)(fp−fl)(a ,...,a ), i∈I, a ∈A, p,l =1,...,n .
i i 1 ni j i
It follows from (7) that A is an ideal in the algebra A. Indeed, for every f ∈Ω
0
α(n)(fk)(a1,...,aj−1,b,aj+1,...,an)=0, b∈A0,
ifj 6=k(f ∈Ω,n=ν(f),k =1,...,n). Forj =k,onemayaddα(n)(fq)(a1,...,aj−1,
b,a ,...,a ) with q 6=k (which is zero) and make sure the result is againin A .
j+1 n 0
Denote A¯=A/A . The morphism α induces a morphism α¯ :P(2) →E(A¯), such
0
that (A¯,α¯) is the quotient P(2)-algebra. In this algebra, the values of algebraic
operations α(n)(fk), f ∈ Ω, n = ν(f), k ∈ {1,...,n} do not depend on k, so this
is actually an algebra of language Ω:
fA¯(a¯ ,...,a¯ )=α(n)(fk)(a ,...,a ), f ∈Ω, n=ν(f),
1 n 1 n
where a ∈ A, a¯ = a + A ∈ A¯. Moreover, it is obvious that A¯ is actually a
j 0
P-algebra.
Consider the formal direct sum of spaces Aˆ=A¯⊕A and define algebraic oper-
ations of language Ω on Aˆ as follows:
gA¯(z ,...,z ), z ∈A¯ for all i=1,...,n;
1 n i

gAˆ(z1,...,zn)=α(n)(gk)(a1,...,an), zi =a¯i ∈A¯ for all i6=k,
z =a ∈A;
k k
0, more than one zi ∈A.
g ∈Ω, ν(g)=n. Denote by αˆ the corresponding morphism from F to E(Aˆ).
Ω
ThedefinitionofthecanonicalsectionρfromLemma2andinductiononnimply
that for every s∈F (n), sAˆ :=αˆ(n)(s), we have
Ω
sAˆ(z ,...,z )=ρ(n)(s⊗e(n))(a ,...,a )∈A⊆Aˆ
1 n k 1 n
if z =a¯ ∈A¯ for all i6=k and z =a ∈A. Therefore, every s ∈S is an identity
i i k k
on Aˆ, so (Aˆ,αˆ) is actually a P-algebra.
Theorem 1. The kernels of π(2) and πˆ(2) coincide, so the operads P(2) and diP
are equivalent.
Proof. We have already seen that the kernel of πˆ(2) is contained in the kernel of
π(2). Conversely,assumethereexistsanidentity thatholdsonalldiP-algebrasbut
does not hold on some P(2)-algebra (A,α).
Consider the P-algebra (Aˆ,αˆ) constructed above and fix a bialgebra H with a
nonzero T ∈ H such that ε(T) = 0. For example, one may consider the group
algebra H =kZ .
2
The current conformal algebra CurAˆ = H ⊗Aˆ is a P-conformal algebra. By
Lemma 1, (CurAˆ)(0) is a diP-algebra. Note that
A→(CurAˆ)(0), a7→1⊗a¯+T ⊗a, a∈A
isaninjectivehomomorphismofΩ(2)-algebras. Hence, AisinfactdiP-algebraand
thus satisfies all identities that hold on the class of such algebras. (cid:3)
ON SPECIAL IDENTITIES FOR DIALGEBRAS 9
Corollary 1. Every diP-algebraA is embedded into(CurAˆ)(0) over an appropriate
bialgebra H, Aˆ is a P-algebra.
Lemma3. Consider t=t(x ,...,x )∈diP(n). Then t=t ⊗e(n)+···+t ⊗e(n),
1 d 1 1 n n
t ∈ P(n). Let (A,α) be a P-algebra. Then the dialgebra (CurA)(0) satisfies the
k
identity t=0 if and only if A satisfies all identities t =0, k =1,...,d.
k
Proof. Itfollowsfromtheconstructionthatifα(n)(t )=0forallk =1,...,nthen
k
(Curα)(n)(t )=0 and hence (Curα)(0)(n)(t)=0.
k
Conversely, consider g =(Curα)(0)(n)(t)∈E(H ⊗A)(n) and compute
b =g(1⊗a ,...,T ⊗a ,...,1⊗a ), k =1,...,n,
k 1 k n
foralla ,...,a ∈A. Ontheonehand,b =0since(CurA)(0) satisfiestheidentity
1 n k
t = 0. On the other hand, b = T ⊗α(n)(t )(a ,...,a ), so A satisfies t = 0 for
k k 1 n k
all k. (cid:3)
7. Morphisms of operads and functors
If P and R are two operads then every morphism α : P → R gives rise to a
functor from the category of R-algebras to the category of P-algebras. Namely,
if (A,β) is an R-algebra then the same space A with respect to the composition
α ◦ β : P → E(A) is a P-algebra. The correspondence (A,β) 7→ (A,α ◦ β) is
obviously functorial. The construction that appears in Problem 1 is a particular
case of such a functor.
Indeed, assume Ω and Ξ are two languages, F and F are two corresponding
Ω Ξ
free operads. Suppose π :F →P and ρ:F →R are two quotient morphisms to
Ω Ξ
operads P and R governing some varieties of algebras.
Letω :F →F beamorphismofoperads. Thefamilyofmapsω(n):F (n)→
Ω Ξ Ω
F (n)determines(andcanbecompletelydeterminedby)aninterpretationofoper-
Ξ
ations from Ω via operations from Ξ. We say that ω induces a morphism ω¯P →R
iff Kerπ(n)⊆Ker(ω(n)◦ρ(n)) for all n∈N.
Example 1. Let Ω and Ξ contain one binary operation denoted by λ in Ω and µ
inΞ,thenthemorphismω determinedbytheruleλ=µ−µ(12),(12)∈S ,induces
2
a morphism ω¯ from the operad Lie (governing the variety of Lie algebras) to the
operadAs (associativealgebras). If (A,α) is anAs-algebrathenthe pair (A,ω¯◦α)
is exactly the adjoint Lie-algebra of A (usually denoted by A(−)).
The Hadamard product ω¯⊗id:diLie→diAs defines the corresponding functor
fromthe categoryof associativedialgebrasto the categoryof Leibniz algebras(Lie
dialgebras) [15].
AsimilarrelationholdsforMal’cevdialgebras[6]andalternativedialgebras[13].
Example 2. LetJTSbethe operadgoverningthe varietyofJordantriplesystems
(see, e.g., [9]). Then there exists ω¯ : JTS →As defined as follows: If τ = (·,·,·) ∈
JTS(3) is the triple operation on JTS-algebras and µ ∈ As(2) is the product on
associative algebras then ω¯(τ) = γ (µ;id,µ)+γ (µ;id,µ)(13). This is the well-
1,2 1,2
known construction of a Jordan triple system on an associative algebra: (a,b,c)=
abc+cba.
In [4], the notion of a Jordan triple disystem (JTD) was introduced in such a
way that JTD=JTS(2), in our notations. Theorem 1 immediately implies JTD=
diJTS=JTS⊗Perm. Hence, ω¯⊗id:JTD→diAs defines a structure of a Jordan
triple disystem on an associative dialgebra (c.f. [4, Theorem 5.10]).
10 P.S.KOLESNIKOVANDV.YU.VORONIN
Example3. LetJordstandfortheoperadgoverningthevarietyofJordanalgebras,
µ ∈ Jord(2) is the commutative operation. Then there exists a morphism ω¯ :
JTS→ Jord defined by ω¯(τ) = γ (µ;µ,id)−γ (µ;µ,id)(23)+γ (µ;id,µ), i.e.,
2,1 2,1 1,2
(a,b,c)=(ab)c−(ac)b+a(bc).
The notion of a Jordan dialgebra was studied in [19, 3, 11], see also [5]. As in
the previousexample, Theorem1 gives a new proofof the relationbetween Jordan
dialgebras and Jordan triple disystems [4, Theorem 7.3].
Let us fix two quotient morphisms π : F → P, ρ : F → R, and a morphism
Ω Ξ
ω : F → F inducing a morphism from P to R which is also denoted by ω for
Ω Ξ
simplicity.
Definition 1. A P-algebra (A,α) is called ω-special if there exists an R-algebra
(A,β) such that α = ω◦β. The same notion makes sense for conformal algebras
over an arbitrary cocommutative bialgebra H.
Lemma 4. If (A,α) is an ω-special P-algebra then (CurA,Curα) is an ω-special
P-conformal algebra.
Proof. SupposethereexistsanR-algebra(A,β),β :R→E(A),suchthatα=ω◦β.
Then the claim follows from the observation
(9) ω◦Curβ =Cur(ω◦β).
Indeed,foreveryf ∈P(n)thepseudo-linearmaps(ω◦Curβ)(n)(f)∈E∗(H⊗A)are
completelydefinedbytheirvaluesat(1⊗a ,...,1⊗a ),a ∈A. Bythedefinitionof
1 n i
Cur,wehave(Curβ)(n)(g)(1⊗a ,...,1⊗a )=(1⊗···⊗1)⊗ (β(n)(g))(a ,...,a )
1 n H 1 n
for every g ∈ R(n), in particular, for g = ω(n)(f). This is now easy to see that
left- and right-hang sides of (9) coincide at every f ∈P(n). (cid:3)
Lemma 5. If (C,α) is an ω-special P-conformal algebra then (C,α(0)) is an (ω⊗
id)-special diP-algebra.
Proof. It is enough to show (ω◦β)(0) = (ω⊗id)◦β(0) for every β : R → E∗(C).
Relation (3) implies
(ω◦β)(0)(n):f ⊗e(n) 7→(ω◦β)(n)(f)◦µk =β(n)(ω(n)(f))◦µk
k n n
for every f ∈P(n), k =1,...,n. On the other hand,
((ω⊗id)◦β(0))(n):f ⊗e(n) 7→β(0)(n)(ω(n)(f)⊗e(n))=β(n)(ω(n)(f))◦µk.
k k n
(cid:3)
The class of all ω-special P-algebrasis closed under Cartesianproducts. There-
fore,the classofallhomomorphicimagesofallsubalgebrasofω-specialP-algebras
isavarietyS. Considerthesetofpolylinearidentitiesthatholdonthisvarietyand
define the corresponding operad SωP. This is a quotient operad of P, and there
existsamorphismSω :P →SωP. EveryP-algebrafromSisanSωP-algebra,but
the converse may not be true if chark>0.
Lemma 6. Consider the class of all quotient operads P′ of P satisfying the fol-
lowing property: For every morphism α : R → E(A) there exists a morphism