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arXiv:2306.02812v3 [math.CT] 6 Feb 2025
WEAK REPRESENTABILITY OF ACTIONS
OF NON-ASSOCIATIVE ALGEBRAS
J. BROX, X. GARCÍA-MARTÍNEZ, M. MANCINI, T. VAN DER LINDEN, AND C. VIENNE
Abstract. We study the categorical-algebraic condition that internal actions
are weakly representable (WRA) in the context of varieties of (non-associative)
algebras over a field.
Our first aim is to give a complete characterization of action accessible, op-
eradic quadratic varieties of non-associative algebras which satisfy an identity
of degree two and to study the representability of actions for them. Here we
prove that the varieties of two-step nilpotent (anti-)commutative algebras and
that of commutative associative algebras are weakly action representable, and
we explain that the condition (WRA) is closely connected to the existence of
a so-called amalgam.
Our second aim is to work towards the construction, still within the context
of algebras over a field, of a weakly representing object EpXqfor the actions
on (or split extensions of) an object X. We actually obtain a partial algebra
EpXq, which we call external weak actor of X, together with a monomorph-
ism of functors SplExtp´, XqHompUp´q,EpXqq, which we study in detail
in the case of quadratic varieties. Furthermore, the relations between the con-
struction of the universal strict general actor USGApXqand that of EpXqare
described in detail. We end with some open questions.
Introduction
In the article [3], F. Borceux, G. Janelidze and G. M. Kelly introduce the concept
of an internal object action, with the aim of extending the correspondence between
actions and split extensions from the context of groups and Lie algebras to arbitrary
semi-abelian categories [19]. In certain of those categories, internal actions are
exceptionally well behaved, in the sense that the actions on each object Xare
2020 Mathematics Subject Classification. 08A35; 08C05; 16B50; 16W25; 17A32; 17A36;
18C05; 18E13.
Key words and phrases. Action representable category, amalgamation property, split extension,
non-associative algebra, partial algebra, quadratic operad.
The first author is supported by a postdoctoral fellowship “Convocatoria 2021” funded by
Universidad de Valladolid, and partially supported by grant PID2022-137283NB-C22 funded by
MCIN/AEI/10.13039/501100011033 and ERDF “A way of making Europe”. The second author
is supported by Ministerio de Economía y Competitividad (Spain) with grant number PID2021-
127075NA-I00. The third author is supported by the University of Palermo, by the “National
Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA – INdAM),
by the National Recovery and Resilience Plan (NRRP), Mission 4, Component 2, Investment
1.1, Call for tender No. 1409 published on 14/09/2022 by the Italian Ministry of University and
Research (MUR), funded by the European Union – NextGenerationEU – Project Title Quantum
Models for Logic, Computation and Natural Processes (QM4NP) – CUP B53D23030160001 –
Grant Assignment Decree No. 1371 adopted on 01/09/2023 by the Italian Ministry of Ministry
of University and Research (MUR), by the SDF Sustainability Decision Framework Research
Project – MISE decree of 31/12/2021 (MIMIT Dipartimento per le politiche per le imprese –
Direzione generale per gli incentivi alle imprese) – CUP: B79J23000530005, COR: 14019279, Lead
Partner: TD Group Italia Srl, Partner: Università degli Studi di Palermo, and he is a postdoctoral
researcher of the Fonds de la Recherche Scientifique–FNRS. The fourth author is a Senior Research
Associate of the Fonds de la Recherche Scientifique–FNRS. The fifth author is supported by the
Fonds Thelam of the Fondation Roi Baudouin.
1
2 J. BROX, X. GARCÍA-MARTÍNEZ, M. MANCINI, T. VAN DER LINDEN, AND C. VIENNE
representable: this means that there exists an object rXs, called the actor of X,
such that the functor Actp´, Xq – SplExtp´, X q, which sends an object Bto
the set of actions/split extensions of Bon/by X, is naturally isomorphic to the
functor Homp´,rXsq. The context of action representable semi-abelian categories
is further studied in [4], where it is for instance explained that the category of
commutative associative algebras over a field is not action representable. Later it
was shown that the only action representable variety of non-associative algebras over
an infinite field Fof characteristic different from 2is the variety of Lie algebras [14].
The relative strength of the notion naturally led to the definition of closely related
weaker notions.
The first of these was the concept of an action accessible category due to D. Bourn
and G. Janelidze [6]: it is weak enough to include all Orzech categories of in-
terest [29], as proved by A. Montoli in [28].
Alternatively, the properties of the representing object rXsmay be weakened;
this is the aim in [8], where it is shown that each Orzech category of interest admits
a so-called universal strict general actor (USGA for short).
Our present article focuses on a concept which was more recently introduced,
by G. Janelidze in [17]: weak representability of actions (WRA). Instead of asking
that for each object Xin a semi-abelian category Cwe have an object rXsand a
natural isomorphism SplExtp´, X q – HomCp´,rXsq, we require the existence of
an object Tand a monomorphism of functors
τ: SplExtp´, X qHomCp´, T q.
Such an object Tis then called a weak actor of X, and when each Xadmits a
weak actor, Cis said to be weakly action representable. For instance, if in an
Orzech category of interest, each USGApXqis an object of the category, then this
category is weakly action representable [10]. This is the case of the category Assoc
of associative algebras [17] or the category Leib of Leibniz algebras [10] over a field.
J. R. A. Gray observed in [15] that an Orzech category of interest need not be
weakly action representable. One of our aims in the present article is to study the
condition (WRA) in the context of varieties of (non-associative) algebras over a
field. (We recall basic definitions and results concerning this setting in Section 1.)
It is known that such a variety is action accessible if and only if it is algebraic-
ally coherent [14], and it is also known [17] that action accessibility is implied by
(WRA). In Section 2 we give a complete classification of the action accessible, op-
eradic quadratic varieties of non-associative algebras with an identity of degree 2
(so commutative or anti-commutative algebras) and we study the representability
of actions of each of them. Moreover, we prove that the variety of commutative
associative algebras, the variety of two-step nilpotent commutative algebras and
that of two-step nilpotent anti-commutative algebras are weakly action represent-
able categories. For the variety of commutative associative algebras, we show that
the existence of a weak representation is closely connected to the amalgamation
property (AP) [20] which already appeared in [4] in relation to action representab-
ility. In Section 3 the study of (WRA) and its relations with the condition (AP) is
extended to a general variety of algebras over a field.
Our second aim is to work towards the construction, still within the context of
algebras over a field, of a weakly representing object EpXqfor the actions on/split
extensions of an object Xof a variety of non-associative algebras V. We believe
that in certain settings, this ob ject may be easier to work with than the more
abstract weak actor. In Definition 3.3 we actually obtain a partial algebra EpXq,
which we call external weak actor of X, together with a monomorphism of functors
SplExtp´, X qHompUp´q,EpXqq, where Uis the forgetful functor from Vto
WEAK REPRESENTABILITY OF ACTIONS OF NON-ASSOCIATIVE ALGEBRAS 3
the category of partial algebras, which we study in detail in the case of quadratic
varieties of algebras (Section 4).
We end the article with some open questions (Section 5).
1. Preliminaries
The present work takes place in semi-abelian categories which were introduced
in [19] in order to capture categorical-algebraic properties of non-abelian algebraic
structures. A category is semi-abelian if it is pointed, admits binary coproducts,
is protomodular and Barr-exact. Well-known examples are the category Grp of
groups, the category Rng of not necessarily unitary rings, any variety Vof non-
associative algebras over a field F, as well as all abelian categories. Throughout
the remainder of the paper, when we consider a category C, we assume it to be
semi-abelian; when we consider a variety V, we assume that Vis a variety of non-
associative algebra over a field F. We fix the field F, so that we may drop it from
our notation.
Internal actions and their representability. A central notion which appears
in the semi-abelian context is that of split extensions. Let X,Bbe objects of a
semi-abelian category C; a split extension of Bby Xis a diagram
0X A B 0
kα
β(1.1)
in Csuch that α˝β“1Band pX, kqis a kernel of α. We observe that since
protomodularity implies that the pair pk, β qis jointly strongly epic, the morphism
αis indeed the cokernel of kand diagram (1.1) represents an extension of Bby X
in the usual sense. Morphisms of split extensions are morphisms of extensions
that commute with the sections. Let us observe that, again by protomodularity, a
morphism of split extensions fixing Xand Bis necessarily an isomorphism. For an
object Xof C, we define the functor
SplExtCp´, X q:Cop ÑSet
which assigns to any object Bof C, the set SplExtCpB, X qof isomorphism classes
of split extensions of Bby Xin C, and to any arrow f:B1ÑBthe change
of base function f˚: SplExtCpB, X q Ñ SplExtCpB1, Xqgiven by pulling back
along f. When there is no ambiguity on the category C, we will use the nota-
tion SplExtp´, Xq.
A feature of semi-abelian categories is that one can define a notion of internal
action. Internal actions correspond to split extensions via a semidirect product
construction; it turns out that, as a result, for our purposes we need no explicit
description of what is an internal action. We refer the interested reader to [4],
where the equivalence between the two concepts is described in detail. For us here,
it suffices to note that if we fix an ob ject X, internal actions on Xin Cgive rise
to a functor
Actp´, Xq “ ActCp´, X q:Cop ÑSet
and a natural isomorphism of functors Actp´, X q – SplExtp´, X q. This justifies
the terminology in the definition that follows.
Definition 1.1 ([4]).A semi-abelian category Cis said to be action repres-
entable if for every object Xin it the functor Actp´, X qis representable. In
other words, there exists an object rXsin C, called the actor of X, and a natural
isomorphism
SplExtp´, Xq – HomCp´,rXsq.
4 J. BROX, X. GARCÍA-MARTÍNEZ, M. MANCINI, T. VAN DER LINDEN, AND C. VIENNE
Basic examples of semi-abelian categories which satisfy action representability
are the category Grp of groups with the actor of Xbeing the group of automorph-
isms AutpXq, the category Lie of Lie algebras with the actor of Xbeing the Lie
algebra of derivations DerpXq, and any abelian category with the actor of Xbe-
ing the zero object. For the categories Assoc of associative algebras and CAssoc
of commutative associative algebras, representability of actions was studied in [4],
where the authors proved that they are not action representable.
It is explained in [3] that action representability is equivalent to the condition
that for every object Xin Cthe category SplExtpXqof split extensions in Cwith
kernel Xhas a terminal object
0,2X,2rXs ˙ X,2rXs
lr,20.
We can weaken this condition assuming instead that for any X, every object
in SplExtpXqis accessible (i.e. it has a morphism into a subterminal or so-called
faithful object, see [6]). In this way, we encompass a wider class of examples
that did not satisfy representability of actions such as the category Pois of (non-
commutative) Poisson algebras, the category Assoc of associative algebras or the
category CAssoc of commutative associative algebras. This notion called action
accessibility was introduced by D. Bourn and G. Janelidze [6] in order to calculate
centralisers of normal subobjects or of equivalence relations. It was then shown by
A. Montoli that any Orzech category of interest is an action accessible category [28].
This explains why all of the varieties of non-associative algebras mentioned above
are action accessible.
Since by definition the existence of a terminal object in SplExtpXqis stronger
than every object being accessible, it is immediate that
action representability ñaction accessibility.
Recently, in [17], G. Janelidze introduced an intermediate notion: weak represent-
ability of actions.
Definition 1.2. A semi-abelian category Cis said to be weakly action repres-
entable (WRA) if for every object Xin it, there exists an object Tof Cand a
monomorphism of functors
τ: SplExtp´, X qHomCp´, T q.
We call such an object Taweak actor of X, and a morphism ϕ:BÑTin the
image of τBan acting morphism.
It is clear from the definitions that every action representable category is weakly
action representable. Also in [17], it is proven that the category Assoc is weakly
action representable with a weak actor of Xgiven by the associative algebra
BimpXq “ tpf˚ ´,´ ˚ fq P EndpXq ˆ EndpXqop |f˚ pxyq “ pf˚xqy,
pxyq ˚ f“xpy˚fq, xpf˚yq “ px˚fqy, @x, y PXu
of bimultipliers of X(see [25]). The case of the category Leib of Leibniz algebras
was studied in [10]. There the authors showed that a weak actor of a Leibniz
algebra Xis the Leibniz algebra
BiderpXq “ tpd, Dq P EndpXq2|dpxyq “ dpxqy`xdpyq,
Dpxyq “ Dpxqy´Dpyqx, xdpyq “ xDpyq,@x, y PXu
of biderivations of X(see [24] and [26]), where the bilinear operation is defined
by
rpd, Dq,pd1, D1qs “ pd˝d1´d1˝d, D ˝d1´d1˝Dq.
WEAK REPRESENTABILITY OF ACTIONS OF NON-ASSOCIATIVE ALGEBRAS 5
In the same paper, the representability of actions in the categories Pois and CPois
of (commutative) Poisson algebras was studied.
Another important observation made by G. Janelidze is that every weakly action
representable category is action accessible. We thus have that
action representability ñweak action representability ñaction accessibility.
J. R. A. Gray proved in [15] that the varieties of n-solvable groups where ną3
are action accessible but not weakly action representable. This partially answers
a question asked by G. Janelidze in [17], whether reasonably mild conditions may
be found on a semi-abelian category under which the second implication may be
reversed: already it makes clear that not all action accessible semi-abelian varieties
are weakly action representable. Our aim here is to study what happens for a
different class of categories, namely varieties of not necessarily associative algebras
over a field.
Varieties of non-associative algebras. We now recall the algebraic setting we
are working in: varieties of non-associative algebras over a field F. We think of
those as collections of algebras satisfying a chosen set of polynomial equations.
The interested reader can find a more detailed presentation of the subject in [33].
By a (non-associative) algebra Awe mean a vector space Aequipped with
a bilinear operation AˆAÑA:px, y q ÞÑ xy which we call the multiplication.
The existence of a unit element is not assumed, nor are any other conditions on the
multiplication besides its bilinearity. Let Alg denote the category of non-associative
algebras, where morphisms are linear maps which preserve the multiplication.
We consider the free algebra functor Set ÑAlg which sends a set Sto the free
algebra generated by elements of S. This functor has the forgetful functor as a right
adjoint. Moreover, it factorises through the free magma functor FM : Set ÑMag,
which sends a set Sto the magma FMpSqof non-associative words in S, and the
magma algebra functor Fr´s :Mag ÑAlg.
Let Sbe a set. An element ϕof FrFMpSqs is called a non-associative poly-
nomial on S. We say that such a polynomial is a monomial when it is a scalar
multiple of an element in FMpSq. For example, if S“ tx, y, z, tu, then pxyqt`pzyqx,
xx`yz and pxtqpyzqare polynomials in Sand only the last one is a monomial. For a
monomial ϕon a set tx1,...,xnu, we define its type as the n-tuple pk1,...,knq P Nn
where kiis the number of times xiappears in ϕ, and its degree as the natural
number k1` ¨ ¨ ¨ ` kn. A polynomial is said to be multilinear if all monomials
composing it have the same type of the form p1,...,1q. Among the examples above,
only the last one is multilinear.
Definition 1.3. An identity of an algebra Ais a non-associative polynomial
ϕ“ϕpx1,...,xnqsuch that ϕpa1,...,anq “ 0for all a1, ..., anPA. We say that
the algebra Asatisfies the identity ϕ.
Let Ibe a subset of FrFMpSqs with Sbeing a set of variables. The variety of
algebras determined by Iis the class of all algebras which satisfy all the identities
in I. We say that a variety satisfies the identities in Iif all algebras in this
variety satisfy the given identities. In particular, if the variety is determined by a
set of multilinear polynomials, then we say that the variety is operadic. If there
exists a set of identities of degree 2or 3that generate all the identities of V, we say
that the variety is quadratic. Recall—see for instance [11] where this is explained
in detail—that an operadic, quadratic variety of algebras over a field can be viewed
as a variety determined by a quadratic operad.
Any variety of non-associative algebras can, of course, be seen as a category
where the morphisms are the same as in Alg. In particular, any such variety is a
semi-abelian category.
6 J. BROX, X. GARCÍA-MARTÍNEZ, M. MANCINI, T. VAN DER LINDEN, AND C. VIENNE
Remark 1.4.Whenever the characteristic of the field Fis zero, any variety of non-
associative algebras over Fis operadic. This is due to the well-known multilinear-
isation process, see [30, Corollary 3.7]. The reason behind the name “operadic” is
explained in [31, Section 2].
Examples 1.5.(1) We write AbAlg for the variety of abelian algebras determ-
ined by the identity xy “0. Seen as a category, this variety is isomorphic to
the category Vec of vector spaces over F. It is the only non-trivial variety
of non-associative algebras which is an abelian category; this explains the
terminology.
(2) We write Assoc for the variety of associative algebras determined by the
identity of associativity which is xpyzq ´ pxyqz“0, or equivalently xpyzq “
pxyqz.
(3) We write AAssoc for the variety of anti-associative algebras, determined
by the anti-associative identity xpyzq “ ´pxyqz.
(4) We write Com for the variety of commutative algebras determined by the
identity of commutativity which is xy ´yx “0, or equivalently xy “yx.
(5) We write ACom for the variety of anti-commutative algebras determined
by anti-commutativity which is xy `yx “0, or equivalently xy “ ´yx.
(6) We write CAssoc for the variety of commutative associative algebras.
(7) We write ACAAssoc for the variety of anti-commutative anti-associative
algebras.
(8) We write Lie for the variety of Lie algebras determined by anti-commut-
ativity and the Jacobi identity, which respectively are xy `yx “0and
xpyzq ` ypzxq ` zpxyq “ 0.
(9) One can see that all the previous examples are operadic varieties. Let us
provide a non-operadic example: the variety Bool of Boolean rings, which
may be seen as associative Z2-algebras satisfying xx “x. This variety is
action representable.
(10) We write JJord for the variety of Jacobi–Jordan algebras which is determ-
ined by commutativity and the Jacobi identity. Jacobi–Jordan algebras,
also known as mock-Lie algebras, are the commutative counterpart of Lie
algebras. The name of Jordan in the definition is justified by the fact that
every Jacobi–Jordan algebra is a Jordan algebra (see [7]).
(11) We write Leib for the variety of (right) Leibniz algebras determined by the
(right) Leibniz identity which is pxyqz´ pxzqy´xpyzq “ 0.
(12) We write Alt for the variety of alternative algebras, which is determined
by the identities pyxqx´yx2“0and xpxyq ´ x2y“0. Every asso-
ciative algebra is obviously alternative and an example of an alternative
algebra which is not associative is given by the octonions O, that is the
eight-dimensional algebra with basis te1, e2, e3, e4, e5, e6, e7, e8uand multi-
plication table
eiej“$
’
&
’
%
ej,if i“1
ei,if j“1
´δij e1`εijk ek,otherwise,
where δij is the Kronecker delta and εijk acompletely antisymmetric tensor
with value 1 when ijk “123,145,176,246,257,347,365. Notice that e1is
the unit of the algebra O.
When charpFq ‰ 2, the multilinearisation process shows that Alt is
equivalent to the variety defined by
pxyqz` pxzqy´xpyzq ´ xpzyq “ 0
WEAK REPRESENTABILITY OF ACTIONS OF NON-ASSOCIATIVE ALGEBRAS 7
and
pxyqz` pyxqz´xpyzq ´ ypxzq “ 0.
(13) Taking any variety V, one can look at a subvariety of it by adding further
identities to be satisfied. For example, let Vbe a variety determined by
a set of identities Iand let kbe any positive natural number, then we
write NilkpVqfor the variety of k-step nilpotent algebras in Vdetermined
by the identities in Iand the identities of the form x1¨¨¨xk`1“0with all
possible choices of parentheses.
We now want to explain how we may describe actions in a variety of non-
associative algebras. As we already mentioned before, in a semi-abelian category,
actions are split extensions.
Definition 1.6. Let
0X A B 0
iπ
s(1.2)
be a split extension in the variety V. The pair of bilinear maps
l:BˆXÑX, r :XˆBÑX
defined by
b˚x“spbqipxq, x ˚b“ipxqspbq,@bPB, @xPX
where b˚ ´ “ lpb, ´q and ´ ˚ b“rp´, bq, is called the derived action of Bon X
associated with (1.2).
Given a pair of bilinear maps
l:BˆXÑX, r :XˆBÑX
with B,Xobjects of V, we may define a multiplication on the direct sum of vector
spaces B‘Xby
pb, xq ¨ pb1, x1q “ pbb1, xx1`b˚x1`x˚b1q(1.3)
with b˚x1:“lpb, x1qand x˚b1:“rpx, b1q. This construction allows us to build the
split extension in Alg
0X B ‘X B 0
i2π1
i1
(1.4)
with i2pxq “ p0, xq,i1pbq “ pb, 0qand π1pb, xq “ b. This is a split extension
in Vif and only if pB‘X, ¨q is an ob ject of V, i.e. it satisfies the identities
which determine V. In other words, we have the following result analogous to [29,
Theorem 2.4] and [14, Lemma 1.8]:
Lemma 1.7. In a variety of non-associative algebras V, given a pair of bilinear
maps
l:BˆXÑX, r :XˆBÑX,
we define the multiplication on B‘Xas above in (1.3). Then, the pair pl, rqis a
derived action of Bon Xif and only if pB‘X, ¨q is in V. In this case, we call B‘X
the semi-direct product of Band X(with respect to the derived action) and we
denote it by B˙X.
Remark 1.8.Notice that, for any split extension (1.2) and the corresponding derived
action pl, rq, there is an isomorphism of split extensions
0X B ˙X B 0
0X A B 0
i2
1X
π1
θ
i1
1B
iπ
s
8 J. BROX, X. GARCÍA-MARTÍNEZ, M. MANCINI, T. VAN DER LINDEN, AND C. VIENNE
where θ:B˙XÑA:pb, xq ÞÑ spbq ` ipxq. Thus, when we write b˚x(resp. x˚b),
one can think of it as the multiplication pb, 0q ¨ p0, xq(resp. p0, xq ¨ pb, 0)) in B˙X.
Categorical consequences. Let Vbe an operadic variety of non-associative al-
gebras. We recall two results which will be useful for understanding the rest of the
paper.
Theorem 1.9 ([12, 13]).The following conditions are equivalent:
(i) Vis algebraical ly coherent [9];
(ii) Vis an Orzech category of interest;
(iii) Vis action accessible;
(iv) there exist λ1, ..., λ8,µ1, ..., µ8in Fsuch that
xpyzq “ λ1pxyqz`λ2pyxqz`λ3zpxyq ` λ4zpyxq
`λ5pxzqy`λ6pzxqy`λ7ypxzq ` λ8ypzxq
and
pyzqx“µ1pxyqz`µ2pyxqz`µ3zpxyq ` µ4zpyxq
`µ5pxzqy`µ6pzxqy`µ7ypxzq ` µ8ypzxq
are identities in V.
We call the two previous identities together the λ{µ-rules. Since (WRA) implies
action accessibility in general, the existence of the λ{µ-rules is a necessary condition
for the variety Vto be weakly action representable.
Theorem 1.10 ([14]).The following conditions are equivalent:
(i) Vis action representable;
(ii) Vis either the variety Lie or the variety AbAlg.
Theorem 1.10 helps motivating our interest in the condition (WRA). In fact,
in our context, there is only one non-trivial example of a variety which is action
representable. This suggests to study a generalisation of the notion of represent-
ability of actions. On the other hand, action accessibility may not be enough to
study some kind of (weak) actor. The next result, which is closely related to [17,
Proposition 4.5], explains one way of understanding weak action representability
for any variety of non-associative algebras over a field.
Proposition 1.11. A variety of non-associative algebras Vis weakly action rep-
resentable if and only if for any object Xin it, there exists an object Tof Vsuch
that for every derived action of an object Bof Von X
l:BˆXÑX, r :XˆBÑX,
there exists a unique morphism ϕPHomVpB, T qand a derived action pl1, r1q
of ϕpBqon Xsuch that
l1pϕpbq, xq “ lpb, xq, r1px, ϕpbqq “ rpx, bq,
for every bPBand for every xPX.
Proof. (ñ) If Vis weakly action representable, then for any object Xin it there
exists a weak representation pT , τq. Let Bbe an object of Vwhich acts on Xand
let ϕ:BÑTbe the corresponding acting morphism. Consider the split extension
diagram
0X B ˙X B 0
0X ϕpBq ˙ X ϕpBq0
i
1X
π
D!f
s
rϕ
i1π1
s1
WEAK REPRESENTABILITY OF ACTIONS OF NON-ASSOCIATIVE ALGEBRAS 9
where rϕis the corestriction of ϕto its image, i1pxq “ p0, xq,s1pϕpbqq “ pϕpcq,0q,
where pc, 0q “ spbq, and fpb, xq “ pϕpbq, xq. Then the action of ϕpBqon Xis defined
by the pair of bilinear maps
l1:ϕpBq ˆ XÑX, r1:XˆϕpBq Ñ X
where
l1pϕpbq, xq “ s1pϕpbqqi1pxq “ spbqipxq “ lpb, xq
and
r1pϕpbq, xq “ ipxqs1pϕpbqq “ ipxqspbq “ rpb, xq,
for every bPBand for every xPX(we use [17, Proposition 4.5] to see that l1and
r1are well defined).
(ð) Conversely, given an object Xof V, a weak representation of SplExtp´, Xq
is given by pT, τ q, where the component
τB: SplExtpB, X qHomVpB, T q
sends every action of Bon Xto the corresponding morphism ϕ. Moreover, τBis
an injection since the morphism ϕis uniquely determined by the action of Bon X.
Thus τis a monomorphism of functors.
Partial Algebras. We end this chapter with a notion we shall use throughout the
text.
Definition 1.12. Let Xbe an F-vector space. A bilinear partial operation on X
is a map
¨: Ω ÑX,
where Ωis a vector subspace of XˆX, which is bilinear on Ω, i.e.
pα1x1`α2x2q ¨ y“α1x1¨y`α2x2¨y
for any α1, α2PFand x1, x2, y PXsuch that px1, yq,px2, yq P Ωand
x¨ pβ1y1`β2y2q “ β1x¨y1`β2x¨y2
for any β1, β2PFand x, y1, y2PXsuch that px, y1q,px, y2q P Ω.
Definition 1.13. Apartial algebra over Fis an F-vector space Xendowed with a
bilinear partial operation
¨: Ω ÑX.
We denote it by pX, ¨,Ωq. When Ω“XˆXwe say that the algebra is total.
Let pX, ¨,Ωqand pX1,˚,Ω1qbe partial algebras over F. A homomorphism of
partial algebras is an F-linear map f:XÑX1such that fpx¨yq “ fpxq ˚ fpyq
whenever px, yq P Ω, which tacitly implies that pfpxq, f pyqq P Ω1(i.e. both x¨yand
fpxq˚ fpyqare defined). We denote by PAlg the category whose objects are partial
algebras and whose morphisms are partial algebra homomorphisms.
Definition 1.14. We say that a partial algebra pX, ¨,Ωqsatisfies an identity when
that identity holds wherever the bilinear partial operation is well defined.
For instance, a partial algebra pX, ¨,Ωqis associative if
x¨ py¨zq “ px¨yq ¨ z
for every x,y,zPXsuch that px, yq,py, z q,px, yzq,pxy, z q P Ω.
Remark 1.15.We observe that any variety of non-associative algebras Vhas an
obvious forgetful functor U:VÑPAlg.
10 J. BROX, X. GARCÍA-MARTÍNEZ, M. MANCINI, T. VAN DER LINDEN, AND C. VIENNE
2. Commutative and anti-commutative algebras
In this section we aim to study the (weak) representability of actions of some
varieties of non-associative algebras which satisfy the commutative law or the anti-
commutative law. As explained in Section 1, we may assume our variety satisfies
the λ{µ-rules, or equivalently is action accessible.
When Vis either a variety of commutative or anti-commutative algebras, i.e.
xy “εyx is an identity of V, with ε“ ˘1, the λ{µ-rules reduce to
xpyzq “ αpxyqz`βpxzqy,
for some α,βPF. The following proposition is a representation theory exercise:
Proposition 2.1. Let Vbe a non-abelian, action accessible, operadic variety of
non-associative algebras.
(1) If Vis a variety of commutative algebras, then Vis a either a subvariety
of CAssoc or a subvariety of JJord.
(2) If Vis a variety of anti-commutative algebras, then Vis either a subvariety
of Lie or a subvariety of ACAAssoc.
Remark 2.2.We observe that Nil2pComqis a subvariety of both CAssoc and
JJord: in fact, from xpyzq “ pxyqz“0we may deduce that associativity holds
and the Jacobi identity is satisfied:
xpyzq ` ypzxq ` zpxyq “ 0`0`0“0.
If charpFq ‰ 3, then Nil2pComqis precisely the intersection of the varieties CAssoc
and JJord. Indeed, let Vbe a subvariety of both CAssoc and JJord. Since
commutativity, associativity and the Jacobi identity hold in V, we have
pxyqz“xpyzq “ ´ypzxq ´ zpxyq “ ´xpyzq ´ pxyqz“ ´2pxyqz
and thus 3pxyqz“3xpyzq “ 0.
An example of an algebra which lies in the intersection of CAssoc and JJord
but which is not two-step nilpotent is the two-dimensional F3-algebra with basis
te1, e2uand bilinear multiplication determined by
e2
1“e1e2“e2e1“e2
2“e2.
Likewise, Nil2pAComqis a subvariety of both Lie and ACAAssoc: from
xpyzq “ pxyqz“0we may deduce anti-associativity and the Jacobi identity. If
charpFq ‰ 3, then Nil2pAComqcoincides with the intersection of the varieties Lie
and ACAAssoc. Indeed, let Vbe a subvariety of both Lie and ACAAssoc.
Since anti-commutativity, anti-associativity and the Jacobi identity hold in V, we
have
pxyqz“ ´xpyz q “ ´pxyqz´ypxzq “ ´pxyqz` pyxqz“ ´2pxyqz
and thus 3pxyqz“ ´3xpyzq “ 0.
When charpFq “ 3, it is possible to construct an algebra that lies in the inter-
section of Lie and ACAAssoc but which is not two-step nilpotent. Let Xbe the
algebra of dimension 7over F3with basis
te1, e2, e3, e4, e5, e6, e7u
and bilinear multiplication determined by
e1e2“ ´e2e1“e4, e1e3“ ´e3e1“ ´e6, e2e3“ ´e3e2“e5
and
e1e5“ ´e5e1“e2e6“ ´e6e2“e3, e4“ ´e4e3“e7.
WEAK REPRESENTABILITY OF ACTIONS OF NON-ASSOCIATIVE ALGEBRAS 11
Then Xis a Lie algebra such that
xpxyq “ 0
for any x,yPXand, using the multi-linearisation process, one can check this
identity is equivalent to anti-associativity if the characteristic of the field is different
from 2. This Xis not two-step nilpotent, since
e1pe2e3q “ e1e5“e7.
Corollary 2.3. Let Vbe an action accessible, operadic, quadratic variety of non-
associative algebras and suppose that Vis not the variety AbAlg of abelian algeb-
ras.
(1) If Vis commutative, then it has to be one of the following varieties: JJord,
CAssoc, their intersection, or Nil2pComq.
(2) If Vis anti-commutative, then it has to be one of the following varieties:
Lie,ACAAssoc, their intersection, or Nil2pAComq.
We already know that Lie is action representable and that the actor of a Lie
algebra Xis the Lie algebra DerpXqof derivations of X. Therefore, we shall
study the representability of actions of the varieties CAssoc,JJord,Nil2pComq,
ACAAssoc and Nil2pAComq.
Commutative associative algebras. The representability of actions of the vari-
ety of commutative associative algebras over a field was studied in [4], where F. Bor-
ceux, G. Janelidze and G. M. Kelly proved that it is not action representable. We
want to extend this result proving that the variety CAssoc is weakly action rep-
resentable. In Section 3 this is further extended to general algebras over a field.
We start by recalling the following result, where U:CAssoc ÑAssoc denotes the
forgetful functor.
Lemma 2.4 ([4], proof of Theorem 2.6).Let Xbe a commutative associative al-
gebra. There exists a natural isomorphism of functors from CAssocop ÑSet
which we denote
ρ: SplExtp´, X q – HomAssocpUp´q,MpXqq,
where SplExtp´, Xq “ SplExtCAssocp´, X qand
MpXq “ tfPEndpXq | fpxyq “ fpxqy, @x, y PXu
is the associative algebra of multipliers of X, endowed with the product induced by
the usual composition of functions (see [8, 25]).
We recall that MpXqin general does not need to be a commutative algebra. For
instance, let X“F2be the abelian two-dimensional algebra, then MpXq “ EndpXq
which is not commutative. However there are special cases where MpXqis an object
of CAssoc, such as when the annihilator of X(which coincides with the categorical
notion of center)
AnnpXq “ txPX|xy “0,@yPXu
is trivial or when X2“X, where X2denotes the subalgebra of Xgenerated by
the products xy where x,yPX. We refer the reader to [8] for further details.
Theorem 2.5 ([4], Theorem 2.6).Let Xbe a commutative associative algebra.
The following statements are equivalent:
(i) MpXqis a commutative associative algebra;
(ii) the functor SplExtp´, X qis representable.
12 J. BROX, X. GARCÍA-MARTÍNEZ, M. MANCINI, T. VAN DER LINDEN, AND C. VIENNE
Since we have examples where MpXqis not commutative, we conclude that
CAssoc is not action representable. We now want to prove that it is a weakly
action representable category. We analyse what this means and then prove that
the category does indeed fulfil these requirements.
For any commutative associative algebra T, the fully faithful embedding Uof
the category CAssoc into Assoc induces a natural isomorphism
i: HomCAssocp´, T q – HomAssoc pUp´q, U pTqq:CAssocop ÑSet.
Lemma 2.6. If the functor SplExtp´, X qadmits a weak representation pT, τ q, then
there exists an injective function j: MpXq Ñ Tsuch that for each commutative
associative algebra B, the square
SplExtpB, X q
ρB
τB,2HomCAssocpB , T q
–iB
HomAssocpUpBq,MpXqq j˝p´q
,2HomAssocpUpBq, U pTqq
commutes.
Proof. The free associative F-algebra on a single generator is the algebra Frxsof
non-constant polynomials in a single variable x, which since it is commutative is
also the free algebra on a single generator in CAssoc. We find jas the injective
function
VpMpXqq – HomAssocpFrxs,MpXqq Ñ HomAsso cpFrxs, U pTqq – VpUpTqq
where VpAqdenotes the underlying set of an algebra Aand the function in the
middle is the Frxs-component of the monomorphism of functors
i˝τ˝ρ´1: HomAssocpUp´q,MpXqq HomAssocpUp´q, U pTqq.
Now each bPBinduces a morphism b:Frxs Ñ B, and the collection of morphisms
pb:Frxs Ñ BqbPBis jointly epic. Hence its image
pHomAssoc pUpBq, U pTqq Ñ HomAssocpUpFrxsq, U pTqqqbPB
through the contravariant functor HomAssoc pUp´q, U pTqq is a jointly monic col-
lection of arrows. It thus suffices that for each bPB, the outer rectangle in the
diagram
SplExtpB, X q
ρB
τB,2HomCAssocpB , T q
–iB
HomAssocpUpBq,MpXqq
p´q˝Upbq
j˝p´q
,2HomAssocpUpBq, U pTqq
p´q˝Upbq
HomAssocpUpFrxsq,MpXqq
–
iFrxs˝τFrxs˝ρ´1
Frxs
,2HomAssocpUpFrxsq, U pTqq
–
VpMpXqq j
,2VpUpTqq
commutes in Set. This is an immediate consequence of the naturality of the trans-
formations involved.
Remark 2.7.The above proof can be modified to show that the function jis in
fact a vector space monomorphism. If it were moreover an algebra monomorphism,
then this would yield a proof that all MpXqare commutative, which is false by
WEAK REPRESENTABILITY OF ACTIONS OF NON-ASSOCIATIVE ALGEBRAS 13
the above-mentioned example. Thus we would be able to conclude that CAssoc
is not weakly action representable. Theorem 2.11 below proves that this is wrong.
Hence jcannot preserve the algebra multiplication in general.
Each action ξof a commutative associative algebra Bon Xgives rise to a
morphism ρBpξq:UpBq Ñ MpXqin Assoc. If the actions in CAssoc are weakly
representable, then ξis also determined by a morphism of commutative associative
algebras τBpξq:BÑT. The above lemma tells us that j˝ρBpξq “ τBpξq. Note
that here we drop the iBfor the sake of clarity.
Each ρBpξq:UpBq Ñ MpXqis the composite of a surjective commutative associ-
ative algebra map ρ1
Bpξq:UpBq Ñ Mξand the canonical inclusion of a subalgebra
Mξof MpXq. We find a diagram of subalgebras of MpXqindexed over the com-
mutative associative algebra actions on X. Note that since trivial actions exist,
the image in Assoc of the diagram pMξqξactually consists of all commutative sub-
algebras of MpXq, with the canonical inclusions between them. We may re-index
and view pMξqξas a diagram in Assoc over the thin category of commutative
subalgebras of MpXq.
By the above, the Mξfurther include into Tvia j. For each ξ, an image factor-
isation of τBpξq:BÑTis given by the surjective algebra map ρ1
Bpξq:UpBq Ñ Mξ
followed by the inclusion of Mξinto MpXqcomposed with j. We denote this func-
tion µMξ:MξÑTand note that it only depends on the object Mξ. (That is to
say, if ξand ψare two B-actions such that Mξ“Mψ, then the induced inclusions
into Tcoincide as well.) A priori this µMξis only an injective map, but since τBpξq
and ρ1
Bpξqare morphisms of algebras and ρ1
Bpξqis a surjection, that injection is a
monomorphism of commutative associative algebras. Furthermore, the µMξform a
cocone on the diagram of all commutative subalgebras of MpXqwith vertex T.
Recall that for a diagram in a category, an amalgam is a monic cocone, i.e.
a cocone which is a monomorphic natural transformation. This means that each
component of that cocone is a monomorphism, which implies that all the morphisms
of the given diagram were monomorphisms to begin with. Note that in a category
with colimits, an amalgam for a diagram exists if and only if its colimit cocone is
such an amalgam. A category is said to have the amalgamation property (AP) when
each span of monomorphisms admits an amalgam; equivalently, for each pushout
square
It,2
s
T
ιT
SιS
,2S`IT
if sand tare monomorphisms then so are ιSand ιT. It is known that neither
the category of associative algebras, nor the category of commutative associative
algebras satisfies the condition (AP)—see [20] for an overview of examples and
references to the rich literature on the subject.
This is as follows related to the problem at hand. The associative algebra MpXq
is an amalgam in Assoc of the diagram consisting of the commutative subalgebras
Mξof MpXq. So if the functor SplExtp´, Xqadmits a weak representation pT , τq,
then the natural transformation τfactors through the diagram pMξqξas explained
above, and we see that pT , τqrestricts to an amalgam of that diagram in the category
CAssoc.
Thus we find a necessary condition for weak representability of actions in the
category CAssoc: we need that for each commutative associative algebra X, the
diagram pMξqξof commutative subalgebras of the associative algebra MpXqnot
14 J. BROX, X. GARCÍA-MARTÍNEZ, M. MANCINI, T. VAN DER LINDEN, AND C. VIENNE
only admits the amalgam MpXqin the category Assoc; it should also admit an
amalgam Tin CAssoc. Actually, the converse also holds:
Proposition 2.8. For a commutative associative algebra X, a weak representation
pT, τ qof SplExtp´, Xqexists if and only if an amalgam in CAssoc exists for the
diagram of commutative subalgebras of MpXq.
Proof. We already explained that any weak representation of SplExtp´, Xqrestricts
to such an amalgam. So let us assume that a commutative amalgam for the diagram
of commutative subalgebras of MpXqexists. For each commutative associative
algebra action ξof an object Bon X, we let τBpξq:BÑTbe the composite of
ρ1
Bpξq:UpBq Ñ Mξwith the inclusion µMξ:MξÑTof Mξinto the amalgam T.
The thus defined τis a natural transformation by the naturality of both ρ1
and the cocone components in the amalgam. Note that if two maps, say ρBpξq
and ρCpψq, to MpXqhave the same image subalgebra Mξ“Mψof MpXq, then by
naturality of ρand the fact that the inclusion of Mξinto MpXqis a monomorphism,
for any equivariant map f:BÑCwe have that the square on the left
B
f
ρ1
Bpξq,2Mξ
µMξ,2T
Cρ1
Cpψq
,2MψµMψ
,2T
commutes. The commutativity of the entire diagram proves naturality of τ.
We still have to prove that the components of τare monomorphisms: two differ-
ent actions ξand ψof Bon Xgive rise to two different maps τBpξq, τBpψq:BÑT.
Suppose, on the contrary, that τBpξq “ τBpψq. Then by uniqueness of image factor-
isations, the images of µMξ:MξÑTand µMψ:MψÑTare isomorphic subob jects
of T. Now the image in Assoc of the diagram pMξqξis a thin category, so that
Mξ“Mψ. Hence µMξ˝ρ1
Bpξq “ τBpξq “ τBpψq “ µMψ˝ρ1
Bpψq “ µMξ˝ρ1
Bpψq,
which implies ρ1
Bpξq “ ρ1
Bpψq. But then the actions ξand ψare equal, since ρis a
natural isomorphism by Lemma 2.4.
Thus we see that the problem of weak representability of actions of CAssoc
amounts to proving that an amalgam in CAssoc exists for the diagram of com-
mutative subalgebras of MpXqfor any object X. We are actually going to prove
something a bit stronger: namely, that an amalgam in CAssoc exists for any dia-
gram of commutative associative algebras for which an amalgam exists in Assoc.
The essence of the proof is contained in the following special case.
Theorem 2.9. If SÐIÑTis a span of commutative associative algebras for
which an amalgam exists in Assoc, then it has an amalgam in CAssoc.
The proof depends on the following lemma.
Lemma 2.10. Let f:XÑYand g:YÑZbe morphisms in a semi-abelian
category. The composite g˝f:XÑZis a monomorphism if and only if fis a
monomorphism and Impfq X Kerpgqis trivial.
Proof. Note that if g˝fis a monomorphism, then so is f. So we may assume
that fis a monomorphism in either case. The composite g˝fis a monomorphism
precisely when Kerpg˝fqis trivial. Now this kernel is a pullback of Kerpgqalong f.
Since fis a monomorphism, that pullback is Impfq X Kerpgq. So Kerpg˝fqis zero
if and only if Impfq X Kerpgqis zero.
Proof of Theorem 2.9. Let SÐIÑTbe such a span. Recall that an amalgam in
either category exists if and only if the Sand Tcomponents of the induced pushout
WEAK REPRESENTABILITY OF ACTIONS OF NON-ASSOCIATIVE ALGEBRAS 15
cocone in either category are monic. We focus on the case ιS:SÑS`IT“
S`Assoc
ITwhich we assume to be monic. The question is whether its composite
with the reflection unit ηS`IT:S`Assoc
ITÑS`CAssoc
ITis still monic. We are
going to prove that the answer is yes, indeed it is.
Consider the following morphism of short exact sequences in Assoc:
0,2J
ηS`T|J
,2S`T
ηS`T
,2S`IT
ηS`IT
,20
0,2K,2S`CAssoc T,2S`CAssoc
IT,20
As a vector space, the coproduct S`Tof Sand Tin Assoc is S‘T‘Uwhere
U“ pSbTq ‘ pTbSq ‘ pSbTbSq ‘ ¨ ¨ ¨ contains all the tensors. The coproduct
S`CAssoc Tof Sand Tin CAssoc is S‘T‘ pSbTq, so that ηS`Tadmits
a canonical splitting σ:S`CAssoc TÑS`Tin Vec which commutes with the
inclusions of S,Tand SbT.
We note that Kis the ideal of S`CAsso c T“S‘T‘ pSbTqgenerated by the
elements of the form i´i, where iis iPIviewed as an element of S, while iis i
viewed as an element of T. Let Gdenote the set of generators ti´i|iPIu. The
algebra Jis generated by Gas well, but now as an ideal of S`T“S‘T‘U.
It follows that σpKq Ď J—even though σis not a morphism of algebras. We give
a detailed proof of this claim. We know that Kconsists of all elements of the
form x1g1` ¨ ¨ ¨ ` xngnwhere x1, ..., xnPS`CAssoc Tand g1, ..., gnPG.
Since σis a morphism of abelian groups, it suffices that σpxgqbelongs to Jfor
all xPS`CAssoc Tand gPG. Now each xPS`CAssoc Tis of the form
ps1, t1, s1
1bt1
1q ` ¨ ¨ ¨ ` psn, tn, s1
nbt1
nqwith s1, ..., sn,s1
1, ..., s1
nPSand t1, ...,
tn,t1
1, ..., t1
nPT. Hence it suffices to prove that σpps, t, s1bt1qgqbelongs to J
for all s,s1PS,t,t1PTand gPG. Next, we see that ps, t, s1bt1q “ ps, 0,0q `
p0, t, 0q ` p0,0, s1bt1q. As a consequence, it suffices to prove that σpps, 0,0qgq,
σpp0, t, 0qgqand σpp0,0, s btqgqbelong to Jfor all sPSand tPT. Writing
g“ pi, ´i, 0q P S`CAssoc T“S‘T‘ pSbTqand S`T“S‘T‘U, we calculate
what happens in each of these three cases:
σpps, 0,0qgq “ σpps, 0,0qpi, ´i, 0qq “ σpsi, 0,´sbiq “ psi, 0,´sbiq
“ ps, 0,0qpi, ´i, 0q P J,
σpp0, t, 0qgq “ σpp0, t, 0qpi, ´i, 0qq “ σp0,´it, i btq “ p0,´it, i btq
“ pi, ´i, 0qp0, t, 0q P J,
σpp0,0, s btqgq “ σpp0,0, s btqpi, ´i, 0qq “ σp0,0, si bt´sbitq
“ p0,0, si bt´sbitq “ psi, 0,´sbiqp0, t, 0q
“ ps, 0,0qpi, ´i, 0qp0, t, 0q P J.
Note that in the above calculations, we used commutativity twice: in the second
equality of the second and third cases.
Thanks to Lemma 2.10, inside the vector space S`T, the intersection SXJis
trivial, by the assumption that the composite ιS:SÑS`TÑS`ITis monic.
But then the smaller space SXσpKqis trivial as well, so that the composite
SÑS`CAssoc TÑS`CAssoc
IT
is a monomorphism by Lemma 2.10.
For arbitrary diagrams of monomorphisms of commutative associative algebras,
the proof stays essentially the same. This allows us to conclude:
16 J. BROX, X. GARCÍA-MARTÍNEZ, M. MANCINI, T. VAN DER LINDEN, AND C. VIENNE
Theorem 2.11. The category CAssoc of commutative associative algebras is
weakly action representable.
Remark 2.12.For a given diagram of commutative associative algebras as above,
the amalgam Tin CAssoc is also an amalgam in Assoc.
Remark 2.13.Note that by its construction as a colimit, the weak representation
pT, τ qis automatically an initial weak representation (see [17, Section 5]). As ex-
plained in [17, Corollary 5.3, Corollary 5.4], the existence of the initial weak repres-
entation also follows from the existence of a weak representation and the fact that
CAssoc, as a semi-abelian variety of universal algebras, is a total category [32].
Jacobi–Jordan algebras. We now want to study the representability of actions
of the variety JJord of Jacobi–Jordan algebras. As already mentioned in Sec-
tion 1, every split extension of Bby Xin Lie is represented by a homomorphism
BÑDerpXq. For Jacobi–Jordan algebras, the role the derivations have in Lie is
played by the so-called anti-derivations.
Definition 2.14. Let Xbe a Jacobi–Jordan algebra. An anti-derivation is a linear
map d:XÑXsuch that
dpxyq “ ´dpxqy´dpyqx, @x, y PX.
The (left) multiplications Lxfor xPXare particular anti-derivations, called
inner anti-derivations. We denote by ADerpXqthe space of anti-derivations of X
and by InnpXqthe subspace of the inner anti-derivations. Anti-derivations play a
significant role in the study of cohomology of Jacobi–Jordan algebras: see [1] for
further details.
We now want to make explicit what are the derived actions in the category JJord
and how they are related with the anti-derivations. The following is an easy ap-
plication of Lemma 1.7.
Proposition 2.15. Let Xand Bbe two Jacobi–Jordan algebras. Given a pair of
bilinear maps
l:BˆXÑX, r :XˆBÑX
defined by
b˚x“lpb, xq, x ˚b“rpx, bq,
we construct pB‘X, ¨q as in (1.3). Then pB‘X, ¨q is a Jacobi–Jordan algebra if
and only if
(1) b˚x“x˚b;
(2) b˚ pxx1q “ ´pb˚xqx1´ pb˚xq ˚ x1;
(3) pbb1q ˚ x“ ´b˚ pb1˚xq ´ b1˚ pb˚xq;
for all b,b1PBand x,x1PX.
In an equivalent way, a derived action of Bon Xin the variety JJord is given
by a linear map
BÑADerpXq:bÞÑ b˚ ´
which satisfies
pbb1q ˚ x“ ´b˚ pb1˚xq ´ b1˚ pb˚xq,@b, b1PB, @xPX. (2.1)
Remark 2.16.The vector space ADerpXqendowed with the anti-commutator
x´,´y : ADerpXq ˆ ADerpXq Ñ EndpXq,xf , f1y “ ´f˝f1´f1˝f
is not in general an algebra, since the anti-commutator of two anti-derivations
is not in general an anti-derivation: in [2, Remark 2.2], the authors proved that
xf, f 1y P ADerpXqif and only if
xf, f 1ypxyq “ ´fpxqf1pyq ´ f1pxqfpyq,@x, y PX.
WEAK REPRESENTABILITY OF ACTIONS OF NON-ASSOCIATIVE ALGEBRAS 17
Moreover, it can happen that the anti-commutator x´,´y is a well defined bilinear
operation on the space ADerpXqbut it does not define a Jacobi–Jordan algebra
structure: for instance, if X“Fis the abelian one-dimensional algebra, then
ADerpXq “ EndpXq – F(every linear endomorphism of Xis of the form ϕα:xÞÑ
αx, for some αPF) and the Jacobi identity is not satisfied. Nevertheless, there
are some subspaces of ADerpXqthat are Jacobi–Jordan algebras. For instance, the
subspace InnpXqof all inner anti-derivations of X. Indeed, the linear map
XÑADerpXq:xÞÑ Lx,
restricts to a Jacobi–Jordan algebra homomorphism XÑInnpXq. This is true in
general for the image of any linear map BÑADerpXqsatisfying equation (2.1).
Thus we need to use an algebraic structure which includes the space of anti-
derivations endowed with the anti-commutator and which allows us to describe
categorically the representability of actions of the variety JJord. One possible
solution is given by partial algebras.
Indeed, the vector space ADerpXqendowed with the anti-commutator x´,´y is
a commutative partial algebra. In this case Ωis the preimage
x´,´y´1pADerpXqq
of the inclusion ADerpXqãÑEndpXq.
Theorem 2.17. Let Xbe a Jacobi–Jordan algebra and let U:JJordop ÑPAlg
denote the forgetful functor.
(1) There exists a natural isomorphism of functors from JJordop ÑSet
ρ: SplExtp´, Xq – HomPAlgpUp´q,ADerpXqq,
where SplExtp´, Xq “ SplExtJJordp´, X q;
(2) if ADerpXqis a Jacobi–Jordan algebra, then the functor SplExtp´, X qis
representable and ADerpXqis the actor of X;
Proof. (1) For a Jacobi–Jordan algebra B, we define the component
ρB: SplExtpB, X q Ñ HomPAlgpUpBq,ADerpXqq
as the functor which sends any split extension
0X A B 0
iπ
s
to the morphism BÑADerpXq:bÞÑ b˚ ´. The transformation ρis natural.
Indeed, for any Jacobi–Jordan algebra homomorphism f:B1ÑB, it is easy to
check that the diagram in Set
SplExtpB, X qHompUpBq,ADerpXqq
SplExtpB1, XqHompUpB1q,ADerpXqq
ρB
SplExtpf,X qHompUpfq,ADerpXqq
ρB1
where HompUp´q,´q “ HomPAlgpUp´q,´q, is commutative. Moreover, for any
Jacobi–Jordan algebra B, the morphism ρBis an injection, as each element of
SplExtpB, X qis uniquely determined by the corresponding action of Bon X. Thus
ρis a monomorphism of functors. Finally ρis a natural isomorphism since, given
any Jacobi–Jordan algebra Band any homomorphism of partial algebras ϕ:BÑ
ADerpXq, the bilinear maps lϕ:BˆXÑX:pb, xq ÞÑ ϕpbqpxq,rϕ“lϕdefine a
(unique) derived action of Bon Xsuch that ρBplϕ, rϕq “ ϕ.
(2) If ADerpXqis a Jacobi–Jordan algebra, then by (1) we have a natural iso-
morphism
SplExtp´, X q – HomJJordp´,ADerpXqq,
18 J. BROX, X. GARCÍA-MARTÍNEZ, M. MANCINI, T. VAN DER LINDEN, AND C. VIENNE
hence ADerpXqis the actor of X.
Two-step nilpotent commutative algebras. We now analyse the case where V
is a subvariety of both CAssoc and JJord, i.e. Vis the variety Nil2pComqof
two-step nilpotent commutative algebras. We recall this means that xyz “0is an
identity of V. An example of such an algebra is the Kronecker algebra k1(see [21]),
which is the three-dimensional algebra with basis te1, e2, e3uand multiplication
determined by e1e2“e2e1“e3.
We shall show that Nil2pComqis an example of a weakly action representable,
operadic, quadratic variety of commutative algebras.
Proposition 2.18. Let Xand Bbe two algebras in Nil2pComq. Given a pair of
bilinear maps
l:BˆXÑX, r :XˆBÑX,
we construct pB‘X, ¨q as in (1.3). Then pB‘X, ¨q is in Nil2pComqif and only if
(1) b˚x“x˚b;
(2) b˚ pxx1q “ pb˚xqx1“0;
(3) pbb1q ˚ x“b˚ pb1˚xq “ 0;
for any b,b1PBand x,x1PX.
The second equation of Proposition 2.18 states that, for every bPB, the linear
map b˚ ´ belongs to the vector space
rXs2“ tfPEndpXq | fpxyq “ fpxqy“0,@xPXu.
Moreover, seeing rXs2as an abelian algebra (i.e. xf, g y “ 0EndpXq, for every f, g P
rXs2), from the third equation we deduce that the linear map
BÑ rXs2:bÞÑ b˚ ´
is an algebra homomorphism.
On the other hand, given a morphism of algebras
ϕ:BÑ rXs2, ϕpbq “ b˚ ´
satisfying
b˚ pb1˚xq “ 0,@b, b1PB, @xPX,
we can consider the split extension
0XpB‘X, ˚ϕqB0
iπ
s
where the two-step nilpotent commutative algebra structure of B‘Xis given by
pb, xq ˚ϕpb1, x1q “ pbb1, xx1`b˚x1`b1˚xq,@pb, xq,pb1, x1q P B‘X.
We can now claim the following result.
Theorem