Algebraic loop structures on algebra comultiplications

Abstract In this paper, we study the algebraic loop structures on the set of Lie algebra comultiplications. More specifically, we investigate the fundamental concepts of algebraic loop structures and the set of Lie algebra comultiplications which have inversive, power-associative and Moufang properties depending on the Lie algebra comultiplications up to all the possible quadratic and cubic Lie algebra comultiplications. We also apply those notions to the rational cohomology of Hopf spaces.


Introduction
The theory of Lie algebras is an outgrowth of the Lie theory of continuous groups and is playing an important role for many reasons. Moreover, it intervenes in other areas of science such as di erent branches of physics and chemistry. It is an active domain of current research and employs at the same time algebra, algebraic topology, geometry and so on. Lie algebras arise as vector spaces of linear transformations equipped with a new operation which is in general neither commutative nor associative. Indeed, Lie algebras are vector spaces endowed with a particular non-associative product which is called a Lie bracket. Recently, the necessary and su cient conditions for a family of functors from the category of partial group entwining modules to the category of modules over a suitable algebra to be separable were given in [1], and the homotopy relationship between the space of proper maps and the space of local maps was constructed in [2].
In the topological point of view, co-H-spaces, which are also called spaces with a comultiplication, are important objects of study for at least two reasons (see [3][4][5][6][7][8]). First of all, they are the duals of H-spaces in the sense of Eckmann and Hilton (see also [9][10][11]). They have played a signi cant and central role in algebraic topology for many years. Secondly, there is a large class of examples, namely the suspensions and the spheres, which are co-H-spaces. They have the co-H-group structures which enable one to add homotopy classes of maps using the suspension structure and which give rise to the homotopy group of a space in algebraic topology.
In this paper, as an algebraic version of comultiplications of a space, we think about the algebraic comultiplications of algebraic objects [12] and we are particularly interested in the algebraic loop structures on the set of some Lie algebra comultiplications. The main purpose of this paper is to construct the algebraic loops derived from a Lie algebra comultiplication, and investigate the basic properties of those algebraic loops and the set of Lie algebra comultiplications which have the inversive, power-associative and Moufang properties depending on the Lie algebra comultiplications. We also apply these notions to those of the rational cohomology of Hopf spaces.
The paper is organized as follows: In Section 2, we de ne certain Lie algebra comultiplications on the Lie algebras and construct the abelian group structure on the set of Lie algebra comultiplications. In Section 3, we determine concretely when the algebraic loop structures on the set of Lie algebra comultiplications have the inversive, power-associative and Moufang properties for up to all the possible quadratic and cubic Lie algebra comultiplications. Finally, in Section 4, we apply these results to those of some algebra comultiplications on the rational cohomology algebras derived from the Hopf structures.

Preliminaries
We begin this section with the basic notions of comultiplications of topological spaces. A pair (X, η) consisting of a pointed space X and a pointed map η : X → X ∨ X is called a co-H-space if p η and p η , where p and p are the projections X ∨ X → X onto the rst and second summands of the wedge, respectively, and is the identity map of X, and is the homotopy relation. In this case, the map η : X → X ∨ X is said to be a comultiplication.
Let A be a set with binary operation '+' and zero element ∈ A; that is, The following result (see [3,Theorem 2.3] and [13,Proposition 1.13]) is the dual of a well known theorem of James [9] for H-spaces and provides a connection between co-H-spaces and algebraic loops. For the moment, we adopt the following notation: If (X, η) is a co-H-space with a comultiplication η : X → X ∨ X and Y is any space, then the set [X, Y] of homotopy classes of maps from X to Y with the binary operation '+η' induced by η will be denoted by [X, Y]η; that is, Moreover, it is well known in [3] that [X, Y]η is also an algebraic loop if (X, η) is a co-H-space and Y is a nilpotent space. We now move on to the algebraic version of the topological comultiplication above. In this section, we consider Lie algebras over a eld F of characteristic . Let V =< v , v , . . . , vn > and W =< w , w , . . . , wm > be graded vector spaces over F, and write |vs| = ds for the degree of vs , s = , , . . . , n with d ≤ d ≤ . . . ≤ dn, and similarly for w t , t = , , . . . , m. Let L(V) and L(W) be the free graded Lie algebras generated by V and W, respectively, with the Lie brackets [ , ]. We then de ne the coproduct L(V) L(W) of Lie algebras L(V) and In particular, if V = W, then the elements of L(V) L(V) in either summand are distinguished from each other by using a prime on elements from the second factor. Therefore, if L(V) = L(v , v , . . . , vn), then Example 2.2. Let X (n) be the nth Postnikov approximation of X [14,15], and let Q be the eld of rational numbers. Then the graded rational homotopy groups of the Postnikov approximations of the function spaces based on the in nite complex and quaternionic projective spaces with the Samelson products as the Lie brackets become the free graded Lie algebras over Q; that is, where Σ is the suspension functor, and ω is the constant map as the base point of the function spaces, and dim(v i ) = i for i = , , . . . , n, and dim(w j ) = j for j = , , . . . , m.

De nition 2.3. A homomorphism of Lie algebras
where π and π are the projections L(V) L(V) → L(V) onto the rst and second factors, respectively.
Throughout this paper, we will make use of polynomials whose multiplications are the Lie brackets [ , ] in the Lie algebras. We now consider the following. . We call P = (P , P , . . . , Pn) the perturbation of φ and write φ = φ P . We call P or φ one-stage if there is an integer s with ≤ s ≤ n such that P i = for all i ≠ s.
We now describe the group structure on the set of Lie algebra comultiplications of the free graded Lie algebras as follows. This allows us to add and subtract Lie algebra comultiplications of L(V) by adding and subtracting the corresponding perturbations as follows: If φ P and φ Q are the two Lie algebra comultiplications of L(V) with perturbations P = (P , P , . . . , Pn) and Q = (Q , Q , . . . , Qn), respectively, then we de ne the operations by is the Lie algebra comultiplication with perturbation P + Q. Indeed, we can show that φ P+Q and −φ are Lie algebra comultiplications of L(V). This gives the set of Lie algebra comultiplications C(L(V)) the abelian group structure with the standard comultiplication φ as the unit, where = ( , , . . . , ) is the perturbation.
More generally, we let L and M be Lie algebras and let φ : L → L L be a Lie algebra comultiplication. Then we can de ne an addition '+φ' on the set of Lie algebra homomorphisms as follows: De nition 2.8. For the Lie algebra homomorphisms f , g : L → M, we de ne the addition f +φ g by the composition of algebra homomorphisms From the de nition above, we note that if i , i : L → L L are the rst and second inclusions, respectively, then φ = i +φ i : L → L L.

Algebraic loop structures
Let A be an algebraic loop, and let l(a) and r(a) be the left and right inverses of a ∈ A, respectively, under the operation '+'; that is, l(a) + a = = a + r(a). [4,18]). It can be shown that an associative binary operation is Moufang.

Then an algebraic loop
Convention. From now on, we substitute the free graded Lie algebra L(V) by L for a notational convenience, where V =< v , v , . . . , vn > is the graded vector space over a eld F of characteristic zero. If φ : L → L L is a Lie algebra comultiplication and M is any Lie algebra, then the set of Lie algebra homomorphisms h : L → M with the binary operation above induced by the Lie algebra comultiplication φ will be denoted by Hom(L, M)φ. Proof. For the Lie algebra homomorphisms f , g ∈ Hom(L, M)φ, we need to nd unique solutions x, y ∈ Hom(L, M)φ for the equations f +φ x = g and y +φ f = g. The Lie algebra homomorphism x : L → M is constructed on generators of L by induction on the degree |vs| = ds of vs. The equation shows that where Ps is the sth perturbation of φ. In other words, The induction gives the existence and uniqueness of x. Similarly, the existence and uniqueness of y follows.
We recall that there is a bijection as sets Here, 1. the left-hand side is an algebraic loop, while the right-hand side is a direct sum of vector spaces over a eld K of characteristic zero; 2. φ| L(vs) : L(vs) → L L is the standard comultiplication de ned by φ| L(vs) (vs) = vs + v s ; and 3. fs ∈ Hom(L(vs), M) φ| L(vs ) for each s = , , . . . , n.
De nition 3.2. Let φ = φ P : L → L L be a Lie algebra comultiplication of L with perturbation P = (P , P , . . . , Pn). Then φ P or P is said to be purely cubic if each P i , i = , , . . . , n has only polynomials of length , and is said to be cubic if each P i , i = , , . . . , n has only polynomials of length ≤ .

De nition 3.3. We de ne decomposable maps
and It is natural for us to consider the following fundamental question: What kind of operations are there in the algebraic loop Hom(L, M)φ? The following lemma gives the answer to this query. We denote by i,j the summation in which i and j run from through n − with d i + d j = ds for s = , , . . . , n. We note that the image of generators of the graded vector space V under the composition (f g)D ij of D ij : L → L L with f g : L L → M M is as follows: Proof. We rst recall that the sum 'f +φ g' is the composite of the homomorphisms where ∇ is the folding homomorphism. It can easily be seen that the following diagram is commutative: where rs : L(vs) → L is the canonical inclusion for each s = , , . . . , n. For s = , , we have . . , fn) (g , g , . . . , gn))(vs + v s ) = ∇ • (fs(vs) + gs(v s )) = fs(vs) + gs(vs).
In general, a Lie algebra comultiplication φ = φ (P ,P ,...,Pn) : L −→ L L containing at least two non-zero perturbations, say Ps ≠ P t , cannot be guaranteed to have the inversive, power-associative and Moufang properties. This is why we restrict our range in Theorem 3.6 to the one-stage quadratic perturbation. From now on, i,j,k denotes the summation in which i, j and k are the elements of the set { , , . . . , n− } with d i + d j + d k = ds for s = , , . . . , n.
De nition 3.8. We de ne the decomposable maps  where hs = fs + gs for s = , , and . . , f n− + g n− + h n− , fn + gn + hn and Proof. The proof follows from Lemma 3.9.

and 2. it has the Moufang property if and only if
Proof. For a notational convenience, we will also make use of the simple notation for i,j,k , and D l for D l ijk in the proof, where l = , , . . . , . 1. For the power-associative property, we calculate the following: and Thus, the algebraic loop Hom(L, M) ψ is power-associative if and only if The inversive property is obtained as in the proof of the power-associative property by applying Lemma 3.9 and Corollary 3.11.
Similarly, by Corollary 3.11, we get the following: . . , f n− + g n− + h n− , fn + gn + hn Therefore, the algebraic loop Hom(L, M) ψ has the Moufang property; that is, the equality as required.

Remark 3.13.
A space X is said to be a rational space if π * (X) is a graded vector space over Q. It is well known that π * (ΩX) ⊗ Q with the Samelson product < , > is a connected graded Lie algebra L X if X is a -connected rational space, where Ω is the loop functor from the pointed homotopy category to itself. Moreover, there is a beautiful theorem, originally due to Quillen, which states that any connected graded Lie algebra over Q is realized as π * (ΩX) ⊗ Q for some simply connected space X. Therefore, we can obtain the results above with a journey starting from a -connected space X, via the Quillen's theorem (or Quillen minimal model), to the connected graded Lie algebra L X with the Samelson product.
Recall that the multiplication m : X Q × X Q → X Q and the cohomology cross product pairing induce a homomorphism of rational cohomology algebras m * : ∧(S) −→ ∧(S) ⊗ ∧(S).
In a similar way, we consider the coproduct of rational cohomology algebras as follows: is commutative; that is, in (H * (X; Q), +, ∪) = ∧(S), where × is the cohomology cross product and d * is an algebra homomorphism induced by the diagonal map d : X → X × X. Therefore, we will write the cohomology cup product x i ∪ x j as 'x i x j ' for convenience. We now construct concrete cohomology algebra comultiplications in more detail below.