Simply complete hom-Lie superalgebras and decomposition of complete hom-Lie superalgebras

Complete hom-Lie superalgebra are considered and some equivalent conditions for a hom-Lie superalgebra to be a complete hom-Lie superalgebra are established. In particular, the relation between decomposition and completeness for a hom-Lie superalgebra is described. Moreover, some conditions that the set of $\alpha^{s}$-derivations of a hom-Lie superalgebra to be complete and simply complete are obtained.


Introduction
Hom-Lie algebras and quasi-hom-Lie algebras were introduced first by Hartwig, Larsson, and Silvestrov in 2003 in [40] devoted to a general method for construction of deformations and discretizations of Lie algebras of vector fields and deformations of Witt and Virasoro type algebras based on general twisted derivations (σ-derivations) obeying twisted Leibniz rule, and motivated also by the examples of q-deformed Jacobi identities in q-deformations of Witt and Visaroro algebras and in related q-deformed algebras discovered in 1990'th in string theory, vertex models of conformal field theory, quantum field theory and quantum mechanics, and q-deformed differential calculi and q-deformed homological algebra [4,[32][33][34][35]37,41,42,[51][52][53].In 2005, Larsson and Silvestrov introduced quasi-Lie and quasi-Leibniz algebras in [48] and graded color quasi-Lie and graded color quasi-Leibniz algebras in [49] incorporating within the same framework thehom-Lie algebras and quasi-hom-Lie algebras, the color hom-Lie algebras and hom-Lie superalgebras, quasi-hom-Lie color algebras, quasi-hom-Lie superalgebras, quasi-Leibniz algebras and graded color quasi-Leibniz algebras.The central extensions and cocycle conditions have been first considered for quasi-hom-Lie algebras and hom-Lie algebras in [40,47] and for graded color quasi-hom-Lie algebras in [62].
In quasi-Lie algebras, the skew-symmetry and the Jacobi identity are twisted by deforming twisting linear maps, with the Jacobi identity in quasi-Lie and quasi-hom-Lie algebras in general containing six twisted triple bracket terms.In hom-Lie algebras, the bilinear product satisfies the non-twisted skew-symmetry property as in Lie algebras, and the hom-Lie algebras Jacobi identity has three terms twisted by a single linear map.Lie algebras are a special case of hom-Lie algebras when the twisting linear map is the identity map.For other twisting linear maps however the hom-Lie algebras are different and in many ways richer algebraic structures with classifications, deformations, representations, morphisms, derivations and homological structures in the fundamental ways dependent on joint properties of the twisting map and bilinear product which are in the intricate way interlinked by hom-Jacobi identity.Hom-Lie admissible algebras have been considered first in [57], where the hom-associative algebras and more general G-hom-associative algebras including the Hom-Vinberg algebras (hom-left symmetric algebras), hom-pre-Lie algebras (hom-right symmetric algebras), and some other new Hom-algebra structures have been introduced and shown to be Hom-Lie admissible, in the sense that the operation of commutator as new product in these hom-algebras structures yields hom-Lie algebras.Furthermore, in [57], flexible hom-algebras have been introduced and connections to hom-algebra generalizations of derivations and of adjoint derivations maps have been considered, investigations of the classification problems for hom-Lie algebras have been initiated with constriction of families of the low-dimensional hom-Lie algebras.
The Hom-Lie superalgebras and the more general color quasi-Lie algebras provide new general parametric families of non-associative structures, extending and interpolating on the fundamental level of defining identities between the Lie algebras, Lie superalgebras, color Lie algebras and some other important related non-associative structures, their deformations and discritizations, in the special interesting ways which may be useful for unification of models of classical and quantum physics, geometry and symmetry analysis, and also in algebraic analysis of computational methods and algorithms involving linear and non-linear descretizations of differential and integral calculi.Investigation of color hom-Lie algebras and hom-Lie superalgebras and n-ary generalizations have been further expanded recently in [1-3, 5, 6, 9-27, 31, 39, 44-46, 55, 56, 58, 59, 61-65, 67].
In [36,38], the complete Lie superalgebras were introduced and studied.Recently the notion of compact hom-Lie superalgebra was introduced in [13].In this article, complete hom-Lie superalgebras are considered and equivalent conditions for a hom-Lie superalgebra to be a complete hom-Lie superalgebra are established.In particular, the relation between decomposition and completness for a hom-Lie superalgebra is described.Moreover, some conditions for the set of α s -derivations of a hom-Lie superalgebra to be complete and simply complete are obtained.In Section 2, some necessary notations and useful definitions and properties of hom-Lie superalgebras are reviewed.In Section 3, the notion of a complete hom-Lie superalgebra is presented, and the equivalent conditions for the completeness of g 0 and g are studied.Then conditions for a hom-Lie superalgebra to be complete are considered by using the notion of holomorph hom-Lie superalgebras and hom-ideals.After that simply complete hom-Lie superalgebras are defined and equivalence of a hom-Lie superalgebra being simply complete or indecomposable is investigated.Finally we discuss the conditions for the Der α s+1 (g) to be complete and simply complete.

Preliminaries on hom-Lie superalgebras and their representation and derivations
Throughout this article, all linear spaces are assumed to be over a field K of characteristic different from 2. A linear space V is said to be a G-graded by an abelian group G if, there exists a family {V g } g∈G of linear subspaces of V such that V = g∈G V g .The elements of V g are said to be homogeneous of degree g ∈ G.The set of all homogeneous elements of V is denoted Homogeneous linear maps of degree zero, f (V g ) ⊆ V ′ g for any g ∈ G, are also called even.An algebra (A, •) is said to be G-graded if its underlying linear space is G-graded, A = g∈G A g , and moreover A g • A h ⊆ A g+h , for all g, h ∈ G.A homomorphism f : A → A ′ of G-graded algebras A and A ′ is an algebra morphism which is even (degree 0 G ).In Z 2 -graded linear spaces A = A 0 ⊕ A 1 , the elements of A j are homogeneous of degree (parity) j ∈ Z 2 , and the set of all homogeneous elements is H(A) = A 0 ∪ A 1 .The parity of a homogeneous element x ∈ H(A) is denoted |x|.
Hom-superalgebras are triples (A, µ, α) in which A = A 0 ⊕ A 1 is a Z 2 -graded linear space (K-superspace), µ : A × A → A is an even bilinear map, and α : A → A is an even linear map.An even linear map f : A → A ′ is said to be a weak morphism of hom-superalgebras In any hom-superalgebra Hom-subalgebras (graded hom-subalgebras) of hom-superalgebra (A, µ, α) are defined as Z 2graded linear subspaces I = (I ∩ A 0 ) ⊕ (I ∩ A 1 ) of A closed under both α and µ, that is α(I) ⊆ I and µ(I, I) ⊆ I.
(ii) Multiplicative hom-Lie algebra is called regular, if α is an automorphism.
In skew-symmetric hom-superalgebras, the super hom-Jacobi identity can be presented equivalently in the form of super hom-Leibniz rule for the maps ad (2.9) since, by super skew-symmetry (2.7), the following equalities are equivalent: Remark 2.3.If skew-symmetry (2.5) does not hold, then (2.8) and (2.9) are not necessarily equivalent, defining different Hom-superalgebra structures.The Hom-superalgebras defined by just super algebras identity (2.9) without requiring super hom-skew-symmetry on homogeneous elements are Leibniz Hom-superalgebras, a special class of general Γ-graded quasi-Leibniz algebras (color quasi-Leibniz algebras) first introduced in [48,49].
Remark 2.5.From the point of view of Hom-superalgebras, Lie superalgebras is an important subclass of Hom-Lie superalgebras obtained when α = id in Definition 2.2.Namely, when α = Id, Definition 2.2 becomes the definition of Lie superalgebras [29,30,43,43] as Z 2 -graded linear spaces g = g 0 ⊕ g 1 , with a graded Lie bracket [., .]: g × g → g of degree zero, that is [., .] is a bilinear map obeying [g i , g j ] ⊂ g i+j(mod2) , and for homogeneous x, y, z ∈ H(g), Super skew-symmetry (2.10) In skew-symmetric superalgebras, the super hom-Jacobi identity can be presented equivalently in the form of super Leibniz rule for the maps ad (2.12) However, for general linear maps α, the Hom-Lie superalgebras are substantially different from Lie superalgebras, as all algebraic structure properties, morphisms, classifications and deformations become dependent fundamentally on the joint simultaneous structure and properties of both operations, the linear map α and the bilinear product [., .]linked in an intricate way via the α-twisted super-Jacoby identity (2.8).
We are going to need the following definition throughout the rest of the article.Definition 2.10 ( [9,49]).A representation of the hom-Lie superalgebra (g, [., .],α) on a Z 2 -graded linear space V = V0 ⊕ V1 with respect to β ∈ gl(V )0 is an even linear map ρ : g → gl(V ), such that for all homogeneous x, y ∈ H(g), A representation V of g is called irreducible or simple, if it has no nontrivial subrepresentations.Otherwise V is called reducible.
For any linear transformation T : X → X of a set X, and any nonnegative integer s, the s-times composition is Next, we recall the notion of α s -derivations.

Complete hom-Lie superalgebras
In [13], the authors introduced the notion of a complete hom-Lie superalgebra and in this section we state some results about it.
Definition 3.3.A hom-Lie superalgebra (g, [., .],α) is called solvable if g n = 0 for some n ∈ N, where g n , the members of the derived series of g, are defined inductively, Note that any commutative hom-Lie superalgebra is solvabe and for a multiplicative hom-Lie superalgebra g, we have α(g n ) ⊆ g n for any n.
Let g be a hom-Lie superalgebra and let Φ be a bilinear form on g.Recall that Φ is called invariant if Φ([x, y], z) = Φ(x, [y, z]) for all x, y, z ∈ g.The invariant bilinear form associated to the adjoint representation of g is called the Killing form on g.Now, we check conditions under which the completeness of g 0 and g are equivalent.
(ii) g is hom-ideal of h(g) and h(g)/g ≃ Der(g).
Therefore C(Der(g)) = 0. Next, g ⊳ g, so g is a hom-ideal of h(g) and h(g)/g ≃ Der(g).Now, let x ∈ g.Then Now, we state some equivalence conditions for a hom-Lie superalgebra to be complete, by using the notion of holomorph hom-Lie superalgebras.Definition 3.8.Let g, h be two hom-Lie superalgebras.We call e an extension of the hom-Lie superalgebra g by h, if there exists a short exact sequence 0 → h → e → g → 0 of hom-Lie superalgebras and their morphisms.Theorem 3.9.For a multiplicative hom-Lie superalgebra (g, [., .],α) with surjective α, the following conditions are equivalent: (i) g is a complete hom-Lie superalgebra; (ii) any splitting extension e by g is a trivial extension and e = g ⊕ C e (g); (iii) h(g) = g ⊕ C h(g) (g).
Therefore I and J are complete hom-Lie superalgebras.Conversly, let I and J are complete, then C(g) = C(I) ⊕ C(J) = 0, by (i), and Der α s+1 (g) = Der α s+1 (I) ⊕ Der α s+1 (J) = ad s (I) ⊕ ad s (J) = ad s (g), by (ii).Definition 3.11.Let g be a complete hom-Lie superalgebra.If any non-trivial hom-ideal of g is not complete, then g is called a simply complete hom-Lie superalgebra.
A simple and complete hom-Lie superalgebra is a simply complete hom-Lie superalgebra.Now, we want to state the relation between simply complete hom-Lie superalgebras and indecomposable complete hom-Lie superalgebras.Theorem 3.12.Let (g, [., .],α) be a complete multiplicative hom-Lie superalgebra with surjective α on g.
(i) g can be decomposed into the direct sum of simply complete hom-ideals.
(ii) g is simply complete if and only if it is indecomposable.
Proof.(i) If g is simply complete, then (i) holds.If g is not simply complete, then by Proposition 3.5, there exists a nonzero minimal complete hom-ideal I of g such that g = I ⊕ C g (I).Since a hom-ideal of C g (I) is also a hom-ideal of g, by continuing this method for C g (I), we reach to the decomposition of g into the simply complete hom-ideals.
(ii) If g is simply complete, then it is indecomposable by (i).Conversely, if g is indecomposable, then it has no non-trivial hom-ideals.Hence, g is simply complete by Definition 3.11.Definition 3.13.A subspace I of a hom-Lie superalgebra g is called a characteristic homideal of g, if D(I) ⊂ I for all D ∈ Der(g).Lemma 3.14.Let (g, [., .],α) be a multiplicative hom-Lie superalgebra, I be a characteristic hom-ideal of g and α is surjective on g and I. Then I is hom-ideal of g.
Proof.Let x, y ∈ I, since α is surjective on I and ad s (g) is a α s+1 -derivation, then Theorem 3.15.Let (g, [., .],α) be a multiplicative hom-Lie superalgebra with surjective α, C(g) = 0 and ad s (g) be a characteristic hom-ideal of Der(g).Then Der(g) is complete.Furthermore, if g is indecomposable and [g, g] = g, then Der(g) is simply complete.
Proof.The Hom-Lie superalgebra g has trivial center, so g ≃ ad s (g).Let p = Der(g), then g ⊳ p.Let q be a splitting extension by p, that is p ⊳ q.Hence for all q ∈ q, we have ad s (q) ∈ Der α s+1 (p).g is a characteristic hom-ideal of p, so there exists p ∈ p such that ad s (p)| g = ad s (q)| g .Then ad s (p − q)| g = 0 and p − q ∈ C q (g).Hence we have q = p + C q (g).On the other hand, p ∩ C q (g) = C p (g) = 0 and p ⊳ q, thus q = p ⊕ C q (g).Hence C q (g) ⊆ C q (p) and we have q = p ⊕ C q (p).Therefore by Theorem 3.9, p = Der(g) is a complete hom-Lie superalgebra.Now, assume that Der(g) is not simply complete.So there exists a simply complete hom-ideal I.By Proposition 3.5, there exists a hom-ideal J such that p = I ⊕ J.For any x, y ∈ g, there exists x 1 , y 1 ∈ I and x 2 , y 2 ∈ J such that x = x 1 + x 2 and y = y 1 + y 2 .Thus [x, y] = [x 1 + x 2 , y] = [x 1 , y] + [x 2 , y] such that [x 1 , y] ∈ I ∩ g and [x 2 , y] ∈ J ∩ g.Hence g = [g, g] = (I ∩ g) ⊕ (J ∩ g).g is indecomposable, then I ∩ g = 0 or J ∩ g = 0. Hence g ⊂ J and I ⊂ C p (g) = 0. Therefore by Theorem 3.12, p = Der(g) is simply complete.

( i )
An extension 0 → h i → e p → g → 0 is called trivial extension if there exists an hom-ideal I ⊂ e such that e = Ker(p) ⊕ I. (ii) An extension 0 → h i → e p → g → 0 is called splitting extension if there exists an hom-supersubspace S ⊂ e such that e = Ker(p) ⊕ S.