Enveloping Lie superalgebras and Killing-ricci forms of Bol superalgebras

In this paper, enveloping Lie superalgebras of Bol superalgebras are introduced. The notion of Killing-Ricci forms and invariant forms of these superalgebras are investigated as generalization of the ones of Bol algebras.


Introduction
Bol algebras were introduced [1] in the context of a study of smooth Bol loops. e algebras play the same role with respect to Bol loops as Lie algebras do with respect to Lie groups or Malcev algebras to Moufang loops [2]. ·(c · d). A related notion is that of a Lie triple algebra, introduced under the name generalized Lie triple systems, by Yamaguti [3], and called later as Lie Yamaguti algebras [4].
From the standard enveloping Lie algebra of a given Bol algebra, the notion of Killing-Ricci form and invariant form for a Bol algebra are introduced and studied in [5].
A Z 2 -graded generalization of Lie algebras, called Lie superalgebras, is considered in [6,7], while a Z 2 -graded generalization of Lie Yamaguti algebras called Lie Yamaguti superalgebras was first considered in [8] and generalizes Lie supertriple systems [9]. e reader may refer to [10] for applications of Lie supertriple systems in physics. As Lie Yamaguti superalgebras, Bol superalgebras first introduced in [11] may also be viewed as a generalization of Lie supertriple systems. For relations between Malcev superalgebras and Bol superalgebras, one may refer to [12].
As a part of the general theory of superalgebras, the notion of Killing form of Lie algebras is extended to one of Lie triple systems [13], Lie superalgebras [7], Lie supertriple systems [14,15], and next Lie Yamaguti superalgebras [16].
In this paper, we define enveloping Lie superalgebras and the Killing-Ricci form of Bol superalgebras and study this Killing-Ricci form, which could be seen as a generalization of the one of Bol algebras [5] and the Killing form of Lie supertriple systems [14,15]. Unlike Bol algebras, in Bol superalgebras, there is an odd subspace, which is not a Bol subalgebra. is complicates the work more than Bol algebras case. e rest of this paper is organized as follows. In section 2, we first recall some basics on Lie and Malcev superalgebras as well as Lie supertriple systems and Bol superalgebras. In Section 3, we define the notion of pseudo-superderivations (Definition 8), study their properties (Lemma 1), and introduce the notion of enveloping Lie superalgebras of Bol superalgebras (Definition 12). In Section 4, the Killing-Ricci form of Bol superalgebras is defined (Definition 13) and some of its properties are investigated ( eorem 1, Proposition 3, and Lemma 3). In the next section, the invariant form of Bol superalgebras is defined (Definition 14) and some results are obtained (Lemma 4 and eorem 2). roughout this paper, all vector superspaces and superalgebras are finite dimensional over a fixed ground field K of characteristic 0.

Some Basics on Superalgebras
We recall here some useful definitions and examples of Lie supertriple systems as well as the one of Bol superalgebras. ese examples are obtained from the relation between Malcev superalgebras and Bol superalgebras, which could be found in [12]. Now, let M be a linear superspace over K, that is, a Z 2 -graded linear space with a direct sum M � M 0 ⊕M 1 . e elements of M j , j ∈ Z 2 , are said to be homogeneous of parity j. e parity of a homogeneous element x is denoted by x. For all i, j ∈ Z 2 , i + j will always means that this sum is then f is said to be an even linear map. An algebra (A, [, ]) is called a superalgebra if the underlying vector space is Z 2 -graded, i.e., A � A 0 ⊕A 1 and if furthermore [A i , A j ] ⊂ A i+j . For any binary operation, we will sometimes use juxtaposition in order to reduce the number of braces; i.e., for "·", xy · z means (x · y) · z.
) satisfying the superskew-symmetry and the super-Jacobi identities that is for all x, y, z ∈ H(A). In terms of the super-Jacobian, e super-Jacobi identity is written as SJ(x, y, z) � 0 for all x, y, z ∈ H(A).
Another class of superalgebras that is of interest in this paper is the one of Malcev superalgebras.

Definition 3
consisting of a Z 2 -graded K -vector space S � S 0 ⊕S 1 and a K -trilinear map [ " ], satisfying[S i , S j , S k ] ⊂ S i+j+k for all i, j, k in Z 2 such that for allx, y, z, ∈ H(S), the following equations hold: (2) A Lie supertriple system [9,19] holds for all x, y, u, v, w ∈ H(S).
In [12], we proved that any Malcev superalgebra for all x, y, z ∈ H(M) becomes a Bol superalgebra . is allows us to get the following example of Bol superalgebra from an example of a Malcev superalgebra [20].

Enveloping Lie Superalgebras of a Bol Superalgebra
As derivations for algebras, superderivations of different superalgebras are an important subject of study in superalgebras and diverse area. ey appear in many fields of mathematics and physics. In particular, they allow the construction of new superalgebras structures. In the case of Bol superalgebras, instead of superderivations, we have the notion of pseudo-superderivations. ey generalize pseudoderivations for Bol algebras [2] and superderivations for Lie supertriple systems [19] and allow the construction of enveloping Lie superalgebras of Bol superalgebras. en, we have the following.

� +(− 1) r(x+s) c · xP(y) +(− 1) rs c · P(x)y +(− 1) r(x+y+s) c · [x, y, a]
+(− 1) rs c · (a · xy) + P(c) · xy +(− 1) r(x+y+s) [c, xy, a] + a · (c · xy), (12) and therefore, by (9) and (12) for P and R, we get and then, we get (10). is ends the proof. Now, we can prove the following result.  x, y ∈ B 0 ∪ B 1 , r ∈ Z 2 , D x,y (B r ) ∈ B r+x+y ; that is, D x,y is a linear map of degree x + y. Moreover, it comes from (BS 4 ) and (BS 5 ) that for any x, y, u, v in B 0 ∪ B 1 . It follows that D x,y is pseudosuperderivation of degree x + y and companion xy, called inner pseudo-superderivation of B. Now, one can reformulate the definition of a Bol superalgebra in the following manner.

and the endomorphism D x,y : z↦[x, y, z] is its pseudo-superderivation with a companion xy for all
Let ipS r (B, B) be the vector space spanned by all inner pseudo-superderivations D x,y (x, y ∈ B 0 ∪ B 1 and x + y � r ∈ Z 2 ).
We can define naturally a Z 2 − gradation by setting ipS (B, B) � ipS 0 (B, B)⊕ipS 1 (B, B). Evidently, ipS(B, B) is a subsuperalgebra of the Lie superalgebra pS(B). Accordingly, IPS(B, B) can be introduced as the set of all pairs (P, c), ipS(B, B) and c ∈ Com(P). Evidently, IPS(B, B) is a subsuperalgebra of the Lie superalgebra PS(B).
en, we have the following.
Next, the other cases when the three elements are in B 0 ∪ B 1 or H 0 ∪ H 1 follow from (HBS 3 ) and the fact that H is a Lie superalgebra. □ Definition 12. An enveloping superalgebra of a Bol superalgebra B is a Lie superalgebra L H (B) defined above. Taking  H � PS(B) , we obtain the maximal enveloping superalgebra, and taking H � IPS(B, B) , we obtain the minimal (standard) enveloping superalgebra.
e following result will be used in the last section.  For any x, y ∈ B 0 ∪ B 1 , define the endomorphism R x,y of the vector superspace B by R x,y (z) � (− 1) z(x+y) [z, x, y] � (− 1) z(x+y) D z,x (y) for all z ∈ B 0 ∪ B 1 . It is clear that R x,y is of degree x + y. e next result gives an explicit expression of β.
Proof. Let e i , f i , u i , and v i be bases for B 0 , B 1 , IPS 0 (B, B), and IPS 1 (B, B), respectively. It suffices to prove (20) for all elements x, y in the basis, i.e., β(e i , e j ) � α(e i , e j ), For these bases, we express the operations of B and ISP(B, B), using the tensor notation (i.e., repeated indices imply summation), as follows: Since R e i ,f j (B 0 )⊆B 1 and R e i ,f j (B 1 )⊆B 0 , we have str(R e i ,f j + R f j ,e i ) � 0 and then β(e i , f j ) � 0 � α(e i , f j ) (consistency property of α ). Hence, it remains to show that β(e i , e j ) � α(e i , e j ) and β(f i , f j ) � α(f i , f j ). e identities (15) and (16) Again, the identities (15) and (16) imply the following: Again, from the relation β( From the other hand, we get ki v m f j � S m ki L s mj e s , and, By interchanging i and j, we get and therefore from (29) and (32), we obtain where by relation (20)  (1) β(x, y) � (− 1) xy β(y, x) for all x, y ∈ B 0 ∪ B 1 (supersymmetry). (43) us, if β is nondegenerate, the restriction of α to  IPS(B, B) × IPS(B, B) is nondegenerate, and therefore, α is nondegenerate. Now by contradiction, suppose that β is degenerate. en, by Lemma 4, B ⊥ is a nonzero ideal of B, and therefore, B ⊥ ⊕IPS(B, B ⊥ ) is a nonzero ideal of L(B) by Lemma 2.
By the identities (42) and (43) It follows that α is nondegenerate, and therefore, L(B) is nondegenerate, which ends the theorem.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that there are no conflicts of interest.