Structures of Not-finitely Graded Lie Superalgebras

For the readers’ convenience, we give some notations used in this paper. Let ⊆ Γ  be an additive subgroup of  and s∈  such that 2sΓ. For simplicity, let Ω be an additive subgroup of  generated by Γ∪{s}. Denote by , *, , +, Γ*, Ω* the sets of complex numbers, nonzero complex numbers, integers, nonnegative integers, nonzero elements of Γ, nonzero elements of Ω, respectively. We assume that all vector spaces are based on , unless otherwise stated. For convenience, the degree of x or φ is denoted by |x| or |φ|. In addition, x is always assumed to be homogeneous when |x| occurs. Let L be a Lie superalgebra, and denote by hg(L) the set of all homogeneous elements of L.


Introduction
For the readers' convenience, we give some notations used in this paper. Let ⊆ Γ  be an additive subgroup of  and s∈  such that 2sΓ. For simplicity, let Ω be an additive subgroup of  generated by Γ∪{s}. Denote by ,  * , ,  + , Γ * , Ω * the sets of complex numbers, nonzero complex numbers, integers, nonnegative integers, nonzero elements of Γ, nonzero elements of Ω, respectively. We assume that all vector spaces are based on , unless otherwise stated. For convenience, the degree of x or φ is denoted by |x| or |φ|. In addition, x is always assumed to be homogeneous when |x| occurs. Let L be a Lie superalgebra, and denote by hg(L) the set of all homogeneous elements of L.
Lie superalgebras as generalizations of Lie algebras were originated from super symmetry in mathematical physics. The theory of Lie superalgebras plays a prominent role in modern mathematics and physics. In recent years, structures of all kinds of Lie superalgebras have aroused many scholars' great interests [1][2][3][4]. In this paper, we shall investigate the structure theory of a class of not-finitely graded Lie superalgebras related to the generalized super-Virasoro algebras (namely, derivations, automorphisms, 2-cocycles).
If an undefined notation appears in an expression, we treat it as zero. For example, Because Ω may not be finitely generated (as a group),  may not be finitely generated as a Lie superalgebra. Hence, some classical techniques [18] cannot be directly applied to this case. We must use some new techniques in order to deal with problems associated with not-finitely generated Lie superalgebras. In addition, in [5,19] some authors studied some structures of not-finitely graded Lie algebras. Thus, some methods about those Lie algebras can be applied in this case. However, it seems to us that little has been known about not-finitely graded aspect of Lie superalgebras. It may be useful and meaningful for promoting the development of not-finitely graded Lie superalgebras. This is the main motivation of this article.
The present paper is organized as follows. In Section 2, we review the basic notions about Lie superalgebras. In Section 3 and Section 4, we determine the derivations and automorphism groups of , respectively. Finally, the second cohomology groups of  are obtained in Section 5. The main results of this paper are summarized in Theorems 3.6, 4.2 and 5.3.

Preliminaries
In this section, we shall summarize some basic concepts about Lie superalgebras in [11,20].

Definition 2.1
A Lie superalgebra is a superalgebra for any x,y∈hg(L), z∈L.
Let L be a Lie superalgebra, then the space gc(L) consisting of all the linear transformations on L has a natural  2 -gradation: , for any β∈  2 }, for any α∈  2 . Now we give the definition of derivations.

Definition 2.2
for any x,y∈hg(L).

Definition 2.3
A bilinear form ψ : L×L→ is called a 2-cocycle on L if ψ satisfies the following conditions: for any x,y∈hg(L), z∈L.

Definition 2.4
For any linear map f:L→, we define a 2-cocycle ψ f in the following way: for any x, y ∈L. We call it a 2-coboundary of L.

Derivations of 
In this section, we shall determine the derivations of .

Lemma 3.2
For any D ∈ Der , replacing D with D − ad y for some y ∈ , we get D(L 0,0 )=0.
By using the following three equations Then, we get e α =d α for any α∈Ω.
The following theorem can be concluded by Lemmas 3.1-3.5 immediately, which is the main result of this section.

Automorphism Groups of 
The aim of this section is to characterize the automorphism groups of . According to [5], we get the following lemma.
We have completed the proof.

Second Cohomology Groups of 
Let ψ∈c 2 ( , ), we define a linear function f: → such that f(L α,i ) and f(G α,i ) are given inductively on i as follows Set φ=ψ−ψ f . According to [9], we obtain the following formula: for any α,β∈Ω,i, j∈ + .
Induction on j, we also obtain (G α,i ,G −α,j )=0 By previous computations, it will suffice to show the result. Now we get the main result of this section.

Conclusion
From the above study one can conlcude that Lie superalgebras as generalizations of Lie algebras were originated from supersymmetry in mathematical physics. The theory of Lie superalgebras plays a prominent role in modern mathematics and physics. The (generalized) super-Virasoro algebra is closely related to the conformal field theory and the string theory. It plays a very important role in mathematics and physics. The paper is devoted in the investigation of the structure theory of class of not finitely graded Lie superalgebras related to generalized super-Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined.