The Classification, Automorphism Group, and Derivation Algebra of the Loop Algebra Related to the Nappi–Witten Lie Algebra

Set L ≔ H 4 ⊗ C R , R ≔ C [ t ± 1 ] , and S ≔ C [ t ± 1/ m ]( m ∈ Z + ) . Then, L is called the loop Nappi–Witten Lie algebra. R -isomorphism classes of S / R forms of L are classiﬁed. The automorphism group and the derivation algebra of L are also characterized.


Introduction
e conformal field theory (CFT) plays a significant role in many different areas of mathematics and physics. Wess-Zumino-Novikov-Witten (WZNW) models [1] provide interesting examples of CFTs. WZNW models were first studied in the setting of semisimple groups. Later, since the WZNW models based on non-abelian nonsemisimple Lie groups were found to be closely related to the construction of exact string backgrounds, the special types received some attention [2,3]. Nappi and Witten showed in [3] that the WZNW model (NW model) based on a central extension of the twodimensional Euclidean group describes the homogeneous fourdimensional space-time corresponding to a gravitational plane wave. e related Lie algebra is called the Nappi-Witten Lie algebra, which is neither abelian nor semisimple. More results on the NW model were given in [4][5][6][7]. e Nappi-Witten Lie algebra H 4 is a four-dimensional Lie algebra over C generated by a, b, c, d { } with the following Lie brackets:  (2) en, L is a Lie algebra over C under the bracket as follows: x ⊗ t n 1 , y ⊗ t n 2 � [x, y] ⊗ t n 1 +n 2 , for x, y ∈ H 4 , n 1 , n 2 ∈ Z.

(4)
We call L the loop Nappi-Witten Lie algebra. Recently, the representation theory of Lie algebras related to L was studied in [8,9].
Reference [10] studied the automorphism groups of vertex operator algebras associated with the affine Nappi-Witten algebra H 4 . e isomorphism of loop algebras was considered in [11] from the perspective of non-abelian Galois cohomology. Reference [12] gave an explicit description of the algebra of derivations for a large class of algebras defined byétale descent. Motivated by the works mentioned above, we study R-isomorphism classes of S/R forms of L, the automorphism group, and the derivation algebra of L in this paper.
To classify R-isomorphism classes of S/R forms of L in Section 2, we recall the results of automorphisms of H 4 in [10]. e following Lemma 1, eorem 1, Lemma 2, and Lemma 3 will be used in the proof of eorem 2.
By eorem 1, any element of G can be viewed as an automorphism of H 4 .
□ e aim of this paper is to study structures of the loop Nappi-Witten Lie algebra L � H 4 ⊗ C R. Some ideas we use come from [10][11][12]. is paper is organized as follows. In Section 2, we classify R-isomorphism classes of S/R forms of L by the first cohomology set. In Section 3, we determine the automorphism group of L. Finally, the derivation algebra of L is also characterized in Section 4. roughout the paper, the sets of the complex numbers, the nonzero complex numbers, the nonnegative integers, the integers, and the positive integers are denoted by C, C * , N, Z, and Z + , respectively.

The First Cohomology
Set H 1 (Γ, Aut S (H 4 (S))) We first review some basic notions and assumptions in [11] and then apply them in the class of H 4 . Let k be an algebraically closed field of characteristic 0. For each m ∈ Z + , we choose a primitive m th root of unity ζ m such that these primitive roots of unity are compatible in the sense that ζ l lm � ζ m for all l, m ∈ Z + , and fix the sequence ζ m m∈Z + from now on. Suppose that g is a Lie algebra over k and that σ is an automorphism of g of period m (the order of σ might not be equal to m, but of course it is a divisor of m ). Let be the group of integers modulo m, where i denotes the congruence class represented by i. Let be two algebras of Laurent polynomials over k and R ⊂ S is a k-subalgebra. If we set z � t 1/m , then t � z m and It is easy to see that S is an R-algebra. ere is an isomorphism of Γ onto Aut R (S) (the group of automorphisms of S which fixes R) such that where g(K) is a K-module with K-action given by is also a Lie algebra over K under the bracket as follows: We now recall the definition of the loop algebra of g relative to σ. Let e loop algebra of g relative to σ is the subalgebra of the Lie algebra Journal of Mathematics Note that L(g, σ) is the set of fixed points in g(S) of the R-algebra automorphism σ ⊗ 1 − 1 , and then L(g, σ) is an R-subalgebra of g(S).
us, L(g, σ) is an R-algebra and hence also a k-algebra. Furthermore, the R-isomorphism class of L(g, σ) does not depend on the choice of the period m for σ (see Lemma 2.3 in [11]).
To interpret loop algebras as S/R forms of the R-algebra g(R), we review the definitions of the S/R form of g(R) and a certain (non-abelian) first cohomology set and relations between them.
An S/R form of g(R) is a Lie algebra F over R such that as Lie S-algebras. We now recall some concepts of a (non-abelian) cohomology set.
Let G be a group. We say that Γ-acts on G if there is a map If Γ acts on a group G, the map u: Γ ⟶ G, i↦u i satisfying for all i ∈ Γ. It is an equivalence relation. For every 1-cocycle u, we denote by [u] the equivalence class containing u. e set of equivalence classes (called cohomology classes) of 1cocycles from Γ to G is denoted by H 1 (Γ, G), and we call it the (non-abelian) first cohomology set.
Define an action of Γ on Aut S (g(S)) by for i ∈ Γ and τ ∈ Aut S (g(S)), where Aut S (g(S)) is the group of automorphisms of g(S) as a Lie S-algebra. e following result comes from Proposition 3.4 of [11].
Proposition 1 (see [11]). R-isomorphism classes of S/R forms of g(R) are in one-to-one correspondence with cohomology classes in H 1 (Γ, Aut S (g(S))). Explicitly this correspondence is as follows: if u is a 1-cocycle on Γ in Aut S (g(S)), then the cohomology class [u] corresponds to the R-isomorphism classes of the S/R form: of g(S).
Furthermore, eorem 3.6 in [11] gives our description of loop algebras as S/R forms of g(R). In view of eorem 3.6 in [11], the loop algebra L(g, σ) is an S/R-form of g(R) and is determined by the 1-cocycle: (32) In the following, set k � C and g � H 4 . Let S and R be the algebras of Laurent polynomials over C as in (19). We will classify loop algebras of H 4 from the view of S/R forms of H 4 (R).

Lemma 4.
Let σ, τ ∈ Aut(H 4 ) and σ m � τ m � id H 4 . We define 1-cocycles u and v on Γ in Aut S (H 4 (S)) by . For any i ∈ Γ and x ⊗ z n ∈ H 4 (S), we have (34) Journal of Mathematics then the corresponding R-isomorphism classes of the S/R form of H 4 (R) are as follows: en, m ∈ 2Z + by Lemma 2, and ψ is en, For We first recall the definition of the centroid in [11] that will be used later. Let K be a unital commutative associative algebra over C and T be a Lie algebra over K. Set where Ctd K (T) is called the centroid of T over K. Define It is clear that λ T (K) ⊆ Ctd K (T). We say that T is central over K if Ctd K (T) � λ T (K) (see [11]).
Proof. Let χ ∈ Ctd C (H 4 ). We can assume that we have μ 1 � λ 2 � 0 and λ 1 � μ 2 . us, Proof. We assume that Journal of Mathematics where λ χ we have So, and It is easy to see that τ is a vector space isomorphism.
Proof. We just need to prove the commutativity of Ctd R (L). Let χ 1 , χ 2 ∈ Ctd R (L). By eorem 3, it is clear that for x ∈ a, b, c { } and n ∈ Z. Moreover, Define the map where λ, ρ ∈ C * , λ χ 0 (t), α χ 0 (t) ∈ R. It is easy to check that η is an R-algebra isomorphism.
Journal of Mathematics Remark 1. Corollary 3 can also follow from Lemma 5 in [12].

Journal of Mathematics
Since for
en, θ c � id R and t k � θ c (t k ) � λ 0 λ k (t) for any k ∈ Z. us, by Lemma 6, we have If δ k (t) − t k δ 0 (t) � 0 for any k ∈ Z, then θ ∈ Aut R (L). If for any k ∈ Z}, and End C (R) is the set of all linear transformations of R.
Proof. For g ∈ Der(H 4 ), we can assume that we have (92) us, the lemma is proved.
Data Availability e data of the Lie algebra relations used to support the findings of this study are included within the article.

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