Quaternifiations and extensions of current algebras on S^3

Let $H$ be the quaternion algebra. Let $g$ be a complex Lie algebra and let $U(g)$ be the enveloping algebra of $g$. We define a Lie algebra structure on the tensor product space of $H$ and $U(g)$, and obtain the quaternification $g^H$ of $g$. Let $S^3g^H$ be the set of $g^H$-valued smooth mappings over $S^3$. The Lie algebra structure on $S^3g^H$ is induced naturally from that of $g^H$. On $S^3$ exists the space of Laurent polynomial spinors spanned by a complete orthogonal system of eigen spinors of the tangential Dirac operator on $S^3$. Tensoring $U(g)$ we have the space of $U(g)$-valued Laurent polynomial spinors, which is a Lie subalgebra of $S^3g^H$. We introduce a 2-cocycle on the space of $U(g)$-valued Laurent polynomial spinors by the aid of a tangential vector field on $S^3$. Then we have the corresponding central extension $\hat g(a)$ of the Lie algebra of $U(g)$-valued Laurent polynomial spinors. Finally we have the a Lie algebra $\hat g=\hat g(a)+Cd$ which is obtained by adding to $\hat g(a)$ a derivation $d$ which acts on $\hat g(a)$ as the radial derivation. When $g$ is a simple Lie algebra with its Cartan subalgebra $h$, We shall investigate the weight space decomposition of $(\hat g, ad(\hat h))$, where $\hat h=h+Ca+Cd$ . The previous versions (v1-v7) of this article contained several incorrect assertions and here we have corrected them.


Introduction
The set of smooth mappings from a manifold to a Lie algebra has been a subject of investigation both from a purely mathematical standpoint and from quantum field theory. In quantum field theory they appear as a current algebra or an infinitesimal gauge transformation group. Loop algebras are the simplest example. Loop algebras and their representation theory have been fully worked out. A loop algebra valued in a simple Lie algebra or its complexification turned out to behave like a simple Lie algebra and the highly developed theory of finite dimensional Lie algebra was extended to such loop algebras. Loop algebras appear in the simplified model of quantum field theory where the space is one-dimensional and many important facts in the representation theory of loop algebra were first discovered by physicists. We aim the three-dimensional generalization of this theory. It turned out that in many applications to field theory one must deal with certain extensions of the associated loop algebra rather than the loop algebra itself. The central extension of a loop algebra is called an affine Lie algebra and the highest weight theory of finite dimensional Lie algebra was extended to this case. [K], [K-W], [P-S] and [W] are good references to study these subjects. In this paper we shall investigate a generalization of affine Lie algebras to the Lie algebra of mappings from three-sphere S 3 to a Lie algebra. As an affine Lie algebra is a central extension of the Lie algebra of smooth mappings from S 1 to the complexification of a Lie algebra, we shall introduce an extension of the Lie algebra of smooth mappings from S 3 to the quaternification of a Lie algebra.
As for the higher dimensional generalization of loop groups, J. Mickelsson introduced an abelian exension of current groups Map(S 3 , SU(N)) for N ≥ 3, [M]. It concerns the Chern-Simons function on the space of SU(N)-connections. One of the author in [Ko4] gave a precise argument of Mickelsson's theory and investigated the adjoint orbit representation of the extended current algebra. Now we shall give a brief explanation of each section.
Let H be the quaternion numbers. In this paper we shall denote a quaternion a + jb ∈ Let ( g , , g ) be a complex Lie algebra. Let U(g) be the enveloping algebra. The quaternification of g is defined as the vector space g H = H ⊗ U(g) endowed with the bracket for z, w ∈ H and X, Y ∈ U(g) . It extends the Lie algebra structure (g, , g ) to g H , , g H . For example we have gl(n, C) H = gl(n, H).
The quaternions H give also a half spinor representation of Spin(4). That is, ∆ = H⊗ C = H ⊕ H gives an irreducible complex representation of the Clifford algebra Clif(R 4 ): Clif(R 4 ) ⊗ C ≃ End(∆), and ∆ decomposes into irreducible representations ∆ ± = H of Spin(4). Let S ± = C 2 × ∆ ± be the trivial even ( respectively odd ) spinor bundle. A section of spinor bundle is called a spinor. The space of even half spinors C ∞ (S 3 , S + ) is identified with the space S 3 H = Map(S 3 , H). Now the space S 3 g H = S 3 H ⊗ U(g) becomes a Lie algebra with respect to the Lie bracket: for X, Y ∈ g and φ, ψ ∈ S 3 H , which is extended to S 3 g H . In the sequel we shall abbreviate the Lie bracket [ , ] H simply to [ , ]. Such an abbreviation will be often adopted for other Lie algebras.
In section 2 we shall review the theory of spinor analysis after [Ko2,Ko3,Ko4]. Let be the polar decomposition on S 3 ⊂ C 2 of the Dirac operator, where ∂ / is the tangential Dirac operator on S 3 and γ + is the Clifford multiplication of the unit normal derivative on S 3 . The eigenvalues of ∂ / are given by { m 2 , − m+3 2 ; m = 0, 1, · · · }, with multiplicity (m + 1)(m + 2). We have an explicitly written formula for eigenspinors φ +(m,l,k) , φ −(m,l,k) 0≤l≤m, 0≤k≤m+1 corresponding to the eigenvalue m 2 and − m+3 2 respectively and they give rise to a complete orthogonal system in L 2 (S 3 , S + ). A spinor φ on a domain G ⊂ C 2 is called harmonic spinor on G if Dφ = 0. Each φ +(m,l,k) is extended to a harmonic spinor on C 2 , while each φ −(m,l,k) is extended to a harmonic spinor on C 2 \ {0}. Every harmonic spinor ϕ on C 2 \ {0} has a Laurent series expansion by the basis φ ±(m,l,k) : If only finitely many coefficients are non-zero it is called a spinor of Laurent polynomial type. The algebra of spinors of Laurent polynomial type is denoted by C[φ ±(m,l,k) ].
C [φ ±(m,l,k) ] is a subspace of S 3 H that is algebraically generated by φ +(0,0,1) = 1 0 , We call the vector res ϕ = −C −(0,0,1) C −(0,0,0) the residue of ϕ at 0. We have the residue formula: Recall that the central extension of a loop algebra Lg = C[z, z −1 ] ⊗g is the Lie algebra (Lg ⊕ Ca , [ , ] c ) given by the bracket with the aid of the 2-cocycle c(P, Q) = res z=0 ( d dz P ·Q) = 1 2 res z=0 ([ d dz , P ]·Q) on C[z, z −1 ]. Here (·|·) is a non-degenerate invariant symmetric bilinear form on g. We shall give in section 3 an analogous 2-cocycle on S 3 H. For φ 1 , φ 2 ∈ S 3 H, we put Then c defines a 2-cocycle on the Lie algebra ( S 3 H, [ , ] ). That is, c satisfies the following equations: Let a be an indefinite number. On the C-vector space S 3 g H = S 3 g H ⊕ Ca we define the following Lie bracket. For X, Y ∈ U(g) and φ, ψ ∈ S 3 H we put Then S 3 g H , [ , ] ∧ becomes an extension of the Lie algebra S 3 g H with 1-dimensional center Ca. The above Lie algebra structure on S 3 g H induces the central extension of the Lie algebra C[φ ±(m,l,k) ] ⊗ U(g) of the spinors of Laurent polynomial type, that will be denoted by ( g(a), [ , ] ∧ ); In section 4 we shall construct the Lie algebra which is obtained by adding to g(a) a derivation d which acts on g(a) as radial derivation ∂ ∂n on S 3 . But, contrary to the case of loop algebra it does not kill the center a. The radial vector field is defined by . For any spinor of Laurent polynomial type φ, we have This yields the following fundamental property of the cocycle c .
Let g = ( C[φ ±(m,l,k) ] ⊗ U(g) ) ⊕ (Ca) ⊕ (Cd). We endow g with the bracket defined by Then ( g , [ , ] g ) is an extension of the Lie algebra g(a) on which d acts as ∂ ∂n . In section 5, when g is a simple Lie algebra with its Cartan subalgebra h , we shall investigate the root space decomposition of g with respect to the Cartan subalgebra h = (φ +(0,0,1) ⊗ h) ⊕ (Cd).
For this purpose we look at the representation of the adjoint action of h on the enveloping algebra U(g). Let g = n + ⊕ h ⊕ n − be the decomposition of g according to the positive and negative roots ∆ ± . We see that the set of weights of (ad, U(g)) is nothing but the set of roots ∆ and the weight space of α ∈ ∆ is given by where α 1 , · · · , α r are the set of simple roots and α = r i=1 (n i − m i )α i with m i , n i ∈ Z >0 . The weight space decomposition becomes Now the dual space h * of h can be regarded naturally as a subspace of h * . So ∆ ⊂ h * is seen to be a subset of h * . We define δ ∈ h * by putting δ, α ∨ i = 0, 1 ≦ i ≦ r and δ, d = 1. Then the root system ∆ of ( g, h ) is The root space decomposition of g is given by Each root space is given as follows: We note that the following decomposition holds The Chevalley generators of g will be given in section 5.3. Let {α i } i=1,··· ,r ⊂ h * be the set of simple roots and {h i } i=1,··· ,r ⊂ h be the set of simple coroots and let (e i , f i , h i ); i = 1, · · · , r, denote the Chevalley generators of g. We put h i = φ +(0,0,1) ⊗ h i ∈ h, e i = φ +(0,0,1) ⊗ e i ∈ g 0δ+α i and f i = φ +(0,0,1) ⊗ f i ∈ g 0δ−α i . Then ( e i , f i , h i ); i = 1, · · · , r, are the part of Chevalley generators that come from g. Let θ be the highest root of g. Let π denote one of the three vectors −φ +(0,0,0) , φ +(1,0,1) and φ −(0,0,0) . We introduce f π = π ⊗ e −θ and e π = π * ⊗ e θ , where π * is the vector that satisfies ππ * = π * π = φ +(0,0,1) and c(π, π * ) = 0. Then the Chevalley generators of g(a) = (C[φ ±(m,l,k) ] ⊗ U(g)) ⊕ (Ca) are given by With d we have the generators of g.

Acknowledgement
The authors would like to express their thanks to Professors Tatsuo Suzuki of Shibaura Institute of Technology, Hideyuki Ishi of Nagoya University and Yasushi Homma of Waseda University for their valuable remarks at the final step of our work.
1 Quaternification of a Lie algebra

Quaternion algebra
The quaternions H are formed from the real numbers R by adjoining three symbols i, j, k satisfying the identities: By taking x 3 = x 4 = 0 the complex numbers C are contained in H if we identify i as the usual complex number. Every quaternion x has a unique expression x = z 1 + jz 2 with z 1 , z 2 ∈ C. This identifies H with C 2 as C-vector spaces. The quaternion multiplication will be from the right x −→ xy where y = w 1 + jw 2 with w 1 , w 2 ∈ C: xy = (z 1 + jz 2 )(w 1 + jw 2 ) = (z 1 w 1 − z 2 w 2 ) + j(z 1 w 2 + z 2 w 1 ). (1. 2) The multiplication of a g = a + jb ∈ H to H from the left yields an endomorphism in H: {x −→ gx} ∈ End H (H). If we look on it under the identification H ≃ C 2 mentioned above we have the C-linear map This establishes the R-linear isomorphism where we defined (1.5) The complex matrices corresponding to i, j, k ∈ H are These are the basis of the Lie algebra su(2). Thus we have the identification of the following objects (1.6) The correspondence between the elements is given by H becomes an associative algebra with the multiplication law defined by which is the rewritten formula of (1.2) and the right-hand side is the first row of the It implies the Lie bracket of two vectors in H, that becomes (1.9) The Lie group corresponding to the Lie algebra H is nothing but the general H-linear group and is isomorphic to GL(2, C).
These expressions are very convenient to develop the analysis on H, and give an interpretation on the quaternion analysis by the language of spinor analysis.
and we have, for z 1 , z 2 , z 3 ∈ H, (1.11) The center of the Lie algebra H is t 0 ∈ H; t ∈ R ≃ R, and (1.6) says that H is the trivial central extension of su (2).
R 3 being a vector subspace of H: we have the action of H on R 3 .

1.2
Lie algebra structure on H ⊗ U (g) Let ( g , , g ) be a complex Lie algebra. Let U(g) be the enveloping algebra of g. Let g H = H ⊗ U(g) and define the following bracket on g H : for z, w ∈ H and X, Y ∈ g . · , · g H is extended naturally to H ⊗ U(g) . In fact the following calculation in H ⊗ U(g) shows that the extension is well defined .
Thus, for X, Y ∈ U(g) and z, w ∈ H , we have (1.15) By the quaternion number notation every element of H ⊗ g may be written as X + jY with X, Y ∈ g. Then the above definition is equivalent to where X is the complex conjugate of X.
Proposition 1.2. The bracket · , · g H defines a Lie algebra structure on H ⊗ U(g) .

Proof
Obviously (1.15) gives an antisymmetric bracket. Next we shall prove the Jacobi identity; where we suppressed the suffix g H . From (1.15) we have Summing up these formulas for cyclic permutation of {1, 2, 3} we find the Jacobi identity (1.17).
If g is a Lie subalgebra of gl(n, C) and satisfies the property:  [G-M], and we follow the calculations developed in [Ko1], [Ko2] and [Ko3].

Harmonic polynomials
The Lie group SU(2) acts on C 2 both from the right and from the left. Let dR(g) and dL(g) denote respectively the right and the left infinitesimal actions of the Lie algebra su(2). We define the following vector fields on C 2 : where {e i ; i = 1, 2, 3} is the normal basis of R 3 . Each of the triple θ i (z), i = 1, 2, 3, and τ i (z), i = 1, 2, 3, gives a basis of the vector fields on the three sphere {|z| = 1} ≃ S 3 .
It is more convenient to introduce the following vector fields: We have the commutation relations; Both Lie algebras spanned by (e + , e − , θ) and (ê + ,ê − ,θ) are isomorphic to sl(2, C).
Therefore the space of harmonic polynomials on C 2 is decomposed by the right action gives an (m+1) dimensional irreducible representation of SU(2) with the highest weight m 2 .
We have the following relations.

Harmonic spinors
is called the even ( resp. odd ) spinor bundle and the sections are called even ( resp. odd ) spinors. The set of even spinors or odd spinors on a set M ⊂ C 2 is nothing but the smooth functions on M valued in H: The Dirac operator is defined by is the bundle homomorphism coming from the Clifford multiplication. By means of the decomposition S = S + ⊕ S − the Dirac operator has the chiral decomposition: We find that D and D † have the following coordinate expressions; An even (resp. odd) spinor ϕ is called a harmonic spinor if Dϕ = 0 ( resp. D † ϕ = 0 ).
We shall introduce a set of harmonic spinors which forms a complete orthonormal basis of L 2 (S 3 , S + ) .
Let ν and µ be vector fields on C 2 defined by (2.20) Then the radial vector field is defined by The vector field θ in (2.4) is also written by θ = 1 2 √ −1 (ν −ν). We shall denote by γ the Clifford multiplication of the radial vector ∂ ∂n , (2.21). γ changes the chirality: The matrix expression of γ becomes as follows: In the sequel we shall write γ + (resp. γ − ) for γ|S + (resp. γ|S + ).
Proposition 2.2. The Dirac operators D and D † have the following polar decompositions: where the tangential (nonchiral) Dirac operator ∂ / is given by Proof. In the matrix expression (2.19) of D and D † , we have ∂ ∂z 1 = 1 |z| 2 (z 1 ν − z 2 e − ) etc., and we have the desired formulas.
The tangential Dirac operator on the sphere S 3 = {|z| = 1}; is a self adjoint elliptic differential operator.

The set of eigenspinors
forms a complete orthonormal system of L 2 (S 3 , S + ).
The constant for normalization of φ ±(m,l,k) is determined by the integral: where σ is the surface measure of the unit sphere S 3 = {|z| = 1}: (2.30) φ +(m,l,k) is a harmonic spinor on C 2 and φ −(m,l,k) is a harmonic spinor on C 2 \{0} that is regular at infinity. If ϕ is a harmonic spinor on C 2 \ {0} then we have the expansion 2. For a spinor of Laurent polynomial type ϕ we call the vector res ϕ = −C −(0,0,1) C −(0,0,0) the residue at 0 of ϕ.
We shall see later that C[φ ±(m,l,k) ] with the multiplication law coming from (1.8) becomes an associative algebra.
We have the residue formula. See, for example, Proposition 4.2 of [Ko3]. (2.32) Here the left hand side is a number in C 2 . We note that if we consider γ 0 = 1 |z| z 1 z 2 ∈ S 3 H then the Clifford multiplication γ + coincides with the left multiplication of γ 0 in the algebra S 3 H : under the identification ∆ + ≃ H ≃ ∆ − . In the sequel we abbreviate to γ 0 = γ + . We have

2.3
Algebraic generators of C[φ ±(m,l,k) ] We investigate the generators of the algebra C[φ ±(m,l,k) ]. First we observe the following facts.
C[φ ±(m,l,k) ] becomes an associative algebra and the Lie algebra structure follows from it.
Hence we find that the algebra C[φ ±(m,l,k) ] is generated by the following I, J, κ, µ: The others are generated by these basis, For example, Here we state a lemma that will be used in section 5. tr Jϕ dσ = 4π 2 Re.C +(0,0,0) .
be the set of smooth even spinors on S 3 . We define the Lie algebra structure on S 3 H after (1.9), that is, for even spinors φ 1 = u 1 v 1 and φ 2 = u 2 v 2 , we have the Lie bracket We define the following symmetric bilinear form on S 3 H : Proposition 3.1.

7)
and We have Then Proposition is easily proved.

Extensions of the Lie algebra C[φ ±(m,l,k) ] ⊗ U (g)
The central extension of the loop algebra Lg C = Map(S 1 , g C ) for a Lie algebra g was studied by many authors both in mathematics and in physics [K], [K-W], [P-S], [W] etc.
In this section we shall construct a central extension for the 3-dimensional loop algebra Map(S 3 , g H ) = S 3 H ⊗ U(g) associated to the above 2-cocycle c , that induces a central extension of C[φ ±(m,l,k) ] ⊗ U(g) . Then we shall obtain the second extension by adding a derivative to the first extension that acts as the radial derivation. The second extension is not central.

A central extension of the Lie algebra
for X, Y ∈ U(g) and φ, ψ ∈ S 3 H .
Theorem 4.2. Let a be an indefinite number . The C vector space: endowed with the following bracket becomes a Lie algebra.

Proof
In the sequel we shall write [ , ] simply by [ , ] and omit the upper wedge indicating on which space the bracket is considered. There would not be any confusion. Since Hence the bracket [ , ] is antisymmetric. Next we shall prove the Jacobi identity: (4.5) We have Here we used the relation [a, φ 3 ⊗ Z] = 0. Then We Therefore we have the Jacobi identity (4.5).
As a subalgebra of S 3 g H we have the central extension of the Lie algebra of Laurent polynomial type spinors valued in g: But this is not the extension we want to have as an analogy of Kac-Moody Lie algebra.
In the following we shall have a Lie algebra g which is obtained by adding to g(a) a derivation d .

Extension of C[φ ±(m,l,k) ] ⊗ U (g)
We shall introduce an analogous concept of the affine Lie algebra, that is, we shall define a Lie algebra extension of the Lie algebra of Laurent polynomial type spinors valued in g. But it is not a central extension.
Lemma 4.4. (4.10) Proof From (2.34), we have Theorem 4.5. Let a and d be indefinite numbers. We consider the C vector space:

Proof
By virtue of Theorem 4.2 it remains for us to prove the following Jacobi identity: In the following we shall abbreviate the bracket [ , ] g simply to [ , ].
We have Similarly here we used the fact [a, d] = 3 2 a. We see that the sum of the first two terms reduces to zero from Lemma 4.3. On the other hand Lemma 4.4 yields Therefore the assertion is proved.
Let ( g , [ , ] g ) be a simple Lie algebra. Let h be a Cartan subalgebra of g and g = h⊕ α∈∆ g α be the root space decomposition with the root space g α = {X ∈ g; [ h, X ] = < α, h > X, ∀h ∈ h} . Here ∆ = ∆(g, h) is the set of roots and dim g α = 1. Let Π = {α i ; i = 1, · · · , r = rank g} ⊂ h * be the set of simple roots and {α ∨ i ; i = 1, · · · , r } ⊂ h be the set of simple coroots. The Cartan matrix A = ( a ij ) i,j=1,··· ,r is given by a ij = α i , α ∨ j . Every root α ∈ ∆ is written uniquely in the form We shall investigate the Lie algebra structure of (5.1) Recall that the Lie bracket was given by the formulas: First we shall investigate the structure of the following vector subspace of g : g ′ is not in general a Lie algebra, that is, the above bracket is not closed in g ′ .
Definition 5.1. A subspace a of g ′ is called a Lie subalgebra of g ′ if a becomes a Lie algebra with respect to the bracket [ , ] g : Since φ +(0,0,1) = 1 0 we identify X ∈ g with φ +(0,0,1) ⊗ X and we look g as a Lie subalgebra of g : h ′ is a subalgebra of g ′ in the above sense, and ad (H) is diagonalizable for any H ∈ h ′ .
h ′ is a maximal abelian subalgebra of g ′ .
Remark 5.2. Since φ +(0,0,0) = 0 −1 the subspace becomes an abelian subalgebra of g ′ that is larger than h ′ , but the adjoint representation ad of k ′ in g ′ is not diagonalizable.
An element λ of the dual space h * of h can be regarded as an element of h * by putting So ∆ ⊂ h * is seen to be a subset of (h ′ ) * . We define the elements δ ∈ (h ′ ) * by Then any λ ∈ (h ′ ) * is written in the form λ = cδ + α with c ∈ C and α ∈ h * , and the set { α 1 , · · · , α r , δ } forms a basis of (h ′ ) * .
For any H ∈ h and X ∈ g, it holds that Hence g ′ is invariant under the adjoint action of h ′ . The adjoint action ad (ĥ) on g ′ is diagonalizable for everyĥ ∈ h ′ , and the vector space g ′ decomposes into a direct sum of the simultaneous eigenspaces of ad (ĥ),ĥ ∈ h ′ .
We proceed now to the root space decomposition of g with respect to h.
First we remember several properties of the enveloping algebra U(g). Let ∆± be the set of positive ( respectively negative ) roots of g and put n ± = α∈∆ ± g α .
Then g = n + ⊕ h ⊕ n − and we have We shall look at the root space decompositions of U(g) under the adjoint action ad(h) of h ∈ h on U(g). Evidently we have We know that, if X i 's are a basis of g then the set { X n 1 1 X n 2 2 · · · ; n i ∈ N ∪ 0 }, where only finitely many n i 's are not zero, forms a basis of the enveloping algebra U(g). We choose with X ±α i ∈ g ±α i and n i , m i ∈ N, i = 0, 1, · · · , r. Then the adjoint action ad(h) is written as follows, [D].
Therefore any α ∈ ∆ is a weight of the representation (U(g), ad(h)) with the weight vetor X α . Of course any weight of (U(g), ad(h)) is a root of g .
Lemma 5.4. 1. The set of weights of the Lie algebra U(g) with respect to ad(h) is ∆ .
2. Let g U α be the weight space of α. Then 3. The weight space decomposition of U(g) with respect to h is given by Turning to the Lie algebra g , the adjoint actions ofĥ = φ +(0,0,1) ⊗ h + td ∈ h on g is written as follows. Take a ξ = φ ⊗ X + µa + νd ∈ g. Then Since h is a commutative subalgebra of g, g is decomposed into the eigenspaces of ad( h). We put g λ = ξ ∈ g ; ξ / ∈ h, and [ĥ, ξ ] = λ,ĥ ξ for ∀ĥ ∈ h .
(5.21) λ = α + kδ ∈ h * for α ∈ ∆ and k ∈ Z is called a root of g if g λ = 0. Let ∆ denote the set of roots of g . g λ is called the root space of λ. It holds also that g λ ⊂ g(a) .
By the same argument as in the previous subsection we have the following  for any h ∈ h. Therefore g m 2 δ+α , g n 2 δ+β g ⊂ gm+n 2 δ+α+β , (5.33) The same calculation for φ ⊗ H ∈ g m 2 δ and ψ ⊗ H ′ ∈ g n 2 δ yields g m 2 δ , g n 2 δ g ⊂ gm+n 2 δ . (5.34) The rests are proved in the same way. The second assertion for the commutativity in case of n = m − 1 or n = m − 2 is proved by virtue of (5.22).
Theorem 5.7. There exists an invariant symmetric bilinear form ( · | · ) on the Lie algebra g. It is given by

Proof
We shall verify the following invariance conditions. The others are immediate consequences of the definition. 1.
From the above discussion we find that the Chevalley generators of g(a) are given by e i , f i , h i , i = 1, 2, · · · , r, e µ , f µ , h θ , e κ , f κ , h θ , e J , f J , h θ . (5.53) With d we have the generators of g. Though we do not have exact relations e κ ∈ g − 3 2 δ+θ nor e µ ∈ g1 2 δ+θ we consider them Chevalley basis up to a base change in the root space.