Simple Finite Jordan Pseudoalgebras

We consider the structure of Jordan $H$-pseudoalgebras which are linearly finitely generated over a Hopf algebra $H$. There are two cases under consideration: $H=U(\mathfrak h)$ and $H=U(\mathfrak h)# \mathbb C[\Gamma ]$, where $\mathfrak h$ is a finite-dimensional Lie algebra over $\mathbb C$, $\Gamma $ is an arbitrary group acting on $U(\mathfrak h)$ by automorphisms. We construct an analogue of the Tits-Kantor-Koecher construction for finite Jordan pseudoalgebras and describe all simple ones.


Introduction
The notion of pseudoalgebra appeared as a natural generalization of the notion of conformal algebra. The last one provides a formal language describing algebraic structures underlying the singular part of the operator product expansion (OPE) in conformal field theory. Roughly speaking, the OPE of two local chiral fields is a formal distribution in two variables presented as N −1 n=−∞ c n (z)(w − z) −n−1 [3]. The coefficients c n , n ∈ Z, of this distribution are considered as new "products" on the space of fields. The algebraic system obtained is called a vertex algebra. Its formal axiomatic description was stated in [4] (see also [7]). The "singular part" of a vertex algebra, i.e., the structure defined by only those operations with non-negative n, is a (Lie) conformal algebra [7].
Another approach to the theory of vertex algebras gives rise to the notion of a pseudotensor category [2] (which is similar to the multicategory of [14]). Given a Hopf algebra H, one may define the pseudotensor category M * (H) [1] (objects of this category are left H-modules). An algebra in this category is called an H-pseudoalgebra. A pseudoalgebra is said to be finite if it is a finitely generated H-module.
In particular, for the one-dimensional Hopf algebra H = k, k is a field, an H-pseudoalgebra is just an ordinary algebra over the field k. For H = k [D], an H-pseudoalgebra is exactly the same as conformal algebra. In a more general case H = k[D 1 , . . . , D n ], n ≥ 2, the notion of an H-pseudoalgebra is closely related with Hamiltonian formalism in the theory of non-linear evolution equations [1]. For an arbitrary Hopf algebra H, an H-pseudoalgebra defines a functor from the category of H-bimodule associative commutative algebras to the category of H-module algebras (also called H-differential algebras).
An arbitrary conformal algebra C can be canonically embedded in a "universal" way into the space of formal power series A[[z, z −1 ]] over an appropriate ordinary algebra A [7,15]. This algebra A = Coeff C is called the coefficient algebra of C. A conformal algebra is said to be associative (Lie, Jordan, etc.) if so is its coefficient algebra. For pseudoalgebras, a construction called annihilation algebra [1] works instead of coefficient algebra. However, the notion of a pseudotensor category provides a direct way to the definition of what is a variety of pseudoalgebras [13].
In the paper [5], the complete description of simple finite Lie conformal algebras over k = C was obtained. Apart from current conformal algebras, the only example of a simple finite Lie conformal algebra is the Virasoro conformal algebra. In the associative case, there are no exceptional examples: A simple finite associative conformal algebra is isomorphic to the current algebra over M n (C), n ≥ 1. It was shown in [19] that there are no exceptional examples of simple finite Jordan conformal algebras.
In [1], the structure theory of finite Lie pseudoalgebras was developed. The classification theorem [1,Theorem 13.2] states that there exist simple finite Lie pseudoalgebras which are not isomorphic to current pseudoalgebras over ordinary simple finite-dimensional Lie algebras. This is not the case for associative pseudoalgebras. In this paper, we show the same for finite  [19] to "multi-dimensional" case.
It was shown in [9] that the structure theory of Jordan conformal superalgebras is richer. The classification of simple finite Jordan superalgebras based on the structure theory of finite conformal Lie superalgebras [6] includes one series and two exceptional algebras [9,Theorem 3.9].
In our proof, we will not use annihilation algebras directly, the TKK construction will be built on the level of pseudoalgebras.
The paper is organized as follows. Section 2 contains the basics of Hopf algebras and pseudoalgebras theory, and some notations that will be used later. In Section 3, we introduce an analogue of the Tits-Kantor-Koecher construction (TKK) for finite Jordan pseudoalgebras. To complete the classification of finite simple Jordan pseudoalgebras, we need some technical results proved in Section 4. The main case under consideration is H = U (h), where h is a finite-dimensional Lie algebra over C. Another case is the smash-product U (h)#C[Γ], where Γ is an arbitrary group. These cases describe all cocommutative Hopf algebras over C with finite-dimensional spaces of primitive elements (see, e.g., [17]).

Preliminaries on Hopf algebras and pseudoalgebras 2.1 Hopf algebras
In this section, we state some notations that will be used later.
An associative algebra H with a unit (over a field k) endowed with coassociative coproduct ∆ : H → H ⊗ H and counit ε : H → k is called a bialgebra. Recall that both ∆ and ε are homomorphisms of algebras and To simplify the notation, we will use the following one which is due to Sweedler [17]: Further, we will omit the symbol Given a bialgebra H, an antihomomorphism S : H → H is called an antipode, if it satisfies A bialgebra with an antipode is called a Hopf algebra.
There exists a natural structure of (right) H-module on the nth tensor power of H (denoted by H ⊗n ): In this paper, we substantially consider cocommutative Hopf algebras, i.e., such that The antipode S of a cocommutative Hopf algebra is involutive, i.e., S 2 = id.
For example, the universal enveloping algebra U (g) of a Lie algebra g over a field of zero characteristic is a cocommutative Hopf algebra. Another series of examples is provided by the group algebra k[Γ] of an arbitrary group Γ and by the general construction of smash-product. Namely, suppose H is a Hopf algebra, and let a group Γ acts on H by algebra automorphisms. Then one may define the following new product on H ⊗ k[Γ]: In other words, the set {h i 1 ⊗ · · · ⊗ h in ⊗ 1 | i 1 , . . . , i n ∈ I} is an H-basis of the H-module H ⊗n+1 (2.1).
To find the presentation (2.2), one may use formal Fourier transformation F and its inverse F −1 [1]: We will use a "left" analogue of the Fourier transformation which is also invertible.

Dual algebras
Suppose H is a cocommutative Hopf algebra, and let X = H * be its dual algebra (i.e., the product on X is dual to the coproduct on H). Let us fix a linear basis {h i | i ∈ I} of H and denote by {x i | i ∈ I} ⊂ X the set of dual functionals: An arbitrary element x ∈ X can be presented as an infinite series in x i : The algebra X is a left and right module over H with respect to the actions given by xh, g = x, gS(h) , hx, g = x, S(h)g , x ∈ X, h, g ∈ H. (2. 3) The actions (2.3) turn X into a differential H-bimodule, i.e., (xy)h = (xh (1) )(yh (2) ), h(xy) = (h (1) x)(h (2) y). The operation ∆ X : X → X ⊗ X := (H ⊗ H) * dual to the product on H is somewhat similar to a coproduct. From the combinatorial point of view, X ⊗ X can be considered as the linear space of all infinite series i,j∈I In order to unify notations, we will use x (1) ⊗ x (2) for ∆ X (x), x ∈ X. In particular, the analogues of Fourier transforms act from X ⊗ X to X ⊗ X.
A local map π : X ⊗ X → V can be naturally continued toπ : X ⊗ X → V . The mapπ is continuous with respect to the topology on X ⊗ X defined by the following family of basic neighborhoods of zero: (we assume V is endowed with discrete topology). Conversely, given a continuous linear map X ⊗ X → V , its restriction to X ⊗ X is local. For example, let us fix h 1 , h 2 ∈ H and consider the map x ⊗ y → x, h 1 y, h 2 . It is clear that this map is local. Obviously, every local map π : X ⊗ X → V is actually the "evaluation" map π(x ⊗ y) = ev a (x, y) := ( x, · ⊗ y, · ⊗ id V )(a) (2.4) for an appropriate a ∈ H ⊗ H ⊗ V .
Proof . Formally speaking, we can not use F −1 as an inverse of F since both F and F −1 are not defined on the entire space X ⊗ X. But it is straightforward to check (see also [1]) that This is easy to deduce that if π = ev a as in (2.4) then For πF −1 the proof is completely analogous.

Pseudoalgebras
In the exposition of the notion of pseudoalgebra we will preferably follow [1].
An H-module P endowed with a pseudoproduct * is called a pseudoalgebra over H (or Hpseudoalgebra). If P is a finitely generated H-module, then P is said to be finite pseudoalgebra.
For every n, m ≥ 1, an H-bilinear map * : This operation allows to consider long terms in P with respect to * .
One of the main features of a cocommutative bialgebra H is that symmetric groups S n act by H-module automorphisms on H ⊗n with respect to (2.1). The action of σ ∈ S n is defined by Let us write down the obvious rules matching the action of S n with the "expanded" pseudo- whereτ ∈ S n+m , kτ = kτ for k = 1, . . . , n, kτ = k for k = n + 1, . . . , n + m; where iσ +n = i for i = 1, . . . , n, (n + j)σ +n = n + jσ for j = 1, . . . , m. A pseudoproduct * : P ⊗ P → (H ⊗ H) ⊗ H P can be completely described by a family of binary algebraic operations. Let P be an H-pseudoalgebra, X = H * . Lemma 1 implies that for every a, b ∈ P their pseudoproduct has a unique presentation of the form for all x ∈ X. The x-products obtained have the following properties: locality Note that the locality property does not depend on the choice of a basis in H: , where t n , D m = n!δ n,m , the correspondence between conformal n-products (n ≥ 0) and the pseudoproduct is provided by In the same way, one may define Fourier coefficients of an arbitrary element A ∈ H ⊗n ⊗ H P , n ≥ 2. By Lemma 1 A can be uniquely presented as . . , i n−1 ) ∈ I n−1 . By abuse of terminology, we will call these aī ∈ P Fourier coefficients of A.
There is a canonical way to associate an ordinary algebra A(P ) with a given pseudoalgebra P [1]. As a linear space, A(P ) coincides with X ⊗ H P , and the product is given by The algebra A(P ) obtained is called the annihilation algebra of P . If P is a torsion-free Hmodule then the structure of P can be reconstructed from A(P ) [1].
In the case of conformal algebras (H = k[D]), there is a slightly different construction called coefficient algebra [7,8,15]. Suppose C is a conformal algebra and consider the space An arbitrary conformal algebra can be embedded into a conformal algebra of formal power series over its coefficient algebra [7].

Varieties of pseudoalgebras
Suppose Ω is a variety of ordinary algebras over a field of zero characteristic. Then Ω is defined by a family of homogeneous polylinear identities. Such an identity can be written as where each t σ (y 1 , . . . , y n ) is a linear combination of non-associative words obtained from y 1 . . . y n by some bracketing.
Let Ω be a variety of ordinary algebras defined by a family of homogeneous polylinear identities of the form (2.10). Then set the Ω variety of pseudoalgebras as the class of pseudoalgebras satisfying the respective "pseudo"-identities of the form where t * σ means the same term t σ with respect to the pseudoproduct operation * .
If P is an Ω pseudoalgebra (or, in particular, conformal algebra) then its annihilation (coefficient) algebra belongs to the Ω variety of ordinary algebras [13]. The converse is also true for conformal algebras.
However, the class of Ω pseudoalgebras is not a variety in the ordinary sense: This class is not closed under Cartesian products.
The main object of our study is the class of Jordan pseudoalgebras. Recall that the variety of Jordan algebras is defined by the following identities: In the polylinear form (if char k = 2, 3) the second identity of (2.12) can be rewritten as follows (see, e.g., [20]): Here [. . . ] and {. . . } stand for the following bracketing schemes: [a 1 . . . a n ] = (a 1 [a 2 . . . a n ]), Therefore, an H-module P (over a cocommutative Hopf algebra H) endowed with a pseudoproduct • is a Jordan pseudoalgebra iff it satisfies the following identities of the form (2.11): As in the case of ordinary algebras, the natural relations hold between associative, Lie, and Jordan pseudoalgebras. An associative pseudoalgebra P with respect to the new pseudoproduct is a Lie pseudoalgebra denoted by P (−) [1]. Similarly, another pseudoproduct • given by makes P into a Jordan pseudoalgebra P (+) [13]. Example 1. Let H ′ be a Hopf subalgebra of H, and let P ′ be an H ′ -pseudoalgebra with respect to a pseudoproduct * ′ . Define a pseudoproduct on P = H ⊗ H ′ P ′ by linearity: is an associative pseudoalgebra. In particular, if h is the 1-dimensional Lie algebra then W (h) is just the Virasoro conformal algebra [7].
Later we will use the classification of simple finite Lie pseudoalgebras [1]. Although the results obtained in [1] are much more explicit, the following statements are sufficient for our purposes.
Theorem 1 ([1]). A simple finite Lie pseudoalgebra L over H = U (h), dim h < ∞, k = C, is isomorphic either to Cur g, where g is a simple finite-dimensional Lie algebra, or to a subalgebra of W (h).
is a finite direct sum of isomorphic simple U (h)-pseudoalgebras such that Γ acts on them transitively.

Conformal linear maps
For example, if P is a pseudoalgebra, a ∈ P , then the operator of left multiplication L a : b → a * b, b ∈ P , belongs to Cend l (P ).
In order to unify notations, we will use ϕ * a for ϕ(a), a ∈ M , ϕ ∈ Cend l (M ). One may consider * here as an H-bilinear map from Cend l (M ) ⊗ M to (H ⊗ H) ⊗ H M . The relation (2.5) allows to expand this map to The correspondence between ϕ * a and (ϕ • x a) (x ∈ X) is given by The space Cend l (M ) can be also endowed with a family of x-products given by for a ∈ M . The x-products (· • x ·) on Cend l (M ) satisfy (2.9), but (2.8) does not hold, in general. To ensure the locality, it is sufficient to assume that M is a finitely generated H-module [1, Section 10]. Therefore, Cend l (M ) for a finitely generated H-module M is an associative Hpseudoalgebra.
For a finite pseudoalgebra P , it is easy to rewrite the identity (2.13) using the operators of left multiplication. Namely, this identity is equivalent to where σ 123 denotes the permutation (1 2 3) ∈ S 4 .

Tits-Kantor-Koecher construction for f inite Jordan pseudoalgebras
The general scheme described in [10,11,12,18] provides an embedding of a Jordan algebra into a Lie algebra. It is called the Tits-Kantor-Koecher (TKK) construction for Jordan algebras. Let us recall the TKK construction for ordinary algebras. For a Jordan algebra j, the set of derivations Der(j) is a Lie subalgebra of End(j) with respect to the commutator of linear maps. Consider the (formal) direct sum S(j) = Der(j) ⊕ L(j), where L(j) is the linear space of all left multiplications L a : b → ab, a ∈ j. It is well-known that [L(j), L(j)] ⊆ Der(j). Then the space S(j) with respect to the new operation [·, ·] given by is a Lie algebra called the structure Lie algebra of j. Finally, consider Let us endow T(j) with the following operation: where Σ * = −L a + D for Σ = L a + D ∈ S(j). This operation makes T(j) to be a Lie algebra called the TKK construction for j.
In the case of conformal algebras, a similar construction was introduced in [19] by making use of coefficient algebras. We are going to get an analogue of TKK construction for finite Jordan pseudoalgebras using the language of pseudoalgebras rather than annihilation algebras. for all a, b ∈ P . The set of all pseudoderivations of P we denote by Der l (P ).
In particular, if P is a finite pseudoalgebra then (3.1) is equivalent to [T * L a ] = L T * a , a ∈ P .

Lemma 3. Suppose that for some
holds for all a, b ∈ P . Then all Fourier coefficients of A belong to in Der l (P ).
Proof . For every B ∈ H ⊗n+1 ⊗ H M (M is an H-module), there exists a unique presentation where Gī form a linear basis of H ⊗n (see Lemma 1). By B x 1 ,...,xn , x i ∈ X, we denote the expression ī x 1 ⊗ · · · ⊗ x n , Gī bī.
It is clear that the map (x 1 , . . . , x n ) → B x 1 ,...,xn is polylinear. If we fix an arbitrary set of n − 2 arguments, then the map X ⊗ X → M obtained is local in the sense of Definition 1.
In the same way, The relations (3.4)-(3.6) together with Lemma 2 imply It means that Lemma 4. For a finite pseudoalgebra P the set of all pseudoderivations is a subalgebra of the Lie pseudoalgebra Cend l (P ) (−) .
Since Cend l (P ) (−) satisfies Jacobi identity, Hence, for every a, b ∈ P we have Lemma 3 implies that [D 1 • x D 2 ] ∈ Der l (P ) for all x ∈ X. Proof . The following relation is easy to deduce from (2.13): So by (2.6), (2.7) It is sufficient to prove that for every a, b, Subtracting (3.9) from (3.10) and using commutativity L a * b = (σ 12 ⊗ H id)L b * a , we obtain (3.8). is called the structure Lie pseudoalgebra of J.
It is straightforward to check that the (pseudo) anticommutativity and Jacobi identities hold for (3.11).
Consider the elements x ∈ X, we denote the Fourier coefficients of U a,b . The linear space S 0 (J) generated by the set {U (a•xb) | a, b ∈ J, x ∈ X} is an H-submodule of S(J).
Note that U * a,b = −(σ 12 ⊗ H id)U b,a , so all Fourier coefficients of U * a,b lie in S 0 (J). If J is a Jordan pseudoalgebra and J 2 = J, i.e., every a ∈ J lies in the subspace generated by the set {(b • x c) | b, c ∈ J, x ∈ X}, then S 0 (J) ⊃ L(J).
Let us consider the direct sum of H-modules where J + and J − are isomorphic copies of J. Given a ∈ J (or A ∈ H ⊗n ⊗ H J), we will denote by a ± (or A ± ) the image of this element in J ± (or H ⊗n ⊗ H J ± ). Define a pseudoproduct on T(J) by the following rule: for a ± , b ± ∈ J ± , Σ ∈ S 0 (J) set Set the pseudoproduct on S 0 (J) to be the same as (3.11). Here we have used Σ * = −L a + D for Σ = L a + D ∈ S(J). Denote the projections of T(J) on J + , J − , S 0 (J) by π + , π − , π 0 , respectively. It is straightforward to check that T(J) is a Lie pseudoalgebra. This is an analogue of the Tits-Kantor-Koecher construction for an ordinary Jordan algebra.
Note that the structure pseudoalgebra is a formal direct sum of the corresponding H-modules, so the condition Proof . Suppose that there exists a non-zero proper ideal I ✁ L. Let Since J 2 = J, we have L ⊃ L(J). Hence, J ± ✁ J.
Analogously, J 0 ± = {a ∈ J | a ± ∈ I ∩ J ± } are also some ideals in J. 1) Consider the case J + = J − = 0 (hence, J 0 , so Σ = 0 as we have shown above, which is a contradiction. 2) Let J + = J, J 0 − = 0. Then for each a ∈ J there exists a + + Σ + d − ∈ I. Consider For every a, b, c ∈ J we have By the definition of commutator, Compare the last relation with (3.13) to get L a * b * c = 0 for all a, b, c ∈ J. Then the condition J 2 = J implies L(J) = 0, which is a contradiction.
3) The case J − = J, J 0 + = 0 is completely analogous. Hereby, if either of the ideals J 0 ± is zero, then at least one of the ideals J ± ✁ J has to be zero, which is impossible. Hence, J 0 + = J 0 − = J, i.e., I ⊃ J + , J − . Since the whole pseudoalgebra L is generated by J + ∪ J − , we have I = L.

Structure of simple Jordan pseudoalgebras
We have shown (Proposition 2), that for a simple finite Jordan pseudoalgebra J its TKK construction L = T(J) is a simple finite Lie pseudoalgebra. This allows to describe simple Jordan pseudoalgebras using the classification of simple Lie pseudoalgebras [1].

The case H = U (h)
Throughout this subsection, H is the universal enveloping Hopf algebra of a finite-dimensional Lie algebra h over C.
Proposition 3. Let J be a simple finite Jordan H-pseudoalgebra. Then the TKK construction T(J) is isomorphic to the current algebra Cur g over a simple finite-dimensional Lie algebra g.
Proof . If J is a simple finite Jordan H-pseudoalgebra, then T(J) is a simple finite Lie pseudoalgebra. Hence, either T(J) = Cur g, where g is a simple finite-dimensional Lie algebra, or T(J) is a subalgebra in W (h) (see Theorem 1 and Example 2). The second case could not be realized since by [1,Proposition 13.6] the pseudoalgebra W (h) does not contain abelian subalgebras. This is not the case for T(J).
It remains to show that if T(J) = J + ⊕S 0 (J)⊕J − = Cur g then J is the current pseudoalgebra over a simple finite-dimensional Jordan algebra.
Suppose e 1 , . . . , e n is a basis of h. Then the set of monomials is a basis of H. In order to simplify notation, we assume e (α) = 0 whenever α contains a negative component.
Theorem 3. Let J be a simple finite Jordan pseudoalgebra over H = U (h), where h is a finitedimensional Lie algebra over the field C. Then J is isomorphic to the current algebra Cur j over a finite-dimensional simple Jordan algebra j. Put ν = β max in (4.2) (i.e., b ν = 0, but b β = 0 for all β > ν). We obtain α γ α,0 ν [a α b ν ] = 0 for each µ. However, To finish the proof, use the induction on β. Suppose that [a µ b β ] = 0 for all µ and β > β 0 .
The second summand is equal to zero by the inductive assumption.
Now, let L = J + ⊕ S 0 (J) ⊕ J − = Cur g. By j ± 0 we denote the spaces spanned by all coefficients a α ∈ g ingoing in the sums α e (α) ⊗ a α ∈ J ± . Lemma 6 implies the spaces j ± 0 are Abelian subalgebras of g such that Lemma 7. Let L = T(J) = Cur g, where g is a finite-dimensional Lie algebra. Suppose that there are no non-zero ideals I ✁ L such that π ± (I) = 0. Then J ± = H ⊗ j ± 0 , respectively.
Proof . It is enough to consider the "+" case. Consider an arbitrary element a ∈ H ⊗ j + 0 , a = π + (a) + π 0 (a) + π − (a). Denote Hence, under the conditions of Lemma 7 one has J = H ⊗ j, j ≃ j ± 0 . Now it is necessary to show that the Jordan pseudoproduct on J may be restricted to an ordinary Jordan product on j.
Proposition 4. Let J be a finite Jordan H-pseudoalgebra such that Ann l (J) := {a ∈ J | a * J = 0} = 0. Assume that J = H ⊗ j, where j is a linear space. If L = T(J) = Cur g then j has a structure of ordinary Jordan algebra such that g ≃ T(j).
Proof of Theorem 3. Let J be a simple finite Jordan pseudoalgebra. Proposition 3 implies that L = T(J) ≃ Cur g, where g is a simple finite-dimensional Lie algebra. By Lemma 7, J = H ⊗ j. Since L satisfies the conditions of Proposition 4, we have J ≃ Cur j, T(j) = g, where j is a simple finite-dimensional Jordan algebra.

Corollary 1 ([19]).
A simple finite Jordan conformal algebra is isomorphic to the current conformal algebra over a simple finite-dimensional Jordan algebra.

The case H = U (h)#C[Γ]
If J is a pseudoalgebra over H = U (h)#C[Γ] then it is in particular a pseudoalgebra over U (h). The structure of H-pseudoalgebra on J is completely encoded by U (h)-pseudoalgebra structure and by the action of Γ on U (h), see [1, Section 5] for details. Proof . By Proposition 2 L = T(J) is a simple H-pseudoalgebra, and it is clear that L is a finitely generated U (h)-module. Theorem 2 and Proposition 3 imply that L = m i=1 Cur U (h) g i where Cur U (h) g i = Cur i are isomorphic simple current Lie U (h)-pseudoalgebras, and Γ acts on them transitively.
Hence, L = Cur U (h)g , whereg = m i=1 g i . The H-pseudoalgebra L could be considered as an U (h)-pseudoalgebra endowed with an action of Γ on it which is compatible with that of U (h): g(ha) = h g (ga), h ∈ U (h), a ∈ L, g ∈ Γ.
Consider L as the current U (h)-pseudoalgebra overg. The condition of Lemma 7 holds for this L. Indeed, if I is an ideal of the U (h)-pseudoalgebra L and π ± (I) = 0, then ΓI is a proper ideal of L (as of an H-pseudoalgebra) such that its projections onto J ± are zero. Moreover, if J as an H-pseudoalgebra has no non-trivial (left) annihilator Ann l (J) then so is J as an U (h)-pseudoalgebra (see [1,Corollary 5.1]).
Therefore, the same arguments as in the proof of Proposition 4 show that J = Cur U (h)j , wherej is a finite-dimensional Jordan algebra.
The explicit expression [1, equation (5.7)] for pseudoproduct over H shows that for every x ∈ X = U (h) * , g ∈ Γ, a, b ∈ J we have where g * , γ = δ g,γ , γ ∈ Γ. Hence, the following relation between Fourier coefficients of U a,b holds: U (a• x⊗g * b) = U (ga•xb) . Here in the left-and right-hand sides we state Fourier coefficients over H * and X, respectively. Therefore, the relations between the H-module S 0 ( H J) and the U (h)-module U (h) S 0 (J) are the same as between H-module H J and U (h)-module U (h) J. Cur U (h) j i , and Γ necessarily acts on these current algebras transitively.