Homology, Homotopy and Applications, vol.4(2), 2002, pp.141–163

NONSPLIT EXTENSIONS OF MODULAR LIE ALGEBRAS OF RANK 2

A.S. DZHUMADIL’DAEV and S.S. IBRAEV

(communicated by Larry Lambe)

Abstract Second cohomology groups of irreducible representations of classical Lie algebras A_{2}, B_{2} and G_{2} over an algebraically closed field of characteristic p > h are calculated. Here h is the Coxeter number.

To Jan–Erik Roos on his sixty–fifth birthday

1. Formulation of the main result

Levi-Mal’cev theorem has cohomological origin. It states that any finite-dimen-sional extension of a finite-dimensional simple Lie algebra over a field of characteristic 0 is split. In case of characteristic p > 0 any Lie algebra has at least one nonsplit extension and the number of irreducible modules with such a property is finite [8]. For example, the 3-dimensional simple Lie algebra A_{1} = sl_{2} has exactly one irreducible module, namely the (p − 1)-dimensional module V_{p−2}, with H^{2}(A_{1}, V_{p−2}) ≠ 0 [7].

The aim of our paper is to calculate second cohomology groups with coefficients in an irreducible module for simple Lie algebras of rank 2: 𝔤 = A_{2}, B_{2} and G_{2}. The field K is algebraically closed and has characteristic p > h, where h is the Coxeter number. An irreducible 𝔤-module V is called 2-peculiar, if H^{2}(𝔤, V) ≠ 0. Let κ_{2}(𝔤) be the number of peculiar modules. From our results it follows that κ_{2}(𝔤) = 2, 3, 3, for 𝔤 = A_{2}, B_{2}, G_{2} respectively. Let L(λ) be an irreducible module with highest weight λ.

The main result of this paper is the following

Theorem 1.1. Let 𝔤 = A_{2}, B_{2}, G_{2}, p > h and V be an irreducible 𝔤-module. Then H^{2}(𝔤, V) is trivial except in the following cases:

(a) 𝔤 = A_{2}, H^{2}(𝔤, L((p − 3)λ_{i})) ≅ L(λ_{i})^{(1)}, i = 1, 2;

(b) 𝔤 = B_{2}, H^{2}(𝔤, L((p − 3)λ_{1} + 2λ_{2})) ≅ L(λ_{1})^{(1)},

H^{2}(𝔤, L(λ_{1} + (p − 4)λ_{2})) ≅ L(λ_{2})^{(1)},
H^{2}(𝔤, L((p − 2)(λ_{1} + λ_{2}))) ≅ L(λ_{2})^{(1)};

Received February 16, 2001, revised February 8, 2002; published on July 12, 2002. 2000 Mathematics Subject Classification: 17B50, 17B56. Key words and phrases: modular Lie algebras, nonsplit extensions, Levi-Mal’cev theorem, second cohomology groups, simple Lie algebras, restricted cohomology groups. \begin{matrix}c \\ ◯ \end{matrix} 2002, A.S. Dzhumadil’daev and S.S. Ibraev. Permission to copy for private use granted.

(c) 𝔤 = G_{2}, H^{2}(𝔤, L((p − 5)λ_{1} + 2λ_{2})) ≅ L(λ_{1})^{(1)},

H^{2}(𝔤, L(4λ_{1} + (p − 3)λ_{2})) ≅ L(λ_{2})^{(1)},
H^{2}(𝔤, L(3λ_{1} + (p − 2)λ_{2}))) ≅ L(0)^{(1)}.

Here we use notations from [11]. Let 𝔤 be a classical Lie algebra over the field K, G an algebraic group of the Lie algebra 𝔤. Recall that the Frobenius map is defined as the morphism G → G of the K-group functor G induced by the map x ↦ x^{p} on the function algebra K[G] [11]. The normal subgroup G_{1} is the scheme-theoretic kernel of this map. Let T be the maximal torus on G, X(T) be the character group of T, R ⊂ X(T) be the root system and R^{+} be the set of positive roots on R. The simple roots α_{1}, α_{2}, …, α_{n} corresponds to the Bourbaki table [3]. Let λ_{1}, λ_{2}, ⋯, λ_{n} be the fundamental weights, X(T)_{+} be the set of dominant weights, X_{1}(T) be the set of restricted dominant weights, i.e., X_{1}(T) = {λ = ∑_{i=1}^{n} r_{i}λ_{i} ∈ X(T) : r_{i} ∈ ℤ, 0 ⩽ r_{i} < p, for all i}. Endow X(T) by the usual order : λ ⩽ μ if and only if, there exist integers r_{i} ⩾ 0 such that μ − λ = ∑_{i=1}^{n} r_{i}α_{i}. For any T -module V and any μ ∈ X(T) denote by V_{μ} its weight subspace in V.

There exists an algebra 𝔤_{ℤ} over ℤ such that 𝔤_{ℤ} ⊗ K ≅ 𝔤. In 𝔤_{ℤ} one can choose a Chevalley basis, that coincides with a basis of the semi-simple complex Lie algebra. To any root α there corresponds a basic vector e_{α} of the Lie algebra 𝔤_{ℤ}. If α, β ∈ R, then [e_{α}, e_{β}] = N_{α,β}e_{α+β} for some integer N_{α,β}. Identify e_{α} with e_{α} ⊗ 1. Note that the p-map e ↦ e^{[p]}, defined on 𝔤, has the property e_{α}^{[p]} = 0 for any α ∈ R.

Recall the definition of a Weyl module. Let 𝔤_{ℂ} be a Lie algebra over the field of complex numbers ℂ. Consider an irreducible 𝔤_{ℂ}-module V (λ)_{ℂ} with highest weight λ. It is known that there exists a ℤ-submodule V (λ)_{ℤ} of the 𝔤_{ℂ}-module V (λ)_{ℂ}. Then V (λ) = V (λ)_{ℤ} ⊗ K is a 𝔤-module. The obtained module is called a Weyl module.

Let B be the Borel subgroup of G corresponding to the negative roots, U be the unipotent radical of B and 𝔲 be the Lie algebra of U. The Lie algebra 𝔲 is a nilpotent subalgebra of the Lie algebra 𝔤 and it spans basic vectors e_{−α}, α ∈ R^{+}. The Cartan subalgebra 𝔥 of the Lie algebra 𝔤 is a Lie algebra of the maximal torus T of G. For any λ ∈ X(T) one can define a one-dimensional module K_{λ} over B using the isomorphism B/U ≅ T. The induced G-module H^{0}(λ) = Ind_{B}^{G}K_{λ} is non-zero if and only if λ ∈ X(T)_{+}. If so, the socle L(λ) of the induced module H^{0}(λ) is a simple G-module with highest weight λ. It can also be constructed as the unique irreducible factor of the Weyl module V (λ).

If λ ∈ X_{1}(T), then L(λ) remains simple as a G_{1}-module. Any simple G_{1}-module is defined uniquely by the highest weight λ ∈ X_{1}(T) and it is isomorphic to L(λ). The theory of restricted representations of the restricted Lie algebra 𝔤 is equivalent to the theory of representations of the group G_{1}.

A composition of a representation of G in a vector space V with the Frobenius map gives us a new representation with trivial action of G_{1}. Denote the obtained module by V^{(1)}. Thus this module as a module over the Lie algebra 𝔤 is a module with a trivial action. To any weight μ ∈ X(T) of the space V there corresponds the weight pμ of the space V^{(1)}. On the other hand, if V_{1} is a G-module with trivial action of G_{1} (or 𝔤 ) then there exists a unique G-module V, such that V_{1} = V^{(1)}.

Denote this G-module V by V^{(−1)}. For example, if L is a G-module, then any cohomology group H^{i}(G_{1}, L) is a G-module with trivial action of G_{1} (or 𝔤). Therefore the module H^{i}(G_{1}, L)^{(−1)} is a G-module with the above mentioned property.

Second cohomology groups of the adjoint representation of the Lie algebra B_{2} in characteristic 3 was studied in [13], [6]. In [16], [11] first cohomology groups of modular Lie algebras with coefficients in irreducible modules are calculated. In [16] the non-triviality of first cohomology groups with coefficients in irreducible restricted modules with highest weights pλ_{i} − α_{i}, i = 1, 2, …, n, are proved. Here α_{i}, λ_{i}, i = 1, 2, …, n are the simple roots and fundamental weights. In [17] a connection between first cohomology groups of irreducible modules and second cohomology groups of restricted Weyl modules are studied.

2. Connection between ordinary and restricted second cohomology groups

Consider the algebra 𝔤 as a restricted Lie algebra with the p-map e ↦ e^{[p]}, e ∈ 𝔤. Let U(𝔤) be the universal enveloping algebra of 𝔤 and U(𝔤)^{+} be a two sided ideal in U(𝔤) such that U(𝔤) is a direct sum of K and U(𝔤)^{+}. Let P (𝔤) be the ideal generated by the elements e^{p} − e^{[p]}, e ∈ 𝔤. The factor-algebra U_{0}(𝔤) = U(𝔤)/P (𝔤) is called the restricted universal enveloping algebra of 𝔤.

Restricted cohomology groups of restricted Lie algebras were introduced by G.Hochschild in ([9]). The cohomology groups H^{i}(G_{1}, V) for a G_{1}-module V are equivalent to the restricted cohomology of the corresponding 𝔤-module ([11], I.9, p.145 ). Let H_{∗}^{i}(𝔤, V) denote the i-th restricted cohomology group of the restricted Lie algebra 𝔤 with coefficients in a restricted 𝔤-module V. By definition H_{∗}^{i}(𝔤, V) = Ext_{U_{0}(𝔤)}^{i}(K, V).

The projection U(𝔤) → K induces the projection U_{0}(𝔤) → K. Denote its kernel by U_{0}(𝔤)^{+}. Then U_{0}(𝔤)^{+} is the image of U(𝔤)^{+} in U_{0}(𝔤) of the canonical map U(𝔤) → U_{0}(𝔤). A map of the corresponding cochain complexes is induced by the

homomorphism ψ ↦ ψ^{0}, where ψ^{0}(s_{1}, s_{2}, …, s_{i}) = ψ(s1^{′}, s2^{′}, …, si^{′}), s_{j} ∈ U(𝔤)^{+}

and sj^{′} are the canonical images in U_{0}(𝔤)^{+}.

Let now C(V) be the cochain complex for the universal enveloping algebra U(𝔤) of the Lie algebra 𝔤 with coefficients in the 𝔤-module V.

Let C^{0}(V) stand for the subcomplex consisting of the cochains of the form ψ^{0}, where ψ is a cochain for U_{0}(𝔤)^{+} with coefficients in V. Then we have an exact sequence of cochain complexes

0 → C^{0}(V) → C(V) → C(V)/C^{0}(V) → 0.

Since the map ψ ↦ ψ^{0} is an isomorphism, we may identify H^{i}(C^{0}(V)) with H_{∗}^{i}(𝔤, V). This gives us the following exact sequence:

⋯ → H_{∗}^{i}(𝔤, V) → H^{i}(𝔤, V) → H^{i}(C(V)/C^{0}(V)) → H_{∗}^{i+1}(𝔤, V) → ⋯

In [9] Hochschild shows that for i = 1, 2

H^{i}(C(V)/C^{0}(V)) ≅ S(𝔤, H^{i−1}(𝔤, V)),

where S(𝔤, H^{i−1}(𝔤, V)) is the space of p-semilinear maps 𝔤 → H^{i−1}(𝔤, V). In [11] it was proved that for i = 1, 2 there is an isomorphism of G-modules

S(𝔤, H^{i−1}(𝔤, V)) ≅ H^{i−1}(𝔤, V) ⊗ 𝔤^{∗}

(proposition 9.20, p. 160). It is evident that H^{0}(C(V)/C^{0}(V)) = 0. The identification of H_{∗}^{i}(𝔤, V) with H^{i}(G_{1}, V) gives us the following exact sequence of G-modules:

0 → H^{1}(G_{1}, V) → H^{1}(𝔤, V) → H^{0}(𝔤, V) ⊗ 𝔤^{∗} → H^{2}(G_{1}, V) →
→ H^{2}(𝔤, V) → H^{1}(𝔤, V) ⊗ 𝔤^{∗} → H^{3}(G_{1}, V).  (1)

Lemma 2.1. Let V be a nontrivial irreducible 𝔤-module and H^{1}(𝔤, V) = 0. Then H^{2}(𝔤, V) ≅ H^{2}(G_{1}, V) as G-modules.

Proof. Since V is a nontrivial irreducible 𝔤-module, H^{0}(𝔤, V) = 0. The isomorphism follows from the exact sequence (1).

3. Peculiar irreducible modules

Call an irreducible 𝔤-module V peculiar, if H^{∗}(𝔤, V) ≠ 0. Let 𝔤 be a simple classical Lie algebra, p > 0, U_{0}(𝔤) its restricted universal enveloping algebra, Z_{0}(𝔤) be the center of U_{0}(𝔤). The central character c_{V} : Z(𝔤) → K, maps each element C ∈ Z_{0}(𝔤) to its unique eigenvalue c_{V} (C) on V.

Let λ, μ ∈ X(T). We will say, that λ and μ are connected, if λ = w(μ + ρ) − ρ for some w ∈ W. If λ and μ are connected, then according to the linkage principal, L(μ) is a composition factor of Weyl module V (λ) ([1], Corollary 3 of theorem 1). This means that the maximal submodule of Weyl module V (λ) is generated by highest vectors with weights connected with λ. If M(λ) is a maximal submodule of the Weyl module V (λ), then the following sequence is exact

0 → M(λ) → V (λ) → V (λ)/M(λ) → 0.

The corresponding long exact cohomological sequence shows that the highest weights of peculiar modules are connected.

Lemma 3.1. Let L(λ) be a peculiar module. Then λ ∈ X_{1}(T) and λ = w(ρ)−ρ+pν where ν ∈ X(T), w ∈ W.

Proof. According to ([10], theorem 2.1) two modules with connected highest weights have just the same central characters. The trivial module is peculiar. It is evident that the central character of the trivial module is equal to zero. According to the linkage principle highest weights of peculiar modules are connected with the highest weight 0. It is known that cohomologies of non-restricted modules are trivial ([7]). Thus, the highest weight of a peculiar module has the form λ = w(ρ) − ρ + pν ∈ X_{1}(T), where ν ∈ X(T) and w runs through elements of Weyl group W.

Corollary 3.2. The lists of possible highest weights of any peculiar module of a simple classical Lie algebra 𝔤 of rank two are given below

𝔤 = A_{2}
0, (p − 2)λ_{1} + λ_{2}, λ_{1} + (p − 2)λ_{2}, (p − 3)λ_{1}, (p − 3)λ_{2}, (p − 2)(λ_{1} + λ_{2});

𝔤 = B_{2}
0, (p − 2)λ_{1} + 2λ_{2}, λ_{1} + (p − 2)λ_{2}, (p − 3)λ_{1} + 2λ_{2},
λ_{1} + (p − 4)λ_{2}, (p − 3)λ_{1}, (p − 4)λ_{2}, (p − 2)(λ_{1} + λ_{2});

𝔤 = G_{2}
0, (p − 2)λ_{1} + λ_{2}, 3λ_{1} + (p − 2)λ_{2}, (p − 5)λ_{1} + 2λ_{2}, 4λ_{1} + (p − 3)λ_{2},
(p − 6)λ_{1} + 2λ_{2}, 4λ_{1} + (p − 4)λ_{2}, (p − 6)λ_{1} + λ_{2},
3λ_{1} + (p − 4)λ_{2}, (p − 5)λ_{1}, (p − 3)λ_{2}, (p − 2)(λ_{1} + λ_{2}).

Proof. We show detailed calculations only in the case of A_{2}. For other algebras the calculations are similar. So, let 𝔤 = A_{2}. The Weyl group has 6 elements 1, s_{1}, s_{2}, s_{1}s_{2}, s_{2}s_{1}, s_{1}s_{2}s_{1}. Here s_{i} corresponds to s_{i}(μ) = μ − \frac{2(μ,α_{i})}{(α_{i},α_{i})}α_{i}. The halfsum of positive roots is equal to ρ = α_{1} + α_{2}. It is evident that, to the neutral element 1 corresponds a peculiar highest weight λ = 0. For s_{1} we have

λ = s_{1}(ρ) − ρ + pν =
s_{1}(α_{1} + α_{2}) − α_{1} − α_{2} + pν = −α_{1} + pν = −2λ_{1} + λ_{2} + pν.

Since λ is restricted and dominant, ν = λ_{1}. So λ = s_{1}(ρ) − ρ + pν = (p − 2)λ_{1} + λ_{2} may be a peculiar weight corresponding to s_{1}. Similarly,

λ = s_{2}(ρ) − ρ + pν = −α_{2} + pν = λ_{1} − 2λ_{2} + pν = λ_{1} + (p − 2)λ_{2},
λ = s_{1}s_{2}(ρ) − ρ + pν = −2α_{1} − α_{2} + pν = −3λ_{1} + pν = (p − 3)λ_{1},
λ = s_{2}s_{1}(ρ) − ρ + pν = −α_{1} − 2α_{2} + pν = −3λ_{2} + pν = (p − 3)λ_{2},
λ = s_{1}s_{2}s_{1}(ρ) − ρ + pν = −2α_{1} − 2α_{2} + pν = −2λ_{1} − 2λ_{2} + pν =
(p − 2)(λ_{1} + λ_{2}).

Lemma 3.3. Let 𝔤 = A_{2}. Then as G-modules,

H^{0}(0) = L(0), H^{0}((p − 3)λ_{1}) = L((p − 3)λ_{1}), H^{0}((p − 3)λ_{2}) =
L((p − 3)λ_{2});

H^{0}((p − 2)λ_{1} + λ_{2})/L((p − 2)λ_{1} + λ_{2}) ≅ L((p − 3)λ_{1});
H^{0}(λ_{1} + (p − 2)λ_{2})/L(λ_{1} + (p − 2)λ_{2}) ≅ L((p − 3)λ_{2});
H^{0}((p − 2)(λ_{1} + λ_{2}))/L((p − 2)(λ_{1} + λ_{2})) ≅ L(0).

Proof. See [4], [14], [12].

Lemma 3.4. Let 𝔤 = B_{2} and p > 3. Then

H^{0}(0) = L(0), H^{0}((p − 3)λ_{1}) = L((p − 3)λ_{1}), H^{0}((p − 4)λ_{2}) =
L((p − 4)λ_{2});
H^{0}((p − 2)λ_{1} + 2λ_{2})/L((p − 2)λ_{1} + 2λ_{2}) ≅ L((p − 3)λ_{1} + 2λ_{2});
H^{0}(λ_{1} + (p − 2)λ_{2})/L(λ_{1} + (p − 2)λ_{2}) ≅ L(λ_{1} + (p − 4)λ_{2});
H^{0}((p − 3)λ_{1} + 2λ_{2})/L((p − 3)λ_{1} + 2λ_{2}) ≅ L((p − 3)λ_{1});
H^{0}(λ_{1} + (p − 4)λ_{2})/L(λ_{1} + (p − 4)λ_{2}) ≅ L((p − 4)λ_{2});
H^{0}((p − 2)(λ_{1} + λ_{2}))/L((p − 2)(λ_{1} + λ_{2})) ≅ L(λ_{1} + (p − 2)λ_{1}).

Proof. Recall that the element of maximal length of the Weyl group is w_{0} = −1 for 𝔤 = B_{2}. Since in this case, V (λ) = H^{0}(−w_{0}(λ))^{∗} = H^{0}(λ)^{∗}, the maximal submodule of the Weyl module is isomorphic to the factor-module H^{0}(λ)/L(λ). Therefore it is enough to prove that for any of the considered modules H^{0}(λ), the maximal submodule of the corresponding Weyl module V (λ) coincides with an irreducible module mentioned in the lemma.

Let { e_{1}, e_{2}, e_{3}, e_{4}, h_{1}, h_{2}, f_{1}, f_{2}, f_{3}, f_{4} } be the Chevalley basis of the Lie algebra 𝔤. Vectors in the module V (λ) can be presented as linear combinations of monomials like

v_{i,j,k,s} := \frac{f_{4}^{s}f_{1}^{k}f_{3}^{j}f_{2}^{i}}{s!k!j!i!} ⊗ v_{λ},

where v_{λ} is the highest vector and { f_{1}, f_{2}, f_{3}, f_{4} } is the basis of 𝔲. The actions of the elements e_{1}, e_{2} on the monomials v_{i,j,k,s} are defined by

e_{1}v_{i,j,k,s} = (s + 1)v_{i,j−2,k,s+1} − (i + 1)v_{i+1,j−1,k,s} + (m_{1} + 1 + i − j − k)v_{i,j,k−1,s},

e_{2}v_{i,j,k,s} = 2(k + 1)v_{i,j−1,k+1,s} − (j + 1)v_{i,j+1,k,s−1} + (m_{2} + 1 − i)v_{i−1,j,k,s},

where λ = m_{1}λ_{1} + m_{2}λ_{2}.

Let

v_{i,k}^{m_{1},m_{2}} = ∑_{0⩽j+s⩽k, 0⩽j+2s⩽i} a_{j,s}v_{i−j−2s,j,k−j−s,s}

be the vector in the space V (λ) with weight λ−kα_{1}−iα_{2}. Call it normal, if a_{0,0} ≠ 0. It is known that highest vectors of proper submodules of V (λ) are normal ([15]).

It is evident that normal vectors of the modules V ((p−3)λ_{1}), V ((p−4)λ_{2}) cannot serve as highest vectors. So, the modules V ((p − 3)λ_{1}), V ((p − 4)λ_{2}) have no proper submodules. Therefore, they are irreducible and are equal to the corresponding induced modules.

Suppose now that λ = (p−3)λ_{1} +2λ_{2}. Then highest vectors can be found among the normal vectors v_{i,k}^{p−3,2}, i ⩽ 2, k ⩽ p − 3.

We now show that v_{1,1}^{p−3,2}, v_{2,2}^{p−3,2} cannot serve as a highest vector. Suppose that

v_{1,1}^{p−3,2} = a_{1}v_{1,0,1,0} + b_{1}v_{0,1,0,0},
v_{2,2}^{p−3,2} = a_{2}v_{2,0,2,0} + b_{2}v_{1,1,1,0} + c_{2}v_{2,0,2,0} + d_{2}v_{1,1,1,0}

are highest vectors. Then

e_{1}v_{1,1}^{p−3,2} = e_{1}(a_{1}v_{1,0,1,0} + b_{1}v_{0,1,0,0}) = 0 ⇒ −2a_{1} − b_{1} = 0,
e_{2}v_{1,1}^{p−3,2} = e_{2}(a_{1}v_{1,0,1,0} + b_{1}v_{0,1,0,0}) = 0 ⇒ 2a_{1} + 2b_{1} = 0;
e_{1}v_{2,2}^{p−3,2} = e_{1}(a_{2}v_{2,0,2,0} + b_{2}v_{1,1,1,0} + c_{2}v_{2,0,2,0} + d_{2}v_{1,1,1,0}) = 0 ⇒
c_{2} = b_{2} = a_{2} = 0;
e_{2}v_{2,2}^{p−3,2} = e_{2}(a_{2}v_{2,0,2,0} + b_{2}v_{1,1,1,0} + c_{2}v_{2,0,2,0} + d_{2}v_{1,1,1,0}) = 0 ⇒
a_{2} + 4b_{2} = 0, 2b_{2} + 2c_{2} − d_{2} = 0.

Therefore, a_{1} = b_{1} = 0, a_{2} = b_{2} = c_{2} = d_{2} = 0.

From the condition e_{1}v_{2,k}^{p−3,2} = 0 it follows that k ⩽ 2. Since the normal vector v_{2,2}^{p−3,2} cannot be a highest vector, we have that k = 1. Since

e_{1}v_{2,1}^{p−3,2} = 2(p − 1)v_{2,0,0,0} + 2v_{0,0,0,0} = 0,
e_{2}v_{2,1}^{p−3,2} = 2v_{1,0,1,0} − 2v_{0,1,0,0} − 2v_{0,1,0,0} + 2v_{0,1,0,0} = 0,

we obtain the unique (up to scalar) highest vector v_{2,1}^{p−3,2} = 2v_{2,0,1,2} − v_{1,1,0,0} − 2v_{0,0,0,1}.

Since the module V ((p − 3)λ_{1} + 2λ_{2}) has no other highest vectors except v_{2,1}^{p−3,2}, the submodule generated by this vector is irreducible. The weight of the highest vector v_{2,1}^{p−3,2} is (p − 3)λ_{1} + 2λ_{2} − α_{1} − 2α_{2} = (p − 3)λ_{1}. Therefore, the maximal submodule of V ((p−3)λ_{1} +2λ_{2}) is a module isomorphic to L((p−3)λ_{1}). Analogous calculations show that the vectors

v_{1,1}^{p−2,2} = v_{1,0,1,0} − v_{0,1,0,0},
v_{2,1}^{1,p−2} = 2v_{2,0,1,0} + 3v_{1,1,0,0} − 6v_{0,0,0,1},
v_{1,1}^{1,p−4} = v_{1,0,1,0} + 2v_{0,1,0,0},
v_{p−3,p−3}^{p−2,p−2} = v_{p−3,0,p−3,0}+

∑_{1⩽j+2s⩽p−2}(−1)^{s}(p − 3 − j − s)!(p − 1) ⋯ (p − j − s)v_{p−3−j−2s,j,p−3−j−s,s}

are unique (up to scalar) highest vectors of the modules V ((p − 2)λ_{1} + 2λ_{2}), V (λ_{1} + (p − 2)λ_{2}), V (λ_{1} + (p − 4)λ_{2}), V ((p − 2)(λ_{1} + λ_{2})) correspondingly.

Therefore, the submodules generated by one of these vectors are irreducible. Their highest weights are respectively (p − 3)λ_{1} + 2λ_{2}, λ_{1} + (p − 4)λ_{2}, (p − 4)λ_{2}, λ_{1}+(p−2)λ_{2}. Thus maximal submodules of the following modules V ((p−2)λ_{1}+2λ_{2}), V (λ_{1} +(p−2)λ_{2}), V (λ_{1} +(p−4)λ_{2}), V ((p−2)(λ_{1} +λ_{2})) are the irreducible modules L((p − 3)λ_{1} + 2λ_{2}), L(λ_{1} + (p − 4)λ_{2}), L((p − 4)λ_{2}), L(λ_{1} + (p − 2)λ_{2}). The lemma is proved completely.

By analogous methods the following lemma can be proved.

Lemma 3.5. Let 𝔤 = G_{2} and p > 5. Then

H^{0}(0) = L(0), H^{0}((p − 5)λ_{1}) = L((p − 5)λ_{1}), H^{0}((p − 3)λ_{2}) =
L((p − 3)λ_{2});
H^{0}((p − 2)λ_{1} + λ_{2})/L((p − 2)λ_{1} + λ_{2}) ≅ L((p − 5)λ_{1} + 2λ_{2});
H^{0}(3λ_{1} + (p − 2)λ_{2})/L(3λ_{1} + (p − 2)λ_{2}) ≅ L(4λ_{1} + (p − 3)λ_{2});
H^{0}((p − 5)λ_{1} + 2λ_{2})/L((p − 5)λ_{1} + 2λ_{2}) ≅ L((p − 6)λ_{1} + 2λ_{2});
H^{0}(4λ_{1} + (p − 3)λ_{2})/L(4λ_{1} + (p − 3)λ_{2}) ≅ L(4λ_{1} + (p − 4)λ_{2});
H^{0}((p − 6)λ_{1} + λ_{2})/L((p − 6)λ_{1} + λ_{2}) ≅ L((p − 5)λ_{1});
H^{0}(4λ_{1} + (p − 4)λ_{2})/L(4λ_{1} + (p − 4)λ_{2}) ≅ L((p − 3)λ_{2});

H^{0}((p − 2)(λ_{1} + λ_{2}))/L((p − 2)(λ_{1} + λ_{2})) ≅ L((2p − 6)λ_{1} + 2λ_{1}).

4. G_{1}-cohomology

Let S(𝔲^{∗}) be the symmetric algebra of the Lie algebra 𝔲^{∗}, w an element of Weyl group W, l(w) length of the element w, ρ the half sum of positive roots and w(ρ) − ρ + pν ∈ X_{1}(T). Below we use the following known facts about first cohomology groups of G_{1} ([12], proposition 4.9(b) and 4.3) and Andersen-Jantzen general formula ([2], corollary 3.7(a),(b)):

H^{1}(G_{1}, L(pλ_{i} − α_{i}))^{(−1)} ≅ H^{0}(λ_{i}), i = 1, 2, …, n,  (2)
H^{1}(G_{1}, L(λ))^{(−1)} ≅ (H^{0}(λ)/L(λ))^{G_{1}},
where λ ≠ pλ_{i} − α_{i} , i = 1, 2, …, n,  (3)
H^{i}(G_{1}, K)^{(−1)} ≅ H^{0}(S^{i/2}(𝔲^{∗})).  (4)
H^{i}(G_{1}, H^{0}(K_{w(ρ)−ρ+pν}))^{(−1)} ≅ H^{0}(S^{(i−l(w))/2}(𝔲^{∗}) ⊗ K_{ν}).  (5)

Proposition 4.1. Let 𝔤 = A_{2}, B_{2}, G_{2} and p > h. Then H^{2}(G_{1}, H^{0}(λ)) = 0, except in the following cases

(a) 𝔤 = A_{2}

H^{2}(G_{1}, H^{0}(0))^{(−1)} ≅ 𝔤^{∗} ≅ H^{0}(λ_{1} + λ_{2}) = L(λ_{1} + λ_{2}),
H^{2}(G_{1}, H^{0}((p − 3)λ_{1}))^{(−1)} ≅ H^{0}(λ_{1}),
H^{2}(G_{1}, H^{0}((p − 3)λ_{2}))^{(−1)} ≅ H^{0}(λ_{2});

(b) 𝔤 = B_{2}

H^{2}(G_{1}, H^{0}(0))^{(−1)} ≅ H^{0}(λ_{1}) ⊕ H^{0}(2λ_{2}),
H^{2}(G_{1}, H^{0}((p − 3)λ_{1} + 2λ_{2}))^{(−1)} ≅ H^{0}(λ_{1}),
H^{2}(G_{1}, H^{0}(λ_{1} + (p − 4)λ_{2}))^{(−1)} ≅ H^{0}(λ_{2});

(c) 𝔤 = G_{2}

H^{2}(G_{1}, H^{0}(0))^{(−1)} ≅ H^{0}(λ_{1}) ⊕ H^{0}(λ_{2}),
H^{2}(G_{1}, H^{0}((p − 5)λ_{1} + 2λ_{2}))^{(−1)} ≅ H^{0}(λ_{1}),
H^{2}(G_{1}, H^{0}(4λ_{1} + (p − 3)λ_{2}))^{(−1)} ≅ H^{0}(λ_{2}).

Proof. follows from (4) and (5) .

For any λ ∈ X_{1}(T) \ {0} the following exact sequence holds

0 → L(λ) → H^{0}(λ) → H^{0}(λ)/L(λ) → 0.  (6)

Consider the corresponding long exact sequence of G_{1}-cohomology groups

⋯ → H^{i}(G_{1}, L(λ)) → H^{i}(G_{1}, H^{0}(λ)) → H^{i}(G_{1}, H^{0}(λ)/L(λ)) →
H^{i+1}(G_{1}, L(λ)) → H^{i+1}(G_{1}, H^{0}(λ)) → H^{i+1}(G_{1}, H^{0}(λ)/L(λ)) → ⋯

The triviality of H^{0}(G_{1}, L(λ)) is evident. The module H^{0}(G_{1}, H^{0}(λ)) is an invariant space for G_{1} and a submodule of the G-module H^{0}(λ). If it is non-zero, it contains the simple socle L(λ) of the G-module H^{0}(λ). Furthermore, G_{1} acts on L(λ) in a trivial way if and only if, λ ∈ pX(T). For the restricted weight λ ∈ X_{1}(T) this is possible only in the case λ = 0. Therefore,

H^{0}(λ)^{G_{1}} = 0  (7)

for any λ ∈ X_{1}(T) \ {0}. Then the exact sequence of G_{1}-cohomology groups looks like

0 → H^{0}(G_{1}, H^{0}(λ)/L(λ)) → H^{1}(G_{1}, L(λ)) → H^{1}(G_{1}, H^{0}(λ)) →
H^{1}(G_{1}, H^{0}(λ)/L(λ)) → H^{2}(G_{1}, L(λ)) → H^{2}(G_{1}, H^{0}(λ)) →
H^{2}(G_{1}, H^{0}(λ)/L(λ)) → H^{3}(G_{1}, L(λ)) → H^{3}(G_{1}, H^{0}(λ)) → ⋯

Proposition 4.2. Let 𝔤 = A_{2}, B_{2}, G_{2} and p > h. Then H^{1}(G_{1}, L(λ)) = 0, except in the following cases

(a) 𝔤 = A_{2}

H^{1}(G_{1}, L((p − 2)(λ_{1} + λ_{2})))^{(−1)} ≅ L(0),
H^{1}(G_{1}, L((p − 2)λ_{1} + λ_{2}))^{(−1)} ≅ L(λ_{1}),
H^{1}(G_{1}, L(λ_{1} + (p − 2)λ_{2}))^{(−1)} ≅ L(λ_{2});

(b) 𝔤 = B_{2}

H^{1}(G_{1}, L((p − 2)λ_{1} + 2λ_{2}))^{(−1)} ≅ L(λ_{1}),
H^{1}(G_{1}, L(λ_{1} + (p − 2)λ_{2}))^{(−1)} ≅ L(λ_{2});

(c) 𝔤 = G_{2}

H^{1}(G_{1}, L((p − 2)λ_{1} + λ_{2}))^{(−1)} ≅ L(λ_{1}),
H^{1}(G_{1}, L(3λ_{1} + (p − 2)λ_{2}))^{(−1)} ≅ L(λ_{2}).

Proof. As we mentioned above, the first ordinary cohomology groups and the corresponding cohomology groups for G_{1} coincide. Statement (a) was proved in ([5],(3.6), p.112) and ([12], 6.10, p.314).

We now prove (b) and (c). We will use (2) and (3). Let us consider the induced modules H^{0}(λ) corresponding to the weights from the list of corollary 3.2.

By lemmas 3.4 and 3.5 the factor-modules H^{0}(λ)/L(λ) for the Lie algebras 𝔤 = B_{2}, G_{2} are simple and the highest weight of H^{0}(λ)/L(λ) is not an element of pX(T), therefore (H^{0}(λ)/L(λ))^{G_{1}} = 0 for peculiar modules. So, nontrivial first cohomology groups can appear only for modules of the form L(pλ_{i} − α_{i}) and they are given by (2).

Let 𝔤 = B_{2}. We have pλ_{1} − α_{1} = pλ_{1} − 2λ_{1} + λ_{2} = (p − 2)λ_{1} + λ_{2}, pλ_{2} − α_{2} = pλ_{2} + λ_{1} − 2λ_{2} = λ_{1} + (p − 2)λ_{2}. So, according to (2) we obtain (b).

If 𝔤 = G_{2}, then pλ_{1} − α_{1} = pλ_{1} − 2λ_{1} + λ_{2} = (p − 2)λ_{1} + λ_{2}, pλ_{2} − α_{2} = pλ_{2} + 3λ_{1} − 2λ_{2} = 3λ_{1} + (p − 2)λ_{2}. So, by (2) we obtain (c).

Proposition 4.3. Let 𝔤 = A_{2}, B_{2}, G_{2} and p > h. Then H^{2}(G_{1}, L(λ)) = 0, except in the following cases

(a) 𝔤 = A_{2}

H^{2}(G_{1}, L(0))^{(−1)} ≅ 𝔤^{∗} ≅ H^{0}(λ_{1} + λ_{2}) = L(λ_{1} + λ_{2}),
H^{2}(G_{1}, L((p − 3)λ_{1}))^{(−1)} ≅ L(λ_{1}),
H^{2}(G_{1}, L((p − 3)λ_{2}))^{(−1)} ≅ L(λ_{2});

(b) 𝔤 = B_{2}

H^{2}(G_{1}, L(0))^{(−1)} ≅ L(λ_{1}) ⊕ L(2λ_{2}),
H^{2}(G_{1}, L((p − 3)λ_{1} + 2λ_{2}))^{(−1)} ≅ L(λ_{1}),
H^{2}(G_{1}, L(λ_{1} + (p − 4)λ_{2}))^{(−1)} ≅ L(λ_{2});

H^{2}(G_{1}, L((p − 2)(λ_{1} + λ_{2}))^{(−1)} ≅ L(λ_{2});

(c) 𝔤 = G_{2}

H^{2}(G_{1}, L(0))^{(−1)} ≅ L(λ_{1}) ⊕ L(λ_{2}),
H^{2}(G_{1}, L((p − 5)λ_{1} + 2λ_{2}))^{(−1)} ≅ L(λ_{1}),
H^{2}(G_{1}, L(4λ_{1} + (p − 3)λ_{2}))^{(−1)} ≅ L(λ_{2}).

Proof. (a) follows from the exact sequence (8), lemma 3.3 and propositions 4.1, 4.2, part (a).

(b) follows from the exact sequence (8), lemma 3.4 and propositions 4.1 and 4.2, part (b).

(c) follows from the exact sequence (8), lemma 3.5 and propositions 4.1 and 4.2, part (c).

5. 𝔤-cohomology

To prove theorem 1.1 we need some lemmas.

Lemma 5.1. Let 𝔤 be a Lie algebra, V be a restricted 𝔤-module. For an associative 2-cocycle ψ, let ψ′ be the function defined by ψ′_{x}(y) = ψ(x^{p} −x^{[p]}, y)−ψ(y, x^{p} −x^{[p]}). Then the map ψ → ψ′ induces a K-linear map of H^{2}(𝔤, V) into S(𝔤, H^{1}(𝔤, V)).

Proof. [9], Theorem 3.1.

Lemma 5.2. Let 𝔤 = A_{2}, B_{2}, G_{2}. Suppose that V is a restricted irreducible 𝔤−module and H^{1}(𝔤, V) ≠ 0. Then the lists of possible weights of the G-module H^{2}(𝔤, V) and the lists of possible dominant weights are the following

𝔤 V  weights of H^{2}(𝔤, V)  dominants
A_{2} L((p − 2)λ_{1} + λ_{2}) pλ_{1}, p(−λ_{1} + λ_{2}), p(−λ_{2})  pλ_{1}
A_{2} L(λ_{1} + (p − 2)λ_{2}) pλ_{2}, p(λ_{1} − λ_{2}), −pλ_{1}  pλ_{2}
A_{2} L((p − 2)(λ_{1} + λ_{2})) 0, p(2λ_{1} − λ_{2}), p(−λ_{1} + 2λ_{2}),  p(λ_{1} + λ_{2})
p(λ_{1} + λ_{2}), p(−2λ_{1} + λ_{2}),
p(λ_{1} − 2λ_{2}), p(−λ_{1} − λ_{2})
B_{2} L((p − 2)λ_{1} + 2λ_{2}) 0, pλ_{1}, −pλ_{1}, p(−λ_{1} + 2λ_{2}),  0, pλ_{1}
p(λ_{1} − 2λ_{2})
B_{2} L(λ_{1} + (p − 2)λ_{2}) pλ_{2}, −pλ_{2}, p(λ_{1} − λ_{2})  0, pλ_{2}
p(−λ_{1} + λ_{2})
G_{2} L((p − 2)λ_{1} + λ_{2}) 0, pλ_{1}, −pλ_{1}, p(−λ_{1} + λ_{2}),  0, pλ_{1}
p(2λ_{1} − λ_{2}), p(λ_{1} − λ_{2}),
p(−2λ_{1} + λ_{2})
G_{2} L(3λ_{1} + (p − 2)λ_{2}) 0, pλ_{2}, −pλ_{2}, p(2λ_{1} − λ_{2}),  0, pλ_{1}, pλ_{2}
p(−3λ_{1} + 2λ_{2}), p(−λ_{1} + λ_{2}),
p(3λ_{1} − λ_{2}), pλ_{1}, p(−2λ_{1} + λ_{2}),
p(3λ_{1} − 2λ_{2}), p(λ_{1} − λ_{2}),
p(−3λ_{1} + λ_{2}), −pλ_{1}

Proof. It follows from lemma 5.1 and proposition 4.2.

Lemma 5.3. Let 𝔤 = B_{2} and p ⩾ 5. Then

H^{3}(G_{1}, L((p − 3)λ_{1} + 2λ_{2})) = H^{3}(G_{1}, L(λ_{1} + (p − 4)λ_{2})) = 0.

Proof. By lemma 3.4 the following sequence is exact

0 → L((p − 3)λ_{1} + 2λ_{2}) → H^{0}((p − 3)λ_{1} + 2λ_{2}) → L((p − 3)λ_{1}) → 0.

The corresponding exact sequence of G_{1}-cohomology groups gives us that the following sequence is exact

H^{2}(G_{1}, L((p − 3)λ_{1})) → H^{3}(G_{1}, L((p − 3)λ_{1} + 2λ_{2})) →
H^{3}(G_{1}, H^{0}((p − 3)λ_{1} + 2λ_{2}))

(9)

By (5) H^{3}(G_{1}, H^{0}((p − 3)λ_{1} + 2λ_{2})) = 0, since i = 3, l(w) = 2. By proposition 4.3 we have H^{2}(G_{1}, L((p−3)λ_{1})) = 0. Therefore, from the exact sequence (9) we obtain H^{3}(G_{1}, L((p − 3)λ_{1} + 2λ_{2})) = 0.

The second statement H^{3}(G_{1}, L(λ_{1} + (p − 4)λ_{2})) = 0 can be proved in an analogous way.

Lemma 5.4. Let 𝔤 = G_{2} and p > 5. Then

H^{3}(G_{1}, L((p − 5)λ_{1} + 2λ_{2})) = 0.

Proof. By lemma 3.5 the following sequence is exact

0 → L((p − 5)λ_{1} + 2λ_{2}) → H^{0}((p − 5)λ_{1} + 2λ_{2}) → L((2p − 6)λ_{1} + 2λ_{2}) → 0.

Therefore, the following sequence of G_{1}-cohomology groups is exact

H^{2}(G_{1}, L((2p − 6)λ_{1} + 2λ_{2})) → H^{3}(G_{1}, L((p − 5)λ_{1} + 2λ_{2})) →
H^{3}(G_{1}, H^{0}((p − 5)λ_{1} + 2λ_{2}))  (10)

By (5) H^{3}(G_{1}, H^{0}((p − 5)λ_{1} + 2λ_{2})) = 0, since i = 3, l(w) = 2. By proposition 4.3 H^{2}(G_{1}, L((2p − 6)λ_{1} + 2λ_{2})) = 0. Then from the exact sequence (10) it follows that H^{3}(G_{1}, L((p − 5)λ_{1} + 2λ_{2})) = 0.

Proof of theorem 1.1. The proof is divided into two parts. In the first part we prove all isomorphisms mentioned in theorem 1.1. In the second part we establish that for all other weights given in corollary 3.2 the second cohomology groups are trivial.

Part 1. By lemma 2.1 all isomorphisms, except the case 𝔤 = G_{2} and V = L(3λ_{1} + (p − 2)λ_{2}), follow from proposition 4.3.

Let us prove the last isomorphism of (c) . If H^{2}(G_{1}, L(λ)) = 0, then from the exact sequence (1) it follows that H^{2}(𝔤, L(λ)) is isomorphic to the kernel of the map

f : H^{1}(𝔤, L(λ)) ⊗ 𝔤^{∗} → H^{3}(G_{1}, L(λ)).  (11)

By proposition 4.3 H^{2}(G_{1}, L(3λ_{1} + (p − 2)λ_{2})) = 0. Hence H^{2}(𝔤, L(3λ_{1} + (p − 2)λ_{2})) is isomorphic to ker f. By (5)

H^{3}(G_{1}, H^{0}(3λ_{1} + (p − 2)λ_{2}))^{(−1)} ≅

H^{0}(2λ_{1}) ⊕ H^{0}(2λ_{2}) ⊕ H^{0}(3λ_{1}) ⊕ H^{0}(λ_{1} + λ_{2}).  (12)

By lemma 3.5 H^{0}(3λ_{1} + (p − 2)λ_{2})/L(3λ_{1} + (p − 2)λ_{2}) ≅ L(4λ_{1} + (p − 3)λ_{2}) and by proposition 4.3 H^{2}(G_{1}, L(4λ_{1} + (p − 3)λ_{2}))^{(−1)} ≅ L(λ_{2}). Therefore, by the exact sequence (8), H^{3}(G_{1}, L(3λ_{1} + (p − 2)λ_{2})) as a G-module has (possible) composition factors H^{0}(2λ_{1}), H^{0}(2λ_{2}), H^{0}(3λ_{2}), H^{0}(λ_{1} + λ_{2}) and H^{0}(λ_{2}).

By proposition 4.2 H^{1}(𝔤, L(3λ_{1} + (p − 2)λ_{2})) ≅ L(λ_{2}). Therefore,

H^{1}(𝔤, L(3λ_{1} + (p − 2)λ_{2})) ⊗ 𝔤^{∗} ≅ L(λ_{2}) ⊗ 𝔤 ≅
H^{0}(2λ_{1}) ⊕ H^{0}(2λ_{2}) ⊕ H^{0}(3λ_{1}) ⊕ H^{0}(λ_{2}) ⊕ H^{0}(0).  (13)

From the decompositions of H^{1}(𝔤, L(3λ_{1} + (p − 2)λ_{2})) ⊗ 𝔤^{∗} and H^{3}(G_{1}, L(3λ_{1} + (p − 2)λ_{2})) we obtain that H^{0}(0) = L(0) ⊆ ker f.

If ker f contains some of the G-modules H^{0}(2λ_{1}), H^{0}(2λ_{2}), H^{0}(3λ_{1}) and H^{0}(λ_{2}), then the G-module H^{2}(𝔤, L(3λ_{1} + (p − 2)λ_{2})) has nontrivial elements with weights 2pλ_{1}, 2pλ_{2}, 3pλ_{1}, or pλ_{2}. We will prove that this is impossible.

For 𝔤 = G_{2} the list of dominant weights of adjoint G-module is {0, pλ_{1}, pλ_{2}} (see lemma 5.2). Therefore, the only non-zero dominant weights of the G-module H^{2}(𝔤, L(3λ_{1} +(p−2)λ_{2})) are pλ_{1} or pλ_{2}. Thus cocycles in Z^{2}(𝔤, L(3λ_{1} +(p−2)λ_{2})) with weights 2pλ_{1}, 2pλ_{2}, 3pλ_{1} are coboundaries.

Now we prove that the classes of cocycles with weights pλ_{2} are also trivial. To do it we use the realization of the 𝔤-module L(3λ_{1} + (p − 2)λ_{2}) as a factor-module of the Weyl module.

Let f_{i}, h_{j}, e_{i} : i = 1, …, 6, j = 1, 2 be a Chevalley basis of 𝔤, where f_{i} = e_{−α_{i}}, e_{i} = e_{α_{i}} for i = 1, 2 i f_{3} = e_{−α_{1}−α_{2}}, f_{4} = e_{−2α_{1}−α_{2}}, f_{5} = e_{−3α_{1}−α_{2}}, f_{6} = e_{−3α_{1}−2α_{2}}, e_{3} = e_{α_{1}+α_{2}}, e_{4} = e_{2α_{1}+α_{2}}, e_{5} = e_{3α_{1}+α_{2}}, e_{6} = e_{3α_{1}+2α_{2}}. The Weyl module V (m_{1}λ_{1} + m_{2}λ_{2}) can be defined on the vector space

v_{i,j,k,l,s,t} := \frac{f_{6}^{t}f_{5}^{s}f_{4}^{l}f_{3}^{k}f_{2}^{j}f_{1}^{i}}{t!s!l!k!j!i!} ⊗ v_{m_{1}λ_{1}+m_{2}λ_{2}},

where v_{m_{1}λ_{1}+m_{2}λ_{2}} is the highest weight, by

e_{1}v_{i,j,k,l,s,t} = (l + 1)v_{i,j,k+1,l,s−1,t} − 3(t + 1)v_{i,j,k,l−2,s,t+1} −
2(k + 1)v_{i,j,k+1,l−1,s,t} − 3(j + 1)v_{i,j+1,k−1,l,s,t} + (m_{1} + 1 − i)v_{i−1,j,k,l,s,t},

e_{2}v_{i,j,k,l,s,t} = (s + 1)v_{i,j,k,l,s+1,t−1} − (l + 1)v_{i,j,k−2,l+1,s,t} +
(i + 1)v_{i+1,j,k−1,l,s,t} − 2(t + 1)v_{i,j,k−3,l,s,t+1} +
(m_{2} + 1 + i − j − k)v_{i,j−1,k,l,s,t},

f_{1}v_{i,j,k,l,s,t} = −3(s + 1)v_{i,j,k,l−1,s+1,t} − 3(t + 1)v_{i,j,k−2,l,s,t+1}
−2(l + 1)v_{i,j,k−1,l+1,s,t} − (k + 1)v_{i,j−1,k+1,l,s,t} + (i + 1)v_{i+1,j,k,l,s,t},

f_{2}v_{i,j,k,l,s,t} = −(t + 1)v_{i,j,k,l,s−1,t+1} + (j + 1)v_{i,j+1,k,l,s,t}.

If some cocycle ψ of the module L(3λ_{1} + (p − 2)λ_{2}) has weight pλ_{2}, then

ψ(e_{1}, f_{3}) = x_{1}v_{0,0,0,0,0,0}, ψ(e_{5}, f_{6}) = x_{2}v_{0,0,0,0,0,0},

ψ(f_{1}, f_{2}) = x_{3}v_{1,0,0,0,0,0}, ψ(e_{1}, f_{4}) = x_{4}v_{1,0,0,0,0,0},

ψ(e_{4}, f_{6}) = x_{5}v_{1,0,0,0,0,0}, ψ(f_{1}, f_{3}) = x_{6}v_{2,0,0,0,0,0},

ψ(e_{1}, f_{5}) = x_{7}v_{2,0,0,0,0,0}, ψ(e_{3}, f_{6}) = x_{8}v_{2,0,0,0,0,0},

ψ(f_{1}, f_{4}) = x_{9}v_{3,0,0,0,0,0}, ψ(e_{2}, f_{6}) = x_{10}v_{3,0,0,0,0,0},

ψ(f_{2}, f_{3}) = x_{11}v_{1,1,0,0,0,0} + x_{12}v_{0,0,1,0,0,0},

ψ(f_{1}, f_{6}) = x_{13}v_{3,0,1,0,0,0} + x_{14}v_{2,0,0,1,0,0} + x_{15}v_{1,0,0,0,1,0},

ψ(e_{1}, f_{6}) = x_{16}v_{2,1,0,0,0,0} + x_{17}v_{1,0,1,0,0,0} + x_{18}v_{0,0,0,1,0,0},

ψ(f_{2}, f_{4}) = x_{19}v_{2,1,0,0,0,0} + x_{20}v_{1,0,1,0,0,0} + x_{21}v_{0,0,0,1,0,0},

ψ(f_{2}, f_{5}) = x_{22}v_{3,1,0,0,0,0} + x_{23}v_{2,0,1,0,0,0} + x_{24}v_{1,0,0,1,0,0} + x_{25}v_{0,0,0,0,1,0},

ψ(f_{3}, f_{4}) = x_{26}v_{3,1,0,0,0,0} + x_{27}v_{2,0,1,0,0,0} + x_{28}v_{1,0,0,1,0,0} + x_{29}v_{0,0,0,0,1,0},

ψ(f_{2}, f_{6}) = x_{30}v_{3,2,0,0,0,0} + x_{31}v_{2,1,1,0,0,0} + x_{32}v_{1,0,2,0,0,0} + x_{33}v_{1,1,0,1,0,0}+

x_{34}v_{0,0,1,1,0,0} + x_{35}v_{0,1,0,0,1,0} + x_{36}v_{0,0,0,0,0,1},

ψ(f_{3}, f_{5}) = x_{37}v_{3,0,1,0,0,0} + x_{38}v_{2,0,0,1,0,0} + x_{39}v_{1,0,0,0,1,0},

ψ(f_{3}, f_{6}) = x_{40}v_{3,1,1,0,0,0} + x_{41}v_{2,0,2,0,0,0} + x_{42}v_{2,1,0,1,0,0} + x_{43}v_{0,0,0,2,0,0}+

x_{44}v_{1,0,1,1,0,0} + x_{45}v_{1,1,0,0,1,0} + x_{46}v_{0,0,1,0,1,0} + x_{47}v_{1,0,0,0,0,1},

ψ(f_{4}, f_{5}) = x_{48}v_{3,0,0,1,0,0} + x_{49}v_{2,0,0,0,1,0},

ψ(f_{4}, f_{6}) = x_{50}v_{3,0,2,0,0,0} + x_{51}v_{3,1,0,1,0,0} + x_{52}v_{1,0,0,2,0,0} + x_{53}v_{2,0,1,1,0,0}+

x_{54}v_{2,1,0,0,1,0} + x_{55}v_{1,0,1,0,1,0} + x_{56}v_{0,0,0,1,1,0} + x_{57}v_{2,0,0,0,0,1},

ψ(f_{5}, f_{6}) = x_{58}v_{3,0,1,1,0,0} + x_{59}v_{2,0,0,2,0,0} + x_{60}v_{3,1,0,0,1,0} + x_{61}v_{0,0,0,0,2,0}+

x_{62}v_{2,0,1,0,1,0} + x_{63}v_{1,0,0,1,1,0} + x_{64}v_{3,0,0,0,0,1},

for some x_{i} ∈ K (i = 1, …, 64). Non-written components of ψ(X, Y) are equal to 0 (sums of weights of X and Y can not be a weight of the module).

The conditions

dψ(h_{i}, f_{2}, f_{6}) = 0, dψ(h_{i}, f_{2}, f_{5}) = 0, dψ(h_{i}, e_{1}, f_{3}) = 0, i = 1, 2, gives us a system of linear equations. Solving this system (we omit the standard but long calculations) gives us that

ψ(h_{i}, f_{2}) = ψ(h_{i}, f_{3}) = ψ(h_{i}, f_{4}) = ψ(h_{i}, f_{5}) = ψ(h_{i}, f_{6}) = 0, i = 1, 2.

Furthermore, from the following system of linear equations with 16 conditions

dψ(f_{4}, f_{5}, f_{6}) = 0, dψ(f_{3}, f_{5}, f_{6}) = 0, dψ(f_{3}, f_{4}, f_{5}) = 0,
dψ(f_{2}, f_{4}, f_{5}) = 0, dψ(f_{2}, f_{5}, f_{6}) = 0, dψ(f_{1}, f_{4}, f_{6}) = 0,
dψ(e_{1}, f_{5}, f_{6}) = 0, dψ(e_{2}, f_{5}, f_{6}) = 0, dψ(e_{3}, f_{5}, f_{6}) = 0,
dψ(e_{4}, f_{5}, f_{6}) = 0, dψ(e_{5}, f_{4}, f_{5}) = 0, dψ(e_{6}, f_{5}, f_{6}) = 0,
dψ(e_{2}, f_{4}, f_{6}) = 0, dψ(e_{3}, f_{3}, f_{4}) = 0, dψ(e_{1}, f_{3}, f_{4}) = 0,
dψ(e_{1}, f_{4}, f_{6}) = 0

we obtain that ψ should have the following form

ψ(f_{1}, f_{2}) = x_{3}v_{1,0,0,0,0,0}, ψ(f_{1}, f_{3}) = x_{6}v_{2,0,0,0,0,0},
ψ(f_{2}, f_{3}) = x_{11}v_{1,1,0,0,0,0} + x_{12}v_{0,0,1,0,0,0}.

Then from the equation dψ(e_{1}, f_{1}, f_{2}) = 0 we obtain that x_{3} = 0. Since ψ(f_{1}, f_{2}) = 0, then the condition dψ(e_{1}, f_{1}, f_{3}) = 0 gives us that x_{6} = 0. Finally, from the condition dψ(f_{2}, f_{3}, f_{6}) = 0, we have that ψ(f_{2}, f_{3}) = 0. Therefore, any cocycle with weight pλ_{1} is trivial.

So, in the case L(λ) = L(3λ_{1} + (p − 2)λ_{2}), the kernel of f is isomorphic to the G-module H^{0}(0) = L(0). Thus,

H^{2}(G_{2}, L(3λ_{1} + (p − 2)λ_{2})) ≅ L(0)^{(1)}.

Part 2. We now prove that the second cohomology groups of irreducible peculiar modules which are not mentioned in the formulation of theorem 1.1 are trivial.

The lists of highest weights of peculiar modules are given in corollary 3.2. By proposition 4.3 the second cohomology groups of G_{1} for these modules are trivial. Therefore, it is enough to show that ker f = 0 for any of these modules. We will do it by considering each case of algebras separately.

The case (a). By proposition 4.2 only the following 3 modules have nontrivial first cohomology groups L((p − 2)(λ_{1} + λ_{2})), L((p − 2)λ_{1} + λ_{2}), L(λ_{1} + (p − 2)λ_{2}). By (4) and (5) we have

H^{3}(G_{1}, H^{0}((p − 2)(λ_{1} + λ_{2})))^{(−1)} ≅ H^{0}(λ_{1} + λ_{2}),  (14)

H^{3}(G_{1}, H^{0}((p − 2)λ_{1} + λ_{2}))^{(−1)} ≅ H^{0}(2λ_{2}) ⊕ H^{0}(2λ_{1} + λ_{2}),  (15)

H^{3}(G_{1}, H^{0}(λ_{1} + (p − 2)λ_{2}))^{(−1)} ≅ H^{0}(2λ_{1}) ⊕ H^{0}(λ_{1} + 2λ_{2}).  (16)

H^{3}(G_{1}, H^{0}(λ))^{(−1)} = 0 if λ = 0, (p − 3)λ_{1}, (p − 3)λ_{2}.  (17)

According to (14)–(17), lemma 3.3 and proposition 4.3 the exact sequence (8) gives us the following exact sequences

0 → H^{0}(λ_{1} + λ_{2}) → H^{3}(G_{1}, L((p − 2)(λ_{1} + λ_{2}))) → H^{0}(λ_{1} + λ_{2}) → 0, (18)

0 → L(λ_{1}) → H^{3}(G_{1}, L((p − 2)λ_{1} + λ_{2})) →
H^{0}(2λ_{2}) ⊕ H^{0}(2λ_{1} + λ_{2}) → 0,  (19)

0 → L(λ_{2}) → H^{3}(G_{1}, L(λ_{1} + (p − 2)λ_{2})) →
H^{0}(2λ_{1}) ⊕ H^{0}(λ_{1} + 2λ_{2}) → 0.  (20)

On the other hand, according to proposition 4.2, we have

H^{1}(𝔤, L((p − 2)(λ_{1} + λ_{2})) ⊗ 𝔤^{∗} ≅ L(0) ⊗ 𝔤 ≅ 𝔤 ≅ H^{0}(λ_{1} + λ_{2}),  (21)

H^{1}(𝔤, L((p − 2)λ_{1} + λ_{2})) ⊗ 𝔤^{∗} ≅ L(λ_{1}) ⊗ 𝔤 ≅
H^{0}(2λ_{2}) ⊕ H^{0}(2λ_{1} + λ_{2}) ⊕ H^{0}(λ_{1}),  (22)

H^{1}(g, L(λ_{1} + (p − 2)λ_{2})) ⊗ 𝔤^{∗} ≅
L(λ_{2}) ⊗ 𝔤 ≅ H^{0}(2λ_{1}) ⊕ H^{0}(λ_{1} + 2λ_{2}) ⊕ H^{0}(λ_{2}).  (23)

Consider each case of these modules.

Let V = L((p − 2)(λ_{1} + λ_{2})). From (14) and (21) we see that H^{3}(G_{1}, L((p − 2)(λ_{1} + λ_{2})) contains a composition factor H^{0}(λ_{1} + λ_{2}) isomorphic to H^{1}(𝔤, L((p − 2)(λ_{1} + λ_{2}))) ⊗ 𝔤^{∗}.

If ker f contains H^{0}(λ_{1} + λ_{2}), then the second cohomology group H^{2}(𝔤, L((p − 2)(λ_{1} + λ_{2}))) as a G-module contains classes of cocycles of weights p(λ_{1} + λ_{2}). The highest weight p(λ_{1} + λ_{2}) − 2α_{1} − 2α_{2} can not be presented as a sum of p(λ_{1} + λ_{2}) and two roots. Therefore, such a case is impossible. Therefore, H^{0}(λ_{1} + λ_{2}) can not be in ker f. Then the exact sequence (1) for L((p − 2)(λ_{1} + λ_{2})) gives us that ker f = 0.

Let f_{i}, h_{j}, e_{i} : i = 1, 2, 3, j = 1, 2 be the Chevalley basis of 𝔤, where f_{i} = e_{−α_{i}}, e_{i} = e_{α_{i}} for i = 1, 2 and f_{3} = e_{−α_{1}−α_{2}}, e_{3} = e_{α_{1}+α_{2}}. The Weyl module V (m_{1}λ_{1} + m_{2}λ_{2}) can be defined on the vector space with basis

v_{i,j,k} := \frac{f_{3}^{k}f_{2}^{j}f_{1}^{i}}{k!j!i!} ⊗ v_{m_{1}λ_{1}+m_{2}λ_{2}},

where v_{m_{1}λ_{1}+m_{2}λ_{2}} is the highest weight.

The action of 𝔤 is given by

e_{1}v_{i,j,k} = −(j + 1)v_{i,j+1,k−1} + (m_{1} + 1 − i)v_{i−1,j,k},

e_{2}v_{i,j,k} = (i + 1)v_{i+1,j,k−1} + (m_{2} + 1 + i − j − k)v_{i,j−1,k},
f_{1}v_{i,j,k} = −(k + 1)v_{i,j−1,k+1},
f_{2}v_{i,j,k} = (j + 1)v_{i,j+1,k}.

Let now V = L((p − 2)λ_{1} + λ_{2}). From (19) and (22) we see that H^{1}(𝔤, L((p − 2)λ_{1} +λ_{2}))⊗𝔤^{∗} and H^{3}(G_{1}, L((p−2)λ_{1} +λ_{2})) have isomorphic composition factors H^{0}(2λ_{1} + λ_{2}), H^{0}(2λ_{2}), H^{0}(λ_{1}).

By lemma 5.2 any nontrivial cocycle of Z^{2}(𝔤, L((p − 2)λ_{1} + λ_{2})) must have a dominant weight of the form pλ_{1}. Therefore, cocycles of Z^{2}(𝔤, L((p − 2)λ_{1} + λ_{2})) with weights p(2λ_{1} + λ_{2}), 2pλ_{2} are 0. Thus H^{0}(2λ_{1} + λ_{2}), H^{0}(2λ_{2}) can not be in ker f. We now prove that H^{0}(λ_{1}) also can not be in ker f. To do this, we prove that cocycles with weight pλ_{1} are coboundaries.

Let ψ be some cocycle of weight pλ_{1}. Then

ψ(h_{1}, f_{1}) = x_{1}v_{0,0,0}, ψ(h_{2}, f_{1}) = x_{2}v_{0,0,0}, ψ(e_{2}, f_{3}) = x_{3}v_{0,0,0},
ψ(h_{1}, f_{3}) = x_{4}v_{0,1,0}, ψ(h_{2}, f_{3}) = x_{5}v_{0,1,0,},
ψ(f_{1}, f_{2}) = x_{6}v_{0,1,0}, ψ(f_{1}, f_{3}) = x_{7}v_{0,0,1},

for some x_{i} ∈ K (i = 1, …, 7). Non-written components are zero.

From the cocyclicity conditions we obtain one cocycle of weight pλ_{1} with the following non-zero components

ψ(e_{2}, f_{3}) = v_{0,0,0}, ψ(f_{1}, f_{2}) = v_{0,1,0}, ψ(f_{1}, f_{3}) = v_{0,0,1}.

It is easy to see that, ψ is a coboundary dω, where ω(f_{1}) = v_{0,0,0}. So, any cocycle of the weight pλ_{1} is trivial. Thus L(λ_{1}) ⊄ ker f. The case of the dual module V = L(λ_{1} + (p − 2)λ_{2}) may be treated in a similar way.

The case (b). According to proposition 4.2 the first cohomology groups are nonzero only for the following two modules L((p − 2)λ_{1} + 2λ_{2}), L(λ_{1} + (p − 2)λ_{2}). By (5) we have

H^{3}(G_{1}, H^{0}((p − 2)λ_{1} + 2λ_{2}))^{(−1)} ≅ H^{0}(2λ_{1}) ⊕ H^{0}(2λ_{2}) ⊕ H^{0}(λ_{1} + 2λ_{2}), (24)

H^{3}(G_{1}, H^{0}(λ_{1} + (p − 2)λ_{2}))^{(−1)} ≅ H^{0}(3λ_{2}) ⊕ H^{0}(λ_{1} + λ_{2}).  (25)

According to (24), (25), lemma 3.4, 5.3 and proposition 4.3 the exact sequence (8) can be rewritten as follows

0 → L(λ_{1}) → H^{3}(G_{1}, L((p − 2)λ_{1} + 2λ_{2})) →
H^{0}(2λ_{1}) ⊕ H^{0}(2λ_{2}) ⊕ H^{0}(λ_{1} + 2λ_{2}) → 0,  (26)

0 → L(λ_{2}) → H^{3}(G_{1}, L(λ_{1} + (p − 2)λ_{2})) →
H^{0}(3λ_{2}) ⊕ H^{0}(λ_{1} + λ_{2}) → 0.  (27)

By proposition 4.2,

H^{1}(𝔤, L((p − 2)λ_{1} + 2λ_{2})) ⊗ 𝔤^{∗} ≅ L(λ_{1}) ⊗ 𝔤 ≅
H^{0}(2λ_{2}) ⊕ H^{0}(λ_{1} + 2λ_{2}) ⊕ H^{0}(λ_{1}),  (28)

H^{1}(𝔤, L(λ_{1} + (p − 2)λ_{2})) ⊗ 𝔤^{∗} ≅ L(λ_{2}) ⊗ 𝔤 ≅
H^{0}(3λ_{2}) ⊕ H^{0}(λ_{1} + λ_{2}) ⊕ H^{0}(λ_{2}).  (29)

Let V = L((p−2)λ_{1}+2λ_{2}). From (26) and (28) we see that all composition factors H^{0}(λ_{1} + 2λ_{2}), H^{0}(2λ_{2}), H^{0}(λ_{1}) of the G-module H^{1}(𝔤, L((p − 2)λ_{1} + 2λ_{2})) ⊗ 𝔤^{∗} are in the module H^{3}(G_{1}, L((p − 2)λ_{1} + 2λ_{2})). By lemma 5.2 cocycles have dominant weights equal to pλ_{1}. Therefore, Z^{2}(𝔤, L((p − 2)λ_{1} + 2λ_{2})) has no cocycles with weights p(λ_{1}+2λ_{2}), 2pλ_{2}. Therefore, ker f has no submodules isomorphic to H^{0}(λ_{1}+ 2λ_{2}) or H^{0}(2λ_{2}). We now prove that H^{0}(λ_{1}) ⊈ ker f. We prove that any cocycle ψ with weight pλ_{1} is a coboundary.

For any cocycle ψ with weight pλ_{1}

ψ(h_{1}, f_{1}) = x_{1}v_{0,0,0,0}, ψ(h_{2}, f_{1}) = x_{2}v_{0,0,0,0}, ψ(e_{2}, f_{3}) = x_{3}v_{0,0,0,0},
ψ(h_{1}, f_{3}) = x_{4}v_{1,0,0,0}, ψ(h_{2}, f_{3}) = x_{5}v_{1,0,0,0}, ψ(f_{1}, f_{2}) = x_{6}v_{1,0,0,0},
ψ(e_{2}, f_{4}) = x_{7}v_{1,0,0,0}, ψ(h_{1}, f_{4}) = x_{8}v_{2,0,0,0}, ψ(h_{2}, f_{2}) = x_{9}v_{2,0,0,0},
ψ(f_{2}, f_{3}) = x_{10}v_{2,0,0,0}, ψ(f_{1}, f_{3}) = x_{11}v_{0,1,0,0},
ψ(f_{1}, f_{4}) = x_{12}v_{1,1,0,0} + x_{13}v_{0,0,0,1}, ψ(f_{3}, f_{4}) = x_{14}v_{1,0,0,1},

for some x_{i} ∈ K (i = 1, …, 14). Non-written components are 0.

From the cocyclicity conditions we obtain that such cocycles are linear combinations of the following two cocycles

ψ_{1}(e_{2}, f_{3}) = 2v_{0,0,0,0}, ψ_{1}(f_{1}, f_{2}) = v_{1,0,0,0},
ψ_{1}(f_{1}, f_{3}) = v_{0,1,0,0}, ψ_{1}(f_{1}, f_{4}) = v_{0,0,0,1};
ψ_{2}(e_{2}, f_{4}) = −v_{1,0,0,0}, ψ_{2}(f_{2}, f_{3}) = −2v_{2,0,0,0},
ψ_{2}(f_{1}, f_{4}) = v_{0,0,0,1}, ψ_{2}(f_{3}, f_{4}) = v_{1,0,0,1}.

Both cocycles are coboundaries: ψ_{1} = dω_{1} and ψ_{2} = dω_{2}, where non-zero components of ω_{1} and ω_{2} are given by ω_{1}(f_{1}) = −v_{0,0,0,0} and ω_{2}(f_{4}) = −v_{2,0,0,0}. So, any cocycle of weight pλ_{1} is a coboundary. Therefore, L(λ_{1}) ⊈ ker f.

Consider now the case V = L(λ_{1} + (p − 2)λ_{2}).

From (27) and (29) we see that H^{1}(𝔤, L(λ_{1} + (p − 2)λ_{2})) ⊗ 𝔤^{∗} and H^{3}(G_{1}, L(λ_{1} + (p − 2)λ_{2})) have equal composition factors H^{0}(3λ_{2}), H^{0}(λ_{2} + λ_{2}), H^{0}(λ_{2}) = L(λ_{2}). By lemma 5.2 a non-trivial 2-cocycle has dominant weight equal to pλ_{2}. Therefore, all cocycles of Z^{2}(𝔤, L(λ_{1} + (p − 2)λ_{2})), with G-weights 3pλ_{2}, p(λ_{1} + λ_{2}) are trivial. Thus, the modules H^{0}(3λ_{2}), H^{0}(λ_{1} + λ_{2}) are not in ker f.

We now prove that H^{0}(λ_{2}) is also not in ker f. So, we need to prove that any 2-cocycle with weight pλ_{2} is trivial.

Let ψ be some cocycle of weight pλ_{2}. Then all components ψ(X, Y) are zero except the following components

ψ(h_{1}, f_{2}) = x_{1}v_{0,0,0,0}, ψ(h_{2}, f_{2}) = x_{2}v_{0,0,0,0}, ψ(e_{1}, f_{3}) = x_{3}v_{0,0,0,0},
ψ(e_{3}, f_{4}) = x_{4}v_{0,0,0,0}, ψ(h_{1}, f_{3}) = x_{5}v_{0,0,1,0}, ψ(h_{2}, f_{3}) = x_{6}v_{0,0,1,0},
ψ(f_{1}, f_{2}) = x_{7}v_{0,0,,0}, ψ(e_{2}, f_{4}) = x_{8}v_{0,0,1,0},
ψ(h_{1}, f_{4}) = x_{9}v_{1,0,1,0} + x_{10}v_{0,1,0,0},
ψ(h_{2}, f_{4}) = x_{12}v_{1,0,1,0} + x_{12}v_{0,1,0,0}, ψ(f_{2}, f_{3}) = x_{13}v_{1,0,1,0} + x_{14}v_{0,1,0,0},
ψ(f_{2}, f_{4}) = x_{15}v_{1,1,0,0} + x_{16}v_{0,0,0,1}, ψ(f_{1}, f_{4}) = x_{17}v_{0,1,1,0},
ψ(f_{3}, f_{4}) = x_{18}v_{1,1,1,0} + x_{19}v_{0,0,1,1}, ψ(e_{1}, f_{4}) = x_{20}v_{1,0,0,0},

for some x_{i} ∈ K (i = 1, …, 20).

We have

dψ(e_{1}, f_{1}, f_{4}) = 0 ⇒ x_{10} = 0, x_{9} − x_{17} − x_{20} = 0.

Then the equations dψ(h_{1}, e_{1}, f_{3}) = 0, dψ(h_{1}, e_{3}, f_{4}) = 0, dψ(h_{1}, f_{2}, f_{3}) = 0 gives us x_{1} = x_{5} = x_{9} = 0.

Furthermore, from

dψ(f_{2}, f_{3}, f_{4}) = 0, dψ(e_{1}, f_{3}, f_{4}) = 0, dψ(f_{1}, f_{3}, f_{4}) = 0,
dψ(e_{1}, f_{1}, f_{4}) = 0, dψ(e_{1}, f_{2}, f_{4}) = 0, dψ(e_{3}, f_{2}, f_{4}) = 0

we have x_{4} = x_{16} = x_{14} = −x_{3} and x_{18} = x_{13} = x_{15} = x_{20} = x_{17} = 0.

From dψ(e_{2}, f_{2}, f_{4}) = 0, dψ(e_{2}, f_{3}, f_{4}) = 0, dψ(e_{3}, f_{3}, f_{4}) = 0 we obtain x_{8} = x_{11} = x_{12} = x_{16} = x_{14} = x_{13} = 0. Finally, from dψ(e_{1}, f_{1}, f_{2}) = 0, dψ(e_{3}, f_{2}, f_{3}) = 0 we obtain x_{7} = x_{2} = 0. So, ψ = 0, if it has weight pλ_{2}. Thus H^{0}(λ_{2}) ⊈ ker f.

The case (c). By proposition 4.2 only the following two modules have non-zero first cohomology groups, L((p − 2)λ_{1} + λ_{2}) and L(3λ_{1} + (p − 2)λ_{2}). We have proved above that H^{2}(𝔤, L(3λ_{1} + (p − 2)λ_{2})) ≅ L(0)^{(1)}. Therefore, it is enough to check that ker f = 0 for the module L((p − 2)λ_{1} + λ_{2}). According to (5),

H^{3}(G_{1}, H^{0}((p − 2)λ_{1} + λ_{2}))^{(−1)} ≅ H^{0}(λ_{2}) ⊕ H^{0}(2λ_{1}) ⊕ H^{0}(λ_{1} + λ_{2}).  (30)

So, (30), lemma 3.5, 5.4 and proposition 4.3 and the exact sequence (8), gives us the following exact sequence

0 → L(λ_{1}) → H^{3}(G_{1}, L((p − 2)λ_{1} + λ_{2})) →
H^{0}(λ_{2}) ⊕ H^{0}(2λ_{1}) ⊕ H^{0}(λ_{1} + λ_{2}) → 0.  (31)

By proposition 4.2, we have

H^{1}(𝔤, L((p − 2)λ_{1} + λ_{2})) ⊗ 𝔤^{∗} ≅ L(λ_{1}) ⊗ 𝔤 ≅

H^{0}(2λ_{1}) ⊕ H^{0}(λ_{1} + λ_{2}) ⊕ H^{0}(λ_{1}).  (32)

By (31), (32) all composition factors of the G−module H^{1}(𝔤, L((p−2)λ_{1} +λ_{2}))⊗ 𝔤^{∗} appear as a composition factors of H^{3}(G_{1}, L((p − 2)λ_{1} + λ_{2})). If ker f has such a composition factor, then H^{2}(𝔤, L((p − 2)λ_{1} + λ_{2})) has non-trivial cocycles with weights 2pλ_{1}, p(λ_{1} + λ_{2}), pλ_{1}. We will prove that this is impossible.

By lemma 5.2 the dominant weight of a non-trivial cocycle in Z^{2}(𝔤, L((p − 2)λ_{1} + λ_{2})) is pλ_{1}. Therefore, cocycles in Z^{2}(𝔤, L((p − 2)λ_{1} + λ_{2})) with weights 2pλ_{1}, p(λ_{1} + λ_{2}) are 0. we now prove that any cocycle ψ with weight pλ_{1} is 0. All components ψ(X, Y) of such a cocycle except the following components are 0:

ψ(e_{2}, f_{3}) = x_{1}v_{0,0,0,0,0,0}, ψ(e_{3}, f_{4}) = x_{2}v_{0,0,0,0,0,0},
ψ(e_{4}, f_{5}) = x_{3}v_{0,0,0,0,0,0}, ψ(f_{1}, f_{2}) = x_{4}v_{0,1,0,0,0,0},
ψ(e_{1}, f_{4}) = x_{5}v_{0,1,0,0,0,0}, ψ(e_{4}, f_{6}) = x_{6}v_{0,1,0,0,0,0},
ψ(f_{1}, f_{3}) = x_{7}v_{1,1,0,0,0,0} + x_{8}v_{0,0,1,0,0,0},
ψ(e_{1}, f_{5}) = x_{9}v_{1,1,0,0,0,0} + x_{10}v_{0,0,1,0,0,0},
ψ(e_{3}, f_{6}) = x_{11}v_{1,1,0,0,0,0} + x_{12}v_{0,0,1,0,0,0}, ψ(e_{2}, f_{4}) = x_{13}v_{1,0,0,0,0,0},
ψ(e_{3}, f_{5}) = x_{14}v_{1,0,0,0,0,0}, ψ(e_{2}, f_{5}) = x_{15}v_{2,0,0,0,0,0},
ψ(f_{2}, f_{4}) = x_{16}v_{0,1,1,0,0,0}, ψ(f_{2}, f_{6}) = x_{17}v_{0,1,2,0,0,0},
ψ(e_{1}, f_{6}) = x_{33}v_{0,1,1,0,0,0},
ψ(f_{1}, f_{4}) = x_{18}v_{2,1,0,0,0,0} + x_{19}v_{1,0,1,0,0,0} + x_{20}v_{0,0,0,1,0,0},
ψ(e_{2}, f_{6}) = x_{21}v_{2,1,0,0,0,0} + x_{22}v_{1,0,1,0,0,0} + x_{23}v_{0,0,0,1,0,0},
ψ(f_{2}, f_{5}) = x_{24}v_{1,1,1,0,0,0} + x_{25}v_{0,0,2,0,0,0} + x_{26}v_{0,1,0,1,0,0},
ψ(f_{3}, f_{4}) = x_{27}v_{1,1,1,0,0,0} + x_{28}v_{0,0,2,0,0,0} + x_{29}v_{0,1,0,1,0,0},
ψ(f_{1}, f_{5}) = x_{30}v_{2,0,1,0,0,0} + x_{31}v_{1,0,0,1,0,0} + x_{32}v_{0,0,0,0,1,0},
ψ(f_{1}, f_{6}) = x_{34}v_{2,1,1,0,0,0} + x_{35}v_{1,0,2,0,0,0} + x_{36}v_{1,1,0,1,0,0}+
x_{37}v_{0,0,1,1,0,0} + x_{38}v_{0,1,0,0,1,0} + x_{39}v_{0,0,0,0,0,1},
ψ(f_{3}, f_{5}) = x_{40}v_{2,1,1,0,0,0} + x_{41}v_{1,0,2,0,0,0} + x_{42}v_{1,1,0,1,0,0}+
x_{43}v_{0,0,1,1,0,0} + x_{44}v_{0,1,0,0,1,0} + x_{45}v_{0,0,0,0,0,1},
ψ(f_{3}, f_{6}) = x_{46}v_{1,1,2,0,0,0} + x_{47}v_{0,0,3,0,0,0} + x_{48}v_{0,1,1,1,0,0} + x_{49}v_{0,1,0,0,0,1},
ψ(f_{4}, f_{5}) = x_{50}v_{3,1,1,0,0,0} + x_{51}v_{2,0,2,0,0,0} + x_{52}v_{2,1,0,1,0,0}+

x_{53}v_{0,0,0,2,0,0} + x_{54}v_{1,0,1,1,0,0} + x_{55}v_{0,0,1,0,1,0},
ψ(f_{4}, f_{6}) = x_{56}v_{2,1,2,0,0,0} + x_{57}v_{1,0,3,0,0,0} + x_{58}v_{0,1,0,2,0,0} + x_{59}v_{1,1,1,1,0,0}+
x_{60}v_{0,0,2,1,0,0} + x_{61}v_{1,1,0,0,0,1} + x_{62}v_{0,0,1,0,0,1} + +x_{63}v_{0,1,1,0,1,0},
ψ(f_{5}, f_{6}) = x_{64}v_{3,1,2,0,0,0} + x_{65}v_{2,0,3,0,0,0} + x_{66}v_{1,1,0,2,0,0} + x_{67}v_{2,1,1,1,0,0}+
x_{68}v_{1,0,2,1,0,0} + x_{69}v_{0,0,1,2,0,0} + x_{70}v_{1,1,1,0,1,0} + x_{71}v_{0,0,2,0,1,0}+
x_{72}v_{0,1,0,1,1,0} + x_{73}v_{2,1,0,0,0,1} + x_{74}v_{1,0,1,0,0,1} + x_{75}v_{0,0,0,1,0,1},

where x_{i} ∈ K (i = 1, …, 75).

From dψ(h_{i}, f_{1}, f_{6}) = 0, dψ(h_{i}, f_{1}, f_{5}) = 0, dψ(h_{i}, e_{2}, f_{3}) = 0, i = 1, 2, it follows that

ψ(h_{i}, f_{1}) = ψ(h_{i}, f_{3}) = ψ(h_{i}, f_{4}) = ψ(h_{i}, f_{5}) = ψ(h_{i}, f_{6}) = 0, i = 1, 2.

From

dψ(f_{4}, f_{5}, f_{6}) = 0, dψ(f_{3}, f_{5}, f_{6}) = 0, dψ(f_{2}, f_{5}, f_{6}) = 0,
dψ(f_{3}, f_{4}, f_{6}) = 0, dψ(f_{3}, f_{4}, f_{5}) = 0, dψ(f_{1}, f_{2}, f_{6}) = 0,
dψ(f_{,}f_{2}, f_{5}) = 0, dψ(e_{2}, f_{,}f_{6}) = 0, dψ(e_{4}, f_{5}, f_{6}) = 0,

it follows that the following components ψ(f_{5}, f_{6}), ψ(f_{3}, f_{6}), ψ(f_{3}, f_{5}), ψ(f_{2}, f_{6}), ψ(f_{2}, f_{5}), ψ(f_{1}, f_{6}), ψ(f_{1}, f_{5}), ψ(f_{1}, f_{2}), ψ(e_{4}, f_{6}), ψ(e_{4}, f_{5}), ψ(e_{2}, f_{6}) are zero and ψ(f_{4}, f_{6}) = x_{62}v_{0,0,1,0,0,1}, ψ(f_{4}, f_{5}) = x_{62}v_{0,0,1,0,1,0}, ψ(f_{3}, f_{4}) = −2x_{62}v_{0,0,2,0,0,0}.

From the conditions

dψ(e_{4}, f_{3}, f_{4}) = 0, dψ(e_{3}, f_{4}, f_{6}) = 0, dψ(e_{1}, f_{4}, f_{5}) = 0,
dψ(e_{1}, f_{4}, f_{6}) = 0

we see that the following components

ψ(e_{1}, f_{4}), ψ(f_{4}, f_{6}), ψ(f_{1}, f_{6}), ψ(f_{3}, f_{4}), ψ(e_{1}, f_{5}), ψ(f_{2}, f_{4}), ψ(e_{3}, f_{6})

are zero. Finally, from the equations

dψ(f_{1}, f_{2}, f_{4}) = 0, dψ(e_{3}, f_{2}, f_{4}) = 0, dψ(e_{1}, e_{2}, f_{4}) = 0,
dψ(e_{1}, e_{3}, f_{5}) = 0, dψ(e_{6}, f_{3}, f_{5}) = 0, dψ(e_{3}, f_{3}, f_{4}) = 0

we have

ψ(f_{1}, f_{4}) = ψ(e_{2}, f_{4}) = ψ(e_{3}, f_{5}) = ψ(f_{1}, f_{3}) = ψ(e_{3}, f_{4}) =
ψ(e_{2}, f_{3}) = 0.

Therefore, any cocycle in Z^{2}(𝔤, L((p − 2)λ_{1} + λ_{2})) with weight pλ_{1} is zero. Thus ker f = 0.

Theorem 1.1 is proved completely.

Acknowledgements. The authors are grateful to prof. C. Löfwall for attention to our results and INTAS foundation and the Royal Swedish Academy of Sciences for support.

References

[1] H.H. Andersen, The strong linkage principle, J. Reine Angew. Math., 315 (1980), 53-59.

[2] H.H. Andersen, J.C. Jantzen, Cohomology of induced representations for algebraic groups, Math. Ann., 269 (1984), 487-525.

[3] N. Bourbaki, Groupes et algèbres de Lie, Chap. 2 et 3, Paris: Hermann, 1972.

[4] B. Braden, Restricted representations of classical Lie algebras of types A_{2} and B_{2}, Bull. Amer. Math. Soc., 73 (1967), 482-485.

[5] Chiu Sen, Cohomology of graded Lie algebras of Cartan type S(n,m), Chin. Ann. Math., 10B(1), 1989, 105-114.

[6] A.S. Dzhumadil’daev, On the defomations of classical Lie algebras, Uspechi Mat. Nauk, 31(1976), No.3, 211-212.

[7] A.S. Dzhumadil’daev, On the cohomology of modular Lie algebras, 119 (161), 1982, n.9, 132-149.

[8] A.S. Dzhumadil’daev, Cohomology and nonsplit extensions of modular Lie algebras, Contemp.Math. 131(1992), part 2, p.31-43.

[9] G. Hochschild, Cohomology of restricted Lie algebras, Amer. J. Math., 76 (1954),555-580.

[10] J.E. Humphreys, Modular representations of classical Lie algebras and semisimple groups, J. Algebra, 19 (1971), 51-79.

[11] J.C. Jantzen, Representations of algebraic groups, Pure and Applied Mathematics, 131, Boston,1987.

[12] J.C. Jantzen, First cohomology groups for classical Lie algebras, Progress in mathematics, 95 (1991), 289-315.

[13] A.I. Kostrikin, Parametrical family of simple Lie algebras, Izv. AN SSSR, Ser. mat., 34 (1970), 744-756.

[14] L.M. Reed, A positive characteristic resolution problem for SL(3, k), J. Algebra, 198 (1997), 57-100.

[15] A.N. Rudakov, Dimensions of some representations of semisimple Lie algebras of classical types over a field of finite characteristic , Trudy Semin. Petrovskogo, No. 3, 1978, 147-160.

[16] J.B. Sullivan, Lie algebra cohomology at irreducible modules, J. Math., 23 (1979), 363-373.

[17] J.B. Sullivan, The second Lie algebra cohomology group and Weyle modules, J. Math., 86 (1980), 321-326.

[18] F.D. Veldkamp, The center of the universal enveloping algebra of a Lie algebra in characteristic p, Ann. Scient. Norm. Sup., 5, 1972, 217-240.

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A.S. Dzhumadil’daev askar@math.kz S. Demirel University, Toraygirov 19, Almaty, Kazakhstan

S.S. Ibraev Institute of Mathematics, Almaty, Kazakhstan