Centralizer construction of the Yangian of the queer Lie superalgebra

Maxim Nazarov^{1} and Alexander Sergeev^{2}

1 Department of Mathematics, University of York, Heslington, York YO10 5DD,

  England; mln1@york.ac.uk

2 Department of Mathematics, Balakovo Institute of Technology, Balakovo 413800,

  Russia; sergeev@bittu.org.ru

To Professor Anthony Joseph on the occasion of his 60th birthday

Summary. Consider the complex matrix Lie superalgebra 𝔤𝔩_{N|N} with the standard generators E_{ij} where i,j = ±1, …, ± N. Define an involutive automorphism η of 𝔤𝔩_{N|N} by η (E_{ij}) = E_{−i,−j}. The queer Lie superalgebra 𝔮_{N} is the fixed point subalgebra in 𝔤𝔩_{N|N} relative to η. Consider the twisted polynomial current Lie superalgebra

𝔤 = { X(t) ∈ 𝔤𝔩_{N|N}[t] : η (X(t)) = X(−t) } .

The enveloping algebra U(𝔤) of the Lie superalgebra 𝔤 has a deformation, called the Yangian of 𝔮_{N}. For each M = 1,2, … denote by A_{N}^{M} the centralizer of 𝔮_{M} ⊂ 𝔮_{N+M} in the associative superalgebra U(𝔮_{N+M}). We construct a sequence of surjective homomorphisms U(𝔮_{N}) ← A_{N}^{1} ← A_{N}^{2} ← …. We describe the inverse limit of the sequence of centralizer algebras A_{N}^{1}, A_{N}^{2}, … in terms of the Yangian of 𝔮_{N}.

1. Main results

In this article we work with the queer Lie superalgebra 𝔮_{N}. This is perhaps the most interesting super-analogue of the general linear Lie algebra 𝔤𝔩_{N}, see for instance [S2]. We will realize 𝔮_{N} as a subalgebra in the general linear Lie superalgebra 𝔤𝔩_{N|N} over the complex field ℂ. Let the indices i,j run through −N , …, −1,1, …, N. Put ī = 0 if i > 0 and ī = 1 if i < 0. Take the ℤ_{2} - graded vector space ℂ^{N|N}. Let e_{i} ∈ ℂ^{N|N} be the standard basis vectors. The ℤ_{2} -gradation on ℂ^{N|N} is defined so that dege_{i} = ī. Let E_{ij} ∈ End(ℂ^{N|N}) be the matrix units: E_{ij} e_{k} = δ_{jk} e_{i}. The algebra End(ℂ^{N|N}) is ℤ_{2} -graded so that degE_{ij} = ī + j̄. We will also regard E_{ij} as basis elements of the Lie superalgebra 𝔤𝔩_{N|N}. The queer Lie superalgebra 𝔮_{N} is the fixed point subalgebra in 𝔤𝔩_{N|N} with respect to the involutive automorphism η defined by

η : E_{ij} ↦ E_{−i,−j} .  (1.1)

Thus as a vector subspace, 𝔮_{N} ⊂ 𝔤𝔩_{N|N} is spanned by the elements

F_{ij} = E_{ij} + E_{−i,−j} .

Note that F_{−i,−j} = F_{ij} . The elements F_{ij} with i > 0 form a basis of 𝔮_{N}.

The vector subspace of End(ℂ^{N|N}) spanned by the elements F_{ij} is closed with respect to the usual matrix multiplication. Hence we can also regard it as an associative algebra. Denote this associative algebra by Q_{N}, to distinguish its structure from that of the Lie superalgebra 𝔮_{N}. Both End(ℂ^{N|N}) and Q_{N} are simple as associative ℤ_{2} -graded algebras, see [J, Theorem 2.6].

The enveloping algebra U(𝔮_{N}) of the Lie superalgebra 𝔮_{N} is a ℤ_{2} -graded associative unital algebra. In this article we will always keep to the following convention. Let A and B be any two associative ℤ_{2} -graded algebras. Their tensor product A ⊗ B is a ℤ_{2} -graded algebra such that for any homogeneous elements X, X′ ∈ A and Y, Y′ ∈ B

(X ⊗ Y)(X′ ⊗ Y′) = (−1)^{deg X′deg Y} X X′ ⊗ Y Y′,  (1.2)
deg (X ⊗ Y) = degX + degY .  (1.3)

By definition, an anti-homomorphism ω : A → B is any linear map which preserves the ℤ_{2} -gradation and satisfies any homogeneous X, X′ ∈ A

ω(X X′) = (−1)^{deg X deg X′} ω(X′) ω(X) .  (1.4)

For any Lie superalgebra 𝔞, the principal anti-automorphism of the enveloping ℤ_{2} -graded algebra U(𝔞) is determined by the assignment X ↦ −X for X ∈ 𝔞.

The supercommutator of any two homogeneous elements X, Y ∈ A is by definition

[X ,Y ] = XY − (−1)^{deg Xdeg Y} Y X.  (1.5)

This definition extends to arbitrary elements X, Y ∈ A by linearity. It is the bracket (1.5) that defines the Lie superalgebra structure on the vector space A = End(ℂ^{N|N}). Thus for any indices i,j ,k ,l = ±1, …, ±N we have

\begin{aligned}[E_{ij} ,E_{kl}] & = δ_{kj} E_{il} − (−1)^{( ī+ j̄) ( k̄ + l̄)} δ_{il} E_{kj} ; \\ [F_{ij} ,F_{kl}] & = δ_{kj} F_{il} − (−1)^{( ī+ j̄) ( k̄ + l̄)} δ_{il} F_{kj} + \end{aligned}

δ_{−k,j} F_{−i,l} − (−1)^{( ī+ j̄) ( k̄ + l̄)} δ_{i,−l} F_{k,−j} .  (1.6)

For any A and any subset C ⊂ A, by the centralizer of C in A we mean the collection of all elements X ∈ A such that [X ,Y ] = 0 for any Y ∈ C. To remind the reader about this convention, we shall then refer to the ℤ_{2} -graded algebra A as a superalgebra. The centre Z(𝔮_{N}) of the enveloping algebra U(𝔮_{N}) will be always taken in the superalgebra sense. A set of generators of the algebra Z(𝔮_{N}) was given in [S1]. In particular, all central elements of U(𝔮_{N}) were shown to have ℤ_{2} -degree 0. A distinguished basis of the vector space Z(𝔮_{N}) was constructed in [N2].

Let us recall the principal results of [S1] here. For any indices n ⩾ 1 and i,j = ±1, …, ±N denote by F_{ij | N}^{(n)} the element of the algebra U(𝔮_{N})

∑_{k_{1},..., k_{n−1}}(−1)^{k̄_{1} + ... + k̄_{n−1}} F_{i k_{1}}F_{k_{1}k_{2}} … F_{k_{n−2}k_{n−1}}F_{k_{n−1}j}  (1.7)

where each of the indices k_{1} , …, k_{n−1} runs through ±1, …, ± N. Note that

F_{−i,−j | N}^{(n)} = (−1)^{n−1} F_{ij | N}^{(n)} .  (1.8)

Of course, here F_{ij | N}^{(1)} = F_{ij}. Observe that if n > 1, then by the definition (1.7)

F_{ij | N}^{(n)} = ∑_{k} (−1)^{k̄} F_{ik} F_{kj | N}^{(n−1)}  (1.9)

where the index k runs through ±1, …, ± N. Using this observation one proves by induction on n = 1,2, … the following generalization of (1.6): in the ℤ_{2} -graded algebra U(𝔮_{N}) the supercommutator

[F_{ij} ,F_{kl | N}^{(n)} ] = δ_{kj} F_{il | N}^{(n)} − (−1)^{( ī+ j̄) ( k̄ + l̄)} δ_{il} F_{kj | N}^{(n)} +
δ_{−k,j} F_{−i,l | N}^{(n)} − (−1)^{( ī+ j̄) ( k̄ + l̄)} δ_{i,−l} F_{k,−j | N}^{(n)} .  (1.10)

For a more general formula, expressing the supercommutator [F_{ij | N}^{(m)} ,F_{kl | N}^{(n)} ] for any m and n, see Proposition 3.1 and the remark after its proof. Now put

C_{N}^{(n)} = ∑_{k} F_{kk | N}^{(n)}  (1.11)

where the index k runs through ±1, …, ± N. The relations (1.10) immediately imply that C_{N}^{(n)} ∈ Z(𝔮_{N}). Note that C_{N}^{(2)} = C_{N}^{(4)} = … = 0 due to (1.8). The following proposition has been stated in [S1] without proof.

Proposition 1.1. The elements C_{N}^{(1)}, C_{N}^{(3)}, … generate the centre Z(𝔮_{N}).

The dependence of the elements C_{N}^{(n)} and F_{ij | N}^{(n)} of U(𝔮_{N}) on the index N has been indicated for the purposes of the next argument, which extends [S1].

For any integers N ⩾ 0 and M ⩾ 1 consider the Lie superalgebra 𝔮_{N+M}. Now let the indices i,j run through −N −M , …, −1,1, …, N +M. Regard the Lie superalgebras 𝔮_{N} and 𝔮_{M} as the subalgebras of 𝔮_{N+M} spanned by the elements F_{ij} = F_{ij | N+M}^{(1)} where |i|,|j| ⩽ N and |i|,|j| > N respectively. Denote by A_{N}^{M} the centralizer of 𝔮_{M} in the associative superalgebra U(𝔮_{N+M}).

By definition, the centralizer A_{N}^{M} contains the centre Z(𝔮_{N+M}) of the U(𝔮_{N+M}). It also contains the subalgebra U(𝔮_{N}) ⊂ U(𝔮_{N+M}). Moreover, the relations (1.10) imply that the centralizer A_{N}^{M} contains the elements

F_{ij | N+M}^{(1)} , F_{ij | N+M}^{(2)} , … where |i|,|j| ⩽ N.  (1.12)

Theorem 1.2. The elements C_{N+M}^{(1)} , C_{N+M}^{(3)} , … and (1.12) generate A_{N}^{M}.

We prove this theorem in Section 2 of the present article. In the particular case N = 0, we will then obtain Proposition 1.1.

Now take the Lie superalgebra 𝔮_{N+M−1} . As a subalgebra of 𝔮_{N+M}, it is spanned by the elements F_{ij} where |i|,|j| < N + M. In particular, the subalgebras 𝔮_{N} and 𝔮_{M−1} of 𝔮_{N+M−1} are spanned by the elements F_{ij} where |i|,|j| ⩽ N and N < |i|,|j| < N + M respectively. The enveloping algebra U(𝔮_{N+M−1}) and its subalgebra A_{N}^{M−1} will be also regarded as subalgebras in the associative algebra U(𝔮_{N+M}). We assume that M ⩾ 1 and A_{N}^{0} = U(𝔮_{N}).

Denote by I_{N+M} the right ideal in the algebra U(𝔮_{N+M}) generated by the elements

F_{N+M, ±1} , …, F_{N+M, ±(N+M)} .  (1.13)

Lemma 1.3. a) the intersection I_{N+M} ∩ A_{N}^{M} is a two-sided ideal of A_{N}^{M} ;

b) there is a decomposition A_{N}^{M} = A_{N}^{M−1} ⊕ ( I_{N+M} ∩ A_{N}^{M}) .

We prove this lemma in Section 3. Using Part (b) of the lemma, denote by α_{M} the projection of A_{N}^{M} to its direct summand A_{N}^{M−1} . Due to Part (a), the map α_{M} : A_{N}^{M} → A_{N}^{M−1} is a homomorphism of associative algebras. The proof of the next proposition will also be given in Section 3.

Proposition 1.4. For any n ⩾ 1 and any i,j such that |i|,|j| ⩽ N we have

α_{M}(F_{ij | N+M}^{(n)}) = F_{ij | N+M−1}^{(n)} and α_{M}(C_{N+M}^{(n)}) = C_{N+M−1}^{(n)} .

The standard filtration (2.9) on the enveloping algebra U(𝔮_{N+M}) defines a filtration on its subalgebra A_{N}^{M}. By definition, the map α_{M} : A_{N}^{M} → A_{N}^{M−1} preserves that filtration. It also preserves the ℤ_{2} -gradation, inherited from U(𝔮_{N+M}). Using the homomorphisms α_{1} ,α_{2} , … define an algebra A_{N} as the inverse limit of the sequence A_{N}^{0} ,A_{N}^{1} ,A_{N}^{2} , … in the category of associative filtered algebras. The main result of this article is an explicit description of the algebra A_{N} in terms of generators and relations.

By definition, an element of A_{N} is any sequence of elements Z_{0} ,Z_{1} ,Z_{2} , … of the algebras A_{N}^{0} ,A_{N}^{1} ,A_{N}^{2} , … respectively, such that α_{M}(Z_{M}) = Z_{M−1} for each M ⩾ 1, and the filtration degrees of the elements in the sequence are bounded from above. Utilising Proposition 1.4, for any n = 1,2, … and any i,j = ±1, …, ±N define an element F_{ij}^{(n)} ∈ A_{N} as the sequence

F_{ij | N}^{(n)} , F_{ij | N+1}^{(n)} , F_{ij | N+2}^{(n)} , ….  (1.14)

Further, for any n = 1,3, … define an element C^{(n)} ∈ A_{N} as the sequence

C_{N}^{(n)} , C_{N+1}^{(n)} , C_{N+2}^{(n)} , ….  (1.15)

The filtration degree of every element in (1.14) and (1.15) does not exceed n.

Note that the algebra A_{N} is unital, and comes with a ℤ_{2} -gradation, such that all possible indices n and i,j we have

deg C^{(n)} = 0 and deg F_{ij}^{(n)} = ī + j̄.  (1.16)

By their definition, the elements C^{(1)}, C^{(3)}, … ∈ A_{N} are central. Due to (1.8)

F_{−i,−j}^{(n)} = (−1)^{n−1} F_{ij}^{(n)}.  (1.17)

Theorem 1.5. a) The algebra A_{N} is generated by the elements C^{(1)}, C^{(3)}, … and F_{ij}^{(1)}, F_{ij}^{(2)}, ….

b) The central elements C^{(1)}, C^{(3)}, … of A_{N} are algebraically independent.

c) Together with the centrality and algebraic independence of C^{(1)}, C^{(3)}, …, the defining relations of the ℤ_{2} -graded algebra A_{N} are (1.17) and

[F_{ij}^{(m)}, F_{kl}^{(n)} ] = F_{il}^{(m+n−1)} δ_{kj} − (−1)^{( ī+ j̄) ( k̄ + l̄)} δ_{il} F_{kj}^{(m+n−1)} +

(−1)^{m − 1} ( F_{−i,l}^{(m+n−1)} δ_{−k,j} − (−1)^{( ī+ j̄) ( k̄ + l̄)} δ_{i,−l} F_{k,−j}^{(m+n−1)} ) +

(−1)^{j̄k̄ + j̄l̄ + k̄ l̄} ∑_{r=1}^{min(m,n)−1}( F_{il}^{(m+n−r−1)} F_{kj}^{(r)} − F_{il}^{(r)} F_{kj}^{(m+n−r−1)} ) +

(−1)^{j̄k̄ + j̄l̄ + k̄ l̄ + k̄ + l̄} ×
∑_{r=1}^{min(m,n)−1}(−1)^{m + r} ( F_{−i,l}^{(m+n−r−1)} F_{−k,j}^{(r)} − F_{i,−l}^{(r)} F_{k,−j}^{(m+n−r−1)} )  (1.18)

where m,n = 1,2,… and i,j ,k ,l = ±1, …, ±N.

The proof will be given in Section 3. In particular, Theorem 1.5 shows that the algebra A_{N} is isomorphic to the tensor product of its two subalgebras, generated by the elements C^{(1)}, C^{(3)}, … and by the elements F_{ij}^{(1)}, F_{ij}^{(2)}, … respectively. Denote the latter subalgebra by B_{N}, it is a ℤ_{2} -graded subalgebra.

The algebra B_{N} appeared in [N3] in the following guise. Let us consider the associative unital ℤ_{2}-graded algebra Y(𝔮_{N}) over the field ℂ with the countable set of generators T_{ij}^{(n)} where n = 1,2, … and i, j = ±1, …, ±N. The ℤ_{2}- gradation on the algebra Y(𝔮_{N}) is determined by setting deg T_{ij}^{(n)} = ī + j̄ for any n ⩾ 1. To write down the defining relations for these generators, put

T_{ij}(x) = δ_{ij} · 1 + T_{ij}^{(1)} x^{−1} + T_{ij}^{(2)} x^{−2} + …

where x is a formal parameter, so that T_{ij}(x) ∈ Y(𝔮_{N})[[x^{−1}]]. Then for all possible indices i,j ,k ,l we have the relations

T_{−i,−j}(x) = T_{ij}(−x)  (1.19)

and

(x^{2} − y^{2} ) · [ T_{ij}(x), T_{kl}(y)] · (−1)^{īk̄ + īl̄ + k̄ l̄} =
(x + y) · (T_{kj}(x) T_{il}(y) − T_{kj}(y) T_{il}(x)) −
(x − y) · (T_{−k, j}(x) T_{−i, l}(y) − T_{k, −j}(y) T_{i, −l}(x)) · (−1)^{k̄ + l̄}  (1.20)

where y is a formal paramater independent of x, so that (1.20) is an equality in the algebra of formal Laurent series in x^{−1}, y^{−1} with coefficients in Y(𝔮_{N}).

The algebra Y(𝔮_{N}) is called the Yangian of the Lie superalgebra 𝔮_{N}. Note that the centre of the associative superalgebra Y(𝔮_{N}) with N ⩾ 1 is not trivial. For a description of the centre of Y(𝔮_{N}) see [N3, Section 3]. In particular, all central elements of Y(𝔮_{N}) have ℤ_{2} -degree 0. In our Section 3 we will prove

Proposition 1.6. The assignement F_{ij}^{(n)} ↦ (−1)^{ī} T_{ji}^{(n)} for any n = 1,2, … and i, j = ±1, …, ±N extents to an anti-isomorphism of ℤ_{2} -graded algebras

ω : B_{N} → Y(𝔮_{N}) .

Now denote by ω_{N+M} the principal anti-automorphism of the enveloping algebra U(𝔮_{N+M}). It preserves the subalgebra U(𝔮_{M}) ⊂ U(𝔮_{N+M}). Hence it also preserves the centralizer A_{N}^{M} ⊂ U(𝔮_{N+M}) of that subalgebra. For any M ⩾ 0 let π_{M} : A_{N} → A_{N}^{M} be the canonical homomorphism. By definition,

π_{M} (F_{ij}^{(n)}) = F_{ij | N+M}^{(n)}  (1.21)

for any n = 1,2, … and i,j = ±1, …, ±N. Using Proposition 1.6, we can define a homomorphism τ_{M} : Y(𝔮_{N}) → A_{N}^{M} by the equality

τ_{M} ∘ ω = ω_{N+M} ∘ (π_{M} |B_{N}) .

By (1.21),

τ_{M} (T_{ij}^{(n)}) = (−1)^{j̄} ω_{N+M} (F_{ji | N+M}^{(n)} ) .

Using the homomorphisms τ_{M} for all M = 0,1,2, …, one can define a family of irreducible finite-dimensional Y(𝔮_{N})-modules; see [N4, Section 1] and [P]. Another family of irreducible finite-dimensional Y(𝔮_{N})-modules can be defined by using the results of [N1] and [N3, Section 5]. It should be possible to give a parametrization of all irreducible finite-dimensional Y(𝔮_{N})-modules, similarly to the parametrization of the irreducible finite-dimensional Y(𝔤𝔩_{N})- modules as given by V. Drinfeld; see [D2, Theorem 2] and [M].

It was shown in [N3] that the associative ℤ_{2} -graded algebra Y(𝔮_{N}) has a natural Hopf superalgebra structure. In particular, the homomorphism of comultiplication Y(𝔮_{N}) → Y(𝔮_{N}) ⊗ Y(𝔮_{N}) can be defined by

T_{ij}(x) ↦ ∑_{k} T_{ik}(x) ⊗ T_{kj}(x) · (−1)^{( ī+ k̄)( j̄+ k̄)}  (1.22)

where the tensor product is over the subalgebra ℂ[[x^{−1}]] of Y(𝔮_{N})[[x^{−1}]], and k runs through ±1, …, ±N. See [N3, Section 2] for the definitions of the the counit map Y(𝔮_{N}) → ℂ and the antipodal map Y(𝔮_{N}) → Y(𝔮_{N}).

There is a distinguished ascending ℤ-filtration on the associative algebra Y(𝔮_{N}). It is obtained by assigning to every generator F_{ij}^{(n)} the degree n − 1. The corresponding ℤ-graded algebra will be denoted by gr Y(𝔮_{N}). Let G_{ij}^{(n)} be the element of grY(𝔮_{N}) corresponding to the generator F_{ij}^{(n)} ∈ Y(𝔮_{N}). The algebra gr Y(𝔮_{N}) inherits the ℤ_{2} -gradation from the algebra Y(𝔮_{N}), so that

deg G_{ij}^{(n)} = ī + j̄.

Moreover, gr Y(𝔮_{N}) inherits from Y(𝔮_{N}) the Hopf superalgebra structure. It follows from the defintion (1.22) that with respect to the homomorphism of comultiplication gr Y(𝔮_{N}) → gr Y(𝔮_{N}) ⊗ gr Y(𝔮_{N}), for any n ⩾ 1 we have

G_{ij}^{(n)} ↦ G_{ij}^{(n)} ⊗ 1 + 1 ⊗ G_{ij}^{(n)}.

On the other hand, for arbitrary Lie superalgebra 𝔞, a comultiplication map U(𝔞) → U(𝔞) ⊗ U(𝔞) can be defined for X ∈ 𝔞 by X ↦ X ⊗ 1 + 1 ⊗ X, and then extended to a homomorphism of ℤ_{2} -graded associative algebras by using the convention (1.2). Let us now consider the enveloping algebra U(𝔤) of the twisted polynomial current Lie superalgebra

𝔤 = { X(t) ∈ 𝔤𝔩_{N|N}[t] : η (X(t)) = X(−t) } .

Here we employ the automorphism (1.1) of the Lie superalgebra 𝔤𝔩_{N|N}. As a vector space, 𝔤 is spanned by the elements

E_{ij} t^{n} + E_{−i,−j} (−t)^{n}  (1.23)

where n = 0,1,2 … and i,j = ±1, …, ±N. Note that the ℤ_{2} -degree of the element (1.23) equals ī+ j̄. The algebra U(𝔤) also has a natural ℤ-gradation, such that the degree of the element (1.23) is n.

It turns out that U(𝔤) and gr Y(𝔮_{N}) are isomorphic as Hopf superalgebras. By [N3, Theorem 2.3] their isomorphism U(𝔤) → gr Y(𝔮_{N}) can be defined by mapping the element (1.23) of the algebra U(𝔤) to the element (−1)^{ī+1} G_{ji}^{(n+1)} of the algebra gr Y(𝔮_{N}). Moreover, Y(𝔮_{N}) is a deformation of U(𝔤) as a Hopf superalgebra; see the end of [N3, Section 2] for details.

Let us finish this introductory section with a few remarks of a historical nature. Our construction of the algebra A_{N} follows a similar construction for the general linear Lie algebra 𝔤𝔩_{N} instead of the queer Lie superalgebra 𝔮_{N}, due to G. Olshanski [O1, O2]. It was him who first considered the inverse limit of the sequence of centralizers of 𝔤𝔩_{M} in the enveloping algebras U(𝔤𝔩_{N+M}) for M = 1,2, …. Following a suggestion of B. Feigin, he then described the inverse limit in terms of the Yangian Y(𝔤𝔩_{N}) of the Lie algebra 𝔤𝔩_{N}. The latter Yangian is a deformation of the enveloping algebra of the polynomial current Lie algebra 𝔤𝔩_{N}[t] in the class of Hopf algebras [D1].

The elements (1.7) of U(𝔮_{N}) were initially considered by A. Sergeev [S1], in order to describe the centre of the superalgebra U(𝔮_{N}). The homomorphisms α_{M} : A_{N}^{M} → A_{N}^{M−1} for M = 1,2, … and the elements F_{ij}^{(1)}, F_{ij}^{(2)}, … of the inverse limit algebra A_{N} were introduced by M. Nazarov following [O1, O2]. He then identified the algebra defined by the relations (1.19) and (1.20), as a deformation of the enveloping algebra U(𝔤) in the class of Hopf superalgebras [N3]. It was also explained in [N3] why this deformation should be called the Yangian of 𝔮_{N}. However, our Theorems 1.2 and 1.5 were only conjectured by M. Nazarov. The purpose of the present article is to prove these conjectures.

We hope these remarks indicate importance of the role that G. Olshanski played at various stages of our work. We are very grateful to him for friendly advice. This work was finished while M. Nazarov stayed at the Max Planck Institute of Mathematics in Bonn. He is grateful to the Institute for hospitality. M. Nazarov has been also supported by the EC grant MRTN-CT-2003-505078.

2. Proof of Theorem 1.2

We will use basic properties of complex semisimple associative superalgebras and their modules [J]. Most of these properties were first established in [W], in a generality greater than we need in the present article. We will also use the following simple lemma. Its proof carries over almost verbatim from the ungraded case, but we shall include the proof for the sake of completeness. Let A be any finite dimensional ℤ_{2} -graded associative algebra over the complex field ℂ. Let G be a finite group of automorphisms of A. The crossed product algebra G ⋉ A is also ℤ_{2} -graded: for any g ∈ G we have degg = 0 in G ⋉ A.

Lemma 2.1. Suppose the superalgebra A is semisimple. Then the superalgebra G ⋉ A is also semisimple.

Proof. We will write G ⋉ A = B. Let V be any module over the superalgebra B, and let ρ : B → End(V) be the corresponding homomorphism. Here we assume that the homomorphism ρ preserves ℤ_{2} -gradation. Let U ⊂ V be any B-submodule. Since A is semisimple, we have the decomposition V = U ⊕ U′ into a direct sum of A-modules for some U′ ⊂ V. Let P ∈ End(V) be the projection onto U along U′. Put

S = \frac{1}{|G|} ∑_{g∈G} ρ(g)P ρ(g)^{−1} ∈ End(V) .

For any element X ∈ A let X^{g} be its image under the automorphism g. Then

ρ(X)S = \frac{1}{|G|} ∑_{g∈G} ρ(X)ρ(g)P ρ(g)^{−1} = \frac{1}{|G|} ∑_{g∈G} ρ(g)ρ(X^{g})P ρ(g)^{−1}

= \frac{1}{|G|} ∑_{g∈G} ρ(g)P ρ(X^{g})ρ(g)^{−1} = \frac{1}{|G|} ∑_{g∈G} ρ(g)P ρ(g)^{−1} ρ(X) = S ρ(X) .

For any h ∈ G we also have ρ(h)S = S ρ(h) by the definition of S. Note that S ∈ End(V) is of ℤ_{2} -degree 0, as well as P is. So ker S ⊂ V is a B-submodule.

Since U is a B-submodule, we have the equalities P ρ(g)P = ρ(g)P for all g ∈ G. Using the definition of S, these equalities imply that S P = P and P S = S. The latter pair of equalities guarantees that im S = im P = U and S^{2} = S. So V = U ⊕ ker S. Since V is an arbitrary module over the superalgebra B, this superalgebra is semisimple by [J,Proposition 2.4] ⊓ ⊔

We will also need the following general “double centralizer theorem”. Let V be any ℤ_{2} -graded complex vector space. The associative algebra End(V) is then also ℤ_{2} -graded. Take any subalgebra B in the superalgebra End(V). Here we assume that B as a vector space splits into the direct sum of its subspaces of ℤ_{2} -degrees 0 and 1. Denote by B′ the centralizer of B in the superalgebra End(V).

Proposition 2.2. Suppose that the superalgebra B is finite dimensional and semisimple. Also suppose that the B′ -module V is finitely generated. Then B = B′′ in End(V).

Proof. We shall prove that for any homogeneous v_{1} , …, v_{n} ∈ V and X ∈ B′′, there exists Y ∈ B such that Xv_{r} = Y v_{r} for any index r = 1, …, n. Then we will choose v_{1} , …, v_{n} to be homogeneous generators of V over B′. By writing any vector v ∈ V as the sum Z_{1}v_{1} + … + Z_{n}v_{n} for some homogeneous Z_{1} , …, Z_{n} ∈ B′, we will get

Xv = ∑_{r=1}^{n} XZ_{r} v_{r} = ∑_{r=1}^{n} (−1)^{deg X deg Z_{r}} Z_{r} X v_{r}

= ∑_{r=1}^{n} (−1)^{deg X deg Z_{r}} Z_{r} Y v_{r} = ∑_{r=1}^{n} Y Z_{r} v_{r} = Y v .

Along with the obvious embedding B ⊂ B′′, this will prove Proposition 2.2.

Recall that the ℤ_{2} -graded vector space \overline{V} opposite to V is obtained from V by changing the ℤ_{2} -gradation deg to deg+1. Define the action of the algebra B in \overline{V} as the pullback its action in V via the involutive automorphism Y ↦ (−1)^{deg Y} Y, where Y is any homogeneous element of the ℤ_{2} -graded algebra B. Now consider the direct sum of B-modules

W = ⨁_{r=1}^{n} V_{r} ,

where the B-module V_{r} equals V or \overline{V} depending on whether degv_{r} in V is 0 or 1. For each index r = 1, …, n we have an embedding of vector spaces A_{r} : V → W, and a projection B_{r} : W → V. Note that the ℤ_{2} -degrees of the linear maps A_{r} and B_{r} coincide with that of the vector v_{r}. We also have the equality B_{p} A_{q} = δ_{pq} · id in End(V), and the equality

∑_{r=1}^{n} A_{r} B_{r} = id

in End(W) . Any homogeneous element Y ∈ B acts in W as the linear operator

Ỹ = ∑_{r=1}^{n} (−1)^{deg v_{r} deg Y} A_{r} Y B_{r} .

Given X ∈ B′′, put

X̃ = ∑_{r=1}^{n} (−1)^{deg v_{r} deg X} A_{r} X B_{r} .

Any homogeneous element Z ∈ End(W) can be written as

Z = ∑_{p,q=1}^{n} A_{p} Z_{pq} B_{q}

for some homogeneous Z_{pq} ∈ End(V), where degZ_{pq} = degZ +degv_{p}+degv_{q}. Then

ỸZ = ∑_{r=1}^{n} (−1)^{deg v_{r} deg Y} A_{r} Y B_{r} · ∑_{p,q=1}^{n} A_{p} Z_{pq} B_{q}  (2.1)

= ∑_{p,q=1}^{n}(−1)^{deg v_{p} deg Y} A_{p} Y Z_{pq} B_{q} ,

Z Ỹ = ∑_{p,q=1}^{n} A_{p} Z_{pq} B_{q} · ∑_{r=1}^{n} (−1)^{deg v_{r} deg Y} A_{r} Y B_{r}  (2.2)

= ∑_{p,q=1}^{n}(−1)^{deg v_{q} deg Y} A_{p} Z_{pq} Y B_{q} .

Suppose that the element Z ∈ End(W) belongs to the centralizer of the image of the superalgebra B in End(W). Due to (2.1) and (2.2), the assumption [Ỹ, Z] = 0 is equivalent to the collection of equalities [Y, Z_{pq}] = 0 for all p,q = 1, …, n. Since here Y ∈ B is arbitrary and X ∈ B′′, we then have [X ,Z_{pq}] = 0 for all p,q = 1, …, n. A calculation similar to (2.1) and (2.2) then shows that [X̃ ,Z] = 0 in End(W).

Now take the vector

w = ∑_{r=1}^{n} A_{r} v_{r} ∈ W ,

it has ℤ_{2} -degree 0. Let U be the cyclic span of the vector w under the action of B. Since the superalgebra B is finite dimensional semisimple, we have the decomposition W = U ⊕ U′ into direct sum of B-modules for some U′ ⊂ W; see [J,Proposition 2.4]. Choose the element Z ∈ End(W) from the centralizer of the image of B, to be the projector onto U′ along U. Here degZ = 0. Then ZX̃w = X̃Zw = 0, and X̃w ∈ U. So there exists Y ∈ B such that X̃w = Ỹw. The last equality means that Xv_{r} = Y v_{r} for each r = 1, …, n. ⊓ ⊔

In the notation of Proposition 2.2, we have the following corollary.

Corollary 2.3. Suppose that the vector space V is finite dimensional, and that the superalgebra B ⊂ End(V) is semisimple. Then B = B′′ in End(V).

Now for any integer n ⩾ 1 consider the tensor product (End(V))^{⊗ n} of n copies of the ℤ_{2} -graded algebra End(V). This tensor product is a ℤ_{2} -graded associative algebra defined using the conventions (1.2) and (1.3). The proof of the following lemma is also included for the sake of completeness.

Lemma 2.4. Suppose that B contains the identity 1 ∈ End(V). Then the centralizer of B^{⊗ n} in the superalgebra (End(V))^{⊗ n} coincides with (B′)^{⊗ n}.

Proof. We will use the induction on n. If n = 1, the statement of Lemma 2.4 is tautological. Suppose that n > 1. The centralizer of B^{⊗ n} contains (B′)^{⊗ n} due to the conventions (1.2) and (1.3). Now suppose that for some homogeneous elements X_{1} , …, X_{l} ∈ End(V) and Y_{1} , …, Y_{l} ∈ (End(V))^{⊗ (n−1)} the sum X_{1} Y_{1} + … + X_{l} Y_{l} belongs to the centralizer of B^{⊗ n}. In particular, then for any homogeneous Y ∈ B^{⊗ (n−1)} we have

∑_{k=1}^{l} X_{k} ⊗ (Y_{k} Y) = ∑_{k=1}^{l} (X_{k} ⊗ Y_{k})(1 ⊗ Y)

= ∑_{k=1}^{l} (−1)^{(deg X_{k} + deg Y_{k}) deg Y} (1 ⊗ Y)(X_{k} ⊗ Y_{k})

= ∑_{k=1}^{l} (−1)^{deg Y_{k} deg Y} X_{k} ⊗ (Y Y_{k}) .

We may assume that the elements X_{1} , …, X_{l} are linearly independent, then the above equalities imply that for any k the element Y_{k} ∈ (End(V))^{⊗ (n−1)} belongs to the centralizer of B^{⊗ (n−1)}. Then Y_{k} ∈ (B′)^{⊗ (n−1)} by the induction assumption. A similar argument shows that X_{k} ∈ B′ for any index k. ⊓ ⊔

For any N ⩾ 0 and M ⩾ 1 take the ℤ_{2} -graded vector space ℂ^{N+M | N+M}. Identify the ℤ_{2} -graded vector spaces ℂ^{N|N} and ℂ^{M|M} with the subspaces in ℂ^{N+M | N+M} spanned by the vectors e_{i} where i = −N , …, −1,1, …, N and i = −N−M , …, −N− 1,N + 1, …, N + M respectively. The decomposition

ℂ^{N|N} = ℂ^{N|N} ⊕ ℂ^{M|M}  (2.3)

determines the embeddings of the Lie superalgebras 𝔤𝔩_{N|N} and 𝔤𝔩_{M|M} into 𝔤𝔩_{N+M | N+M}, and of their subalgebras 𝔮_{N} and 𝔮_{M} into the Lie algebra 𝔮_{N+M}.

Let us now regard End(ℂ^{N|N}) and Q_{M} as subalgebras in the associative superalgebra End(ℂ^{N+M | N+M}), using the decomposition (2.3). The elements

∑_{|i|>N} E_{ii} and ∑_{|i|>N}(−1)^{ī} E_{i,−i}

of End(ℂ^{N+M | N+M}) span a subalgebra, isomorphic to the associative algebra Q_{1}. Using this isomorphism, a direct calculation shows that the centralizer of Q_{M} in End(ℂ^{N+M | N+M}) coincides with End(ℂ^{N|N}) ⊕ Q_{1} . This centralizer is a semisimple associative ℤ_{2} -graded algebra, denote it by C.

For any integer n ⩾ 1 consider the tensor product V = (ℂ^{N+M | N+M})^{⊗ n} of n copies of the ℤ_{2} -graded vector space ℂ^{N+M | N+M}. Identify the algebras

(End(ℂ^{N+M | N+M}))^{⊗ n} and End((ℂ^{N+M | N+M})^{⊗ n})  (2.4)

so that for any homogeneous elements X_{1} , …, X_{n} ∈ End(ℂ^{N+M | N+M}) and any homogeneous vectors u_{1} , …, u_{n} ∈ ℂ^{N+M | N+M}

(X_{1} ⊗ … ⊗ X_{n}) (u_{1} ⊗ … ⊗ u_{n}) = (−1)^{d} X_{1}u_{1} ⊗ … ⊗ X_{n}u_{n}

where

d = ∑_{1⩽p<q⩽n}deg u_{p} deg X_{q} .

By identifying the two algebras we determine an action on V of the subalgebra

C^{⊗ n} ⊂ (End(ℂ^{N+M | N+M}))^{⊗ n}.  (2.5)

The symmetric group S_{n} acts on V so that for any adjacent transposition σ_{p} = (p, p + 1)

σ_{p} (u_{1} ⊗ … ⊗ u_{p} ⊗ u_{p+1} ⊗ … ⊗ u_{n}) =
(−1)^{deg u_{p} deg u_{p+1}} u_{1} ⊗ … ⊗ u_{p+1} ⊗ u_{p} ⊗ … ⊗ u_{n} .

The action of S_{n} and C^{⊗ n} on V extends to that of S_{n} ⋉ C^{⊗ n}. Here the group S_{n} acts by automorphisms of the algebra (End(ℂ^{N+M | N+M}))^{⊗ n} so that

σ_{p} (X_{1} ⊗ … ⊗ X_{p} ⊗ X_{p+1} ⊗ … ⊗ X_{n}) =
(−1)^{deg X_{p} deg X_{p+1}} X_{1} ⊗ … ⊗ X_{p+1} ⊗ X_{p} ⊗ … ⊗ X_{n} .  (2.6)

Now consider the action of enveloping algebra U(𝔮_{N}) on the vector space V, as of a subalgebra of the ℤ_{2} -graded associative algebra U(𝔤𝔩_{N+M | N+M}). We use the comultiplication

U(𝔤𝔩_{N+M | N+M}) → U(𝔤𝔩_{N+M | N+M})^{⊗ n}  (2.7)

along with the identification of the two algebras in (2.4).

Proposition 2.5. The centralizer of the image of U(𝔮_{M}) in the associative superalgebra End(V) coincides with the image of S_{n} ⋉ C^{⊗ n}.

Proof. The ℤ_{2} -graded algebra C^{⊗ n} is semisimple, see [J, Proposition 2.10]. So is the crossed product S_{n} ⋉ C^{⊗ n}, see our Lemma 2.1. Let B be the image of the crossed product in End(V). The ℤ_{2} -graded algebra B is semisimple too. By Corollary 2.3 it suffices to prove that the centralizer of B in End(V) coincides with the image of U(𝔮_{M}).

Let us describe the latter image. Consider the span in End(ℂ^{N+M | N+M}) of 𝔮_{M} and of the identity element 1. It is a subalgebra in End(ℂ^{N+M | N+M}), denote this subalgebra by D. We will prove that the invariant subalgebra

(D^{⊗ n})^{S_{n}} ⊂ (End(ℂ^{N+M | N+M}))^{⊗ n}

coincides with the image of U(𝔮_{M}). Here we use the definition (2.6), and the comultiplication (2.7). There is no need to identify the two algebras (2.4) here.

Due to the convention (1.2) and the definition of the comultiplication (2.7), the image of U(𝔮_{M}) in (End(ℂ^{N+M | N+M}))^{⊗ n} is contained in (D^{⊗ n})^{S_{n}}. Now for any n elements X_{1} , …, X_{n} ∈ D consider their symmetrized tensor product

⟨X_{1} , …, X_{n}⟩ = \frac{1}{n!} ∑_{σ∈S_{n}} σ (X_{1} ⊗ … ⊗ X_{n}) .  (2.8)

Suppose that for some p ∈ {1, …, n} we have X_{q} ∈ ℂ 1 if and only if p < q. By induction on p, let us prove that ⟨X_{1} , …, X_{p}, 1, …, 1⟩ belongs to the image of U(𝔮_{M}) in (End(ℂ^{N+M | N+M}))^{⊗ n}. This is evident if p = 1. If p > 1, then

⟨X_{1} , …, X_{p−1}, 1, …, 1⟩ ⟨X_{p}, 1, …, 1⟩

equals

\frac{n − p + 1}{n} ⟨X_{1} , …, X_{p}, 1, …, 1⟩

plus a sum of certain symmetrized tensor products in (End(ℂ^{N+M | N+M}))^{⊗ n} which belong to the image of U(𝔮_{M}) by the induction assumption.

To prove that the centralizer of B in End(V) coincides with the image of U(𝔮_{M}), it now suffices to show that the centralizer of the subalgebra (2.5) coincides with D^{⊗ n}. But we have C′ = D in End(ℂ^{N+M | N+M}) by definition. Using Lemma 2.4 we now complete the proof of Proposition 2.5. ⊓ ⊔

Consider the symmetric algebra S(𝔮_{N+M}) of the Lie superalgebra 𝔮_{N+M}. The standard filtration

ℂ = U^{0}(𝔮_{N+M}) ⊂ U^{1}(𝔮_{N+M}) ⊂ U^{2}(𝔮_{N+M}) ⊂ …  (2.9)

on the algebra U(𝔮_{N+M}) determines for each n = 1, 2, … a linear map

U^{n}(𝔮_{N+M}) / U^{n−1}(𝔮_{N+M}) → S^{n}(𝔮_{N+M}) ,  (2.10)

which is bijective by the Poincaré -Birkhoff -Witt theorem for Lie superalgebras [MM, Theorem 5.15]. Using (2.10), for all indices i, j = ±1, …, ±(N + M) define an element f_{ij | N+M}^{(n)} ∈ S^{n}(𝔮_{N+M}) as the image of the element

F_{ij | N+M}^{(n)} ∈ U^{n}(𝔮_{N+M}) ;

the latter element is the sum (1.7) where each of the indices k_{1} , …, k_{n−1} runs through ±1, …, ±(N + M). Define c_{N+M}^{(n)} ∈ S^{n}(𝔮_{N+M}) as the image of

C_{N+M}^{(n)} ∈ U^{n}(𝔮_{N+M}) ;

see (1.11). Note that if n is even, then C_{N+M}^{(n)} = 0 and hence c_{N+M}^{(n)} = 0.

Now consider adjoint action of the Lie superalgebra 𝔮_{N+M} on S(𝔮_{N+M}). In particular, consider the action of 𝔮_{M} on S(𝔮_{N+M}) as that of a subalgebra of 𝔮_{N+M}. Then take the subalgebra of invariants S(𝔮_{N+M})^{𝔮_{M}} ⊂ S(𝔮_{N+M}).

Proposition 2.6. The subalgebra S(𝔮_{N+M})^{𝔮_{M}} is generated by the elements c_{N+M}^{(1)} , c_{N+M}^{(3)} , … and f_{ij | N+M}^{(1)} , f_{ij | N+M}^{(2)} , … where |i|,|j| ⩽ N.

Proof. Consider (End(ℂ^{N+M | N+M}))^{⊗ n} as a vector space. Define a linear map φ_{n} from this space to the nth symmetric power S^{n}(𝔮_{N+M}) by setting

φ_{n} (E_{i_{1}j_{1}} ⊗ … ⊗ E_{i_{n}j_{n}}) = F_{i_{1}j_{1}} … F_{i_{n}j_{n}}  (2.11)

for any indices i, j = ±1, …, ±(N + M). The map φ_{n} commutes with the natural action of the Lie superalgebra 𝔮_{N+M} on (End(ℂ^{N+M | N+M}))^{⊗ n} and S^{n}(𝔮_{N+M}). The map φ_{n} has a right inverse linear map ψ_{n} which commutes with the action of 𝔮_{N+M} as well. Namely, using (2.8) we set

ψ_{n} (F_{i_{1}j_{1}} … F_{i_{n}j_{n}}) = 2^{−n} ⟨F_{i_{1}j_{1}} , …, F_{i_{n}j_{n}}⟩ .

It follows that the subspace of 𝔮_{M} -invariants of the ℤ-degree n in S(𝔮_{N+M}),

S^{n}(𝔮_{N+M})^{𝔮_{M}} = φ_{n} (((End(ℂ^{N+M | N+M}))^{⊗ n})^{𝔮_{M}}) = φ_{n}(B)

by Proposition 2.5; here B is the image in of the crossed product S_{n} ⋉C^{⊗ n} in

(End(ℂ^{N+M | N+M}))^{⊗ n} = End((ℂ^{N+M | N+M})^{⊗ n}) .

The vector subspace C ⊂ End(ℂ^{N+M | N+M}) is spanned by the identity element 1, the elements E_{ij} where |i|,|j| ⩽ N and by the element

J = ∑_{i} (−1)^{ī} E_{i,−i}

where the summation index i runs through ±1, …, ±(N + M). For each p = 1, …, n introduce the elements of the algebra (End(ℂ^{N+M | N+M}))^{⊗ n},

E_{ij}^{(p)} = 1^{⊗ (p−1)} ⊗ E_{ij} ⊗ 1^{⊗ (n−p)} and J_{p} = 1^{⊗ (p−1)} ⊗ J ⊗ 1^{⊗ (n−p)} .

The vector subspace B ⊂ (End(ℂ^{N+M | N+M}))^{⊗ n} is spanned by the products of the form

J_{q_{1}} … J_{q_{b}} E_{i_{1}j_{1}}^{(p_{1})} … E_{i_{a}j_{a}}^{(p_{a})} H  (2.12)

where

1 ⩽ p_{1} < … < p_{a} ⩽ n, 1 ⩽ q_{1} < … < q_{b} ⩽ n,
{p_{1} , …, p_{a}} ∩ {q_{1} , …, q_{b}} = ∅,
|i_{1}|,|j_{1}|, …, |i_{a}|,|j_{a}| ⩽ N

and H is the image in (End(ℂ^{N+M | N+M}))^{⊗ n} of some permutation from S_{n}.

By the definintion of the symmetric algebra S(𝔮_{N+M}) and due to (2.6), we have the identity φ_{n} ∘ σ = φ_{n} for any σ ∈ S_{n}. Therefore it suffices to compute the φ_{n} -image of the element (2.12) where the factor H corresponds to a permutation of the form

(1, 2, …, r_{1}) (r_{1} + 1, r_{1} + 2, …, r_{2}) … (r_{c} + 1, r_{c} + 2, …, n)

where c ⩾ 0 and 0 < r_{1} < r_{2} < … < r_{c} < n. But for any p = 1, …, n − 1 we have the relation

φ_{p}(X) φ_{n−p}(Y) = φ_{n}(X ⊗ Y) ,  (2.13)

X ∈ (End(ℂ^{N+M | N+M}))^{⊗ p} and Y ∈ (End(ℂ^{N+M | N+M}))^{⊗ (n−p)} .

Hence it suffices to consider only the case where H corresponds to the cyclic permutation (1, 2, …, n). Suppose this is the case. Then we may assume that p_{1} = 1 or a = 0. Here we use the identity φ_{n} ∘ σ = φ_{n} for σ = (1, 2, …, n).

For the cyclic permutation (1, 2, …, n) we have

H = ∑_{k_{1},...,k_{n}}(−1)^{k̄_{1} + ... + k̄_{n−1}} E_{k_{n}k_{1}}^{(1)}E_{k_{1}k_{2}}^{(2)} … E_{k_{n−1}k_{n}}^{(n)}  (2.14)

where each of the indices k_{1} , …, k_{n} runs through ±1, …, ±(N +M). Hence

E_{ij}^{(1)}H = ∑_{k_{1},...,k_{n−1}} (−1)^{k̄_{1} + ... + k̄_{n−1}}E_{k_{n}k_{1}}^{(1)}E_{k_{1}k_{2}}^{(2)} … E_{k_{n−1}k_{n}}^{(n)} .  (2.15)

Further, for any p = 1, …, n − 1 we have the equality

E_{ij}^{(1)}E_{kl}^{(p+1)}H = (−1)^{j̄k̄ + j̄l̄+ k̄ l̄}E_{il}^{(1)}E_{kj}^{(p+1)}H′ H′′

where the factors H′ and H′′ are the images in (End(ℂ^{N+M | N+M}))^{⊗ n} of the cyclic permutations (1, …, p) and (p + 1, …, n). Using this equality together with the relation (2.13), we reduce the case a > 1 to the case a = 1.

Suppose that a = 1 and p_{1} = 1. In this case, the φ_{n} -image of (2.12) equals

∑_{k_{1},...,k_{n−1}}(−1)^{k̄_{1} + ... + k̄_{n−1} + b ī+ q_{1} + ... + q_{b}} F_{(−1)^{b} i,k_{1}}F_{k_{1}k_{2}} … F_{k_{n−1}j}

= (−1)^{b ī+ q_{1} + ... + q_{b}} f_{(−1)^{b} i , j | N+M}^{(n)} ;

we use the equality (2.15), the definition (2.11) and the identity F_{−k,−l} = F_{kl}. It remains to consider the case a = 0. Then the φ_{n} -image of (2.12) equals

(−1)^{q_{1} + ... + q_{b}} ∑_{k_{1},...,k_{n}}(−1)^{k̄_{1} + ... + k̄_{n−1} + b k̄_{n}} F_{(−1)^{b} k_{n},k_{1}}F_{k_{1}k_{2}} … F_{k_{n−1}k_{n}} ;

we use the equality (2.14), the definition (2.11) and the identity F_{−k,−l} = F_{kl}. Denote by f the sum over k_{1} , …, k_{n} in the above display. If the number b is even, then f = c_{N+M}^{(n)} . Now suppose that the number b is odd, so that f is

∑_{k_{1},...,k_{n}}(−1)^{k̄_{1} + ... + k̄_{n−1} + k̄_{n}} F_{−k_{n},k_{1}}F_{k_{1}k_{2}} … F_{k_{n−1}k_{n}} .  (2.16)

By changing the signs of the indices k_{1} , …, k_{n} in (2.16) and then using the identity F_{−k,−l} = F_{kl} one shows that f = (−1)^{n}f. So f = 0 if n is odd. Since

deg F_{−k_{n},k_{1}} = k̄_{1} + k̄_{n} + 1 = deg (F_{k_{1}k_{2}} … F_{k_{n−1}k_{n}}) + 1,

the element (2.16) of the symmetric algebra S(𝔮_{N+M}) can be also written as

∑_{k_{1},...,k_{n}}(−1)^{k̄_{1} + ... + k̄_{n−1} + k̄_{n}} F_{k_{1}k_{2}} … F_{k_{n−1}k_{n}} F_{−k_{n},k_{1}} .  (2.17)

By changing the signs of the indices k_{2} , …, k_{n} in (2.17) and then using the identity F_{−k,−l} = F_{kl} one shows that f = (−1)^{n−1}f. So f = 0 if n is even. ⊓ ⊔

By using the fact that for each n ⩾ 1 the linear map (2.10) commutes with the adjoint action of the Lie superalgebra 𝔮_{N+M} on U^{n}(𝔮_{N+M}) and S^{n}(𝔮_{N+M}), we can now complete the proof of Theorem 1.2. Take any element X ∈ A_{N}^{M}. We have X ∈ U^{n}(𝔮_{N+M}) for some n ⩾ 0. Let us demonstrate by induction on n that X belongs to the subalgebra in U(𝔮_{N+M}) generated by the elements C_{N+M}^{(1)} , C_{N+M}^{(3)} , … and (1.12). Since U^{0}(𝔮_{N+M}) = ℂ, here we can take n ⩾ 1 and make the induction assumption. By Proposition 2.6, the image of X in S^{n}(𝔮_{N+M}) under the map (2.10) is a linear combination of the products of the elements c_{N+M}^{(1)} , c_{N+M}^{(3)} , … and f_{ij | N+M}^{(1)} , f_{ij | N+M}^{(2)} , … where |i|,|j| ⩽ N. By replacing these elements of S(𝔮_{N+M}) respectively by C_{N+M}^{(1)} , C_{N+M}^{(3)} , … and (1.12) in the linear combination, we obtain a certain element Y ∈ U(𝔮_{N+M}) such that X − Y ∈ U^{(n−1)}(𝔮_{N+M}). We also have Y ∈ A_{N}^{M}. By applying the induction assumption to the difference X − Y, we complete the the proof.

3. Proof of Theorem 1.5

Here we prove Theorem 1.5 along with Lemma 1.3 and Propositions 1.4 ,1.6.

Proof of Lemma 1.3. Together with the right ideal I_{N+M} generated by the elements (1.13), consider the left ideal J_{N+M} in U(𝔮_{N+M}) generated by the elements

F_{±1, N+M} , …, F_{±(N+M), N+M} .

By the Poincaré -Birkhoff -Witt theorem for Lie superalgebras, every element X ∈ U(𝔮_{N+M}) can be uniquely written as a sum of the products of the form

F_{N+M, 1}^{p_{1}} F_{N+M, −1}^{p_{−1}} … F_{N+M, N+M−1}^{p_{N+M−1}} F_{N+M, −N−M + 1}^{p_{−N−M + 1}} ×
F_{N+M, N+M}^{p_{N+M}} F_{N+M, −N−M}^{p_{−N−M}} Y ×
F_{1, N+M}^{q_{1}} F_{−1, N+M}^{q_{−1}} … F_{N+M−1, N+M}^{q_{N+M − 1}} F_{−N−M + 1, N+M}^{q_{−N−M + 1}}  (3.1)

where each of the exponents p_{k} and q_{k} runs through 0,1,2, … or through 0,1 if k > 0 or k < 0 respectively, whereas the factor Y ∈ U(𝔮_{N+M−1}) depends on these exponents. Note that here

[ F_{N+M, N+M} , Y ] = 0 and [ F_{−N−M, N+M} , Y ] = 0 .  (3.2)

Now suppose that X ∈ A_{N}^{M}. In particular, then we have

[ F_{N+M, N+M} , X] = 0  (3.3)

since M ⩾ 1 by our assumption. Note that if |k | < N + M, then due to (1.6)

\begin{matrix}[ F_{N+M, N+M} , F_{N+M, k}] = F_{N+M,k} , \\ [ F_{N+M, N+M} , F_{k, N+M}] = − F_{N+M,k} . \end{matrix}

If |k | = N + M, then

[ F_{N+M, N+M} , F_{k, N+M}] = 0 .

Hence the condition (3.3) implies that X is a sum of the products (3.1) where

p_{1} + p_{−1} + … + p_{N+M−1} + p_{−N−M + 1} =
q_{1} + q_{−1} + … + q_{N+M−1} + q_{−N−M + 1} .  (3.4)

The intersection I_{N+M} ∩ A_{N}^{M} consists of those elements X ∈ A_{N}^{M} which are sums of the products (3.1) where

p_{1} + p_{−1} + … + p_{N+M} + p_{−N−M} > 0 .

Due to the equality (3.4), the latter inequality is equivalent to

q_{1} + q_{−1} + … + q_{N+M−1} + q_{−N−M + 1} + p_{N+M} + p_{−N−M} > 0 .

So by using (3.2),

I_{N+M} ∩ A_{N}^{M} = J_{N+M} ∩ A_{N}^{M} .  (3.5)

In particular, the intersection I_{N+M} ∩ A_{N}^{M} is a two-sided ideal of A_{N}^{M}. Thus we get Part (a) of Lemma 1.3.

Furthemore, due to (3.4) there is only one summand (3.1) of X ∈ A_{N}^{M} with

p_{1} + p_{−1} + … + p_{N+M} + p_{−N−M} = 0 ,  (3.6)

this summand has the form of Y ∈ U(𝔮_{N+M−1}) . Note that the right ideal I_{N+M} of U(𝔮_{N+M}) is stable under the adjoint action of the subalgebra 𝔮_{N+M−1} ⊂ 𝔮_{N+M} . Indeed, if |i|, |j| < N + M then by (1.6) we have

[ F_{ij} , F_{N+M, l}] = − (−1)^{ī l̄+ j̄ l̄} δ_{il} F_{N+M, j} − (−1)^{ī l̄+ j̄ l̄} δ_{i,−l} F_{N+M, −j}

for any index l = ±1, …, ±(N +M). In particular, I_{N+M} is stable under the adjoint action of 𝔮_{M−1} . So the condition [Z , X] = 0 on X for any Z ∈ 𝔮_{M−1} implies the condition [Z , Y ] = 0 on the summand Y of X corresponding to (3.6). Therefore Y ∈ A_{N}^{M−1}, and we get Part (b) of Lemma 1.3. ⊓ ⊔

Proof of Proposition 1.4. Suppose that |i|,|j| ⩽ N. Let us prove by induction on n = 1,2, … that the differences

F_{ij | N+M}^{(n)} − F_{ij | N+M−1}^{(n)} and C_{N+M}^{(n)} − C_{N+M−1}^{(n)}  (3.7)

belong to the left ideal J_{N+M} in U(𝔮_{N+M}). Due to (3.5), Proposition 1.4 will then follow. Neither of the elements F_{ij | N+M}^{(n)} and C_{N+M}^{(n)} , nor the ideal J_{N+M} depend on the partition of the number N +M into N and M. Hence it suffices to consider the case M = 1. Note that according to the definition (1.11)

C_{N+1}^{(n)} − C_{N}^{(n)} = ∑_{|k|⩽N}( F_{kk | N+1}^{(n)} − F_{kk | N}^{(n)} ) +

F_{N+1,N+1 | N+1}^{(n)} + F_{−N−1,−N−1 | N+1}^{(n)}

where the last two summands belong to J_{N+1} , by their definition. Therefore it suffices to consider only the first of the differences (3.7), where M = 1.

If n = 1, that difference is zero. Now suppose that n > 1, and make the induction assumption. Using the relation (1.9),

F_{ij | N+1}^{(n)} − F_{ij | N}^{(n)} = ∑_{|k|⩽N} (−1)^{k̄} F_{ik} ( F_{kj | N+1}^{(n−1)} − F_{kj | N}^{(n−1)} ) +

F_{i , N+1} F_{N+1, j | N+1}^{(n−1)} − F_{i , −N−1} F_{−N−1, j | N+1}^{(n−1)} .

At the right hand side of the last equality, the summands corresponding to |k| ⩽ N belong to the left ideal J_{N+1} by the induction assumption. Using (1.10) and (1.8), the remainder of the right hand side is equal to the sum (−1)^{īj̄} F_{N+1, j | N+1}^{(n−1)} F_{i , N+1} − (−1)^{(ī + 1) (j̄ + 1)} F_{−N−1, j | N+1}^{(n−1)} F_{i , −N−1} +

(−1)^{īj̄ + 1} ( 1 + (−1)^{ī + j̄ + n}) δ_{ij} F_{N+1, N+1 | N+1}^{(n−1)} +
(−1)^{(ī+1) (j̄+1)} ( 1 + (−1)^{ī + j̄ + n}) δ_{i,−j} F_{−N−1, N+1 | N+1}^{(n−1)} .

But in this sum, every summand evidently belongs to the left ideal J_{N+1} . ⊓ ⊔

Let us now prove Theorem 1.5. Firstly, we will verify the formula (1.18) for the supercommutator [ F_{ij}^{(m)}, F_{kl}^{(n)} ] in the algebra A_{N}. We will use

Proposition 3.1. In U(𝔮_{N}) for m,n = 1,2 , … and i,j ,k ,l = ±1, …, ±N

[F_{ij | N}^{(m)} ,F_{kl | N}^{(n)} ] = F_{il | N}^{(m+n−1)} δ_{kj} − (−1)^{( ī+ j̄) ( k̄ + l̄)} δ_{il} F_{kj | N}^{(m+n−1)} +
(−1)^{m − 1} ( F_{−i,l | N}^{(m+n−1)} δ_{−k,j} − (−1)^{( ī+ j̄) ( k̄ + l̄)} δ_{i,−l} F_{k,−j | N}^{(m+n−1)} ) +

(−1)^{j̄k̄ + j̄l̄ + k̄ l̄} ∑_{r=1}^{m−1} ( F_{il | N}^{(n+r−1)} F_{kj | N}^{(m−r)} − F_{il | N}^{(m−r)} F_{kj | N}^{(n+r−1)} ) +

(−1)^{j̄k̄ + j̄l̄ + k̄ l̄ + k̄ + l̄} ∑_{r=1}^{m−1} (−1)^{r} ( F_{−i,l | N}^{(n+r−1)} F_{−k,j | N}^{(m−r)} − F_{i,−l | N}^{(m−r)} F_{k,−j | N}^{(n+r−1)} ) .

Proof. The formula for the supercommutator [F_{ij | N}^{(m)} ,F_{kl | N}^{(n)} ] in the ℤ_{2} -graded algebra U(𝔮_{N}) displayed above is easy to verify by using the induction on m. When m = 1, this formula coincides with (1.10). Now make the induction assumption. Let the index h run through ±1, …, ±N. Then due to (1.9),

[ F_{ij | N}^{(m+1)} ,F_{kl | N}^{(n)} ] = ∑_{h} (−1)^{h̄} [ F_{ih} F_{hj | N}^{(m)} ,F_{kl | N}^{(n)} ] =

∑_{h} (−1)^{h̄} F_{ih} [ F_{hj | N}^{(m)} ,F_{kl | N}^{(n)} ] + ∑_{h} (−1)^{h̄ + (h̄ + j̄) (k̄ + l̄)} [ F_{ih} ,F_{kl | N}^{(n)} ] F_{hj | N}^{(m)}

= ∑_{h} (−1)^{h̄} F_{ih} ( F_{hl | N}^{(m+n−1)} δ_{kj} − (−1)^{( h̄ + j̄) ( k̄ + l̄)} δ_{hl} F_{kj | N}^{(m+n−1)} +

(−1)^{m − 1} ( F_{−h,l | N}^{(m+n−1)} δ_{−k,j} − (−1)^{( h̄ + j̄) ( k̄ + l̄)} δ_{h,−l} F_{k,−j | N}^{(m+n−1)} ) +

(−1)^{j̄k̄ + j̄l̄ + k̄ l̄} ∑_{r=1}^{m−1} ( F_{hl | N}^{(n+r−1)} F_{kj | N}^{(m−r)} − F_{hl | N}^{(m−r)} F_{kj | N}^{(n+r−1)} ) +

(−1)^{j̄k̄ + j̄l̄ + k̄ l̄ + k̄ + l̄} ∑_{r=1}^{m−1} (−1)^{r}( F_{−h,l | N}^{(n+r−1)} F_{−k,j | N}^{(m−r)} − F_{h,−l | N}^{(m−r)} F_{k,−j | N}^{(n+r−1)} ))

+ ∑_{h} (−1)^{h̄ + (h̄ + j̄) (k̄ + l̄)} ( δ_{kh} F_{il | N}^{(n)} − (−1)^{( ī+ h̄) ( k̄ + l̄)} δ_{il} F_{kh | N}^{(n)} +

δ_{−k,h} F_{−i,l | N}^{(n)} − (−1)^{( ī+ h̄) ( k̄ + l̄)} δ_{i,−l} F_{k,−h | N}^{(n)} ) F_{hj | N}^{(m)} .

Here we used the induction assumption with the index i replaced by h, and the relation (1.10) with the index j replaced by h. Using the relation (1.9) repeatedly, the right hand side of the above displayed equalities equals

F_{il | N}^{(m+n)} δ_{kj} − (−1)^{( ī+ j̄) ( k̄ + l̄)} δ_{il} F_{kj | N}^{(m+n)} +
(−1)^{m} ( F_{−i,l | N}^{(m+n)} δ_{−k,j} − (−1)^{( ī+ j̄) ( k̄ + l̄)} δ_{i,−l} F_{k,−j | N}^{(m+n)} ) +
(−1)^{j̄k̄ + j̄l̄ + k̄ l̄} ( F_{il | N}^{(n)} F_{kj | N}^{(m)} − F_{il | N}^{(1)} F_{kj | N}^{(m+n−1)} ) +

(−1)^{j̄k̄ + j̄l̄ + k̄ l̄ + k̄ + l̄}((−1)^{m+1} F_{i,−l | N}^{(1)} F_{k,−j | N}^{(m+n−1)} − F_{−i,l | N}^{(n)} F_{−k,j | N}^{(m)} ) +

(−1)^{j̄k̄ + j̄l̄ + k̄ l̄} ∑_{r=1}^{m−1} ( F_{il | N}^{(n+r)} F_{kj | N}^{(m−r)} − F_{il | N}^{(m−r+1)} F_{kj | N}^{(n+r−1)} ) +

(−1)^{j̄k̄ + j̄l̄ + k̄ l̄ + k̄ + l̄} ∑_{r=1}^{m−1} (−1)^{r+1}( F_{−i,l | N}^{(n+r)} F_{−k,j | N}^{(m−r)} + F_{i,−l | N}^{(m−r+1)} F_{k,−j | N}^{(n+r−1)} ) .

But the last displayed sum can also be obtained by replacing m by m + 1 at right hand side of the equality in Proposition 3.1. Thus we have made the induction step. ⊓ ⊔

If m > n then at the right hand side of the equality in Proposition 3.1, the summands corresponding to the indices r = 1, …, m−n cancel in each of the two sums over r = 1, …, m − 1. In the first of the two sums, this is obvious. To cancel these summands in the second sum, one utilises the relations (1.8). Hence if m > n, the summation over r = 1, …, m − 1 in Proposition 3.1 can be replaced by the summation over r = m − n + 1, …, m − 1. Thus if we change the running index r to m − r, the latter index should run through 1, …, min(m,n) − 1. Using this remark, the relation (1.18) in Theorem 1.5 follows from Proposition 3.1.

In the remainder of the proof of Theorem 1.5, we will also make use of the next proposition. For any integers M ⩾ 0 and n ⩾ 1 consider the elements

c_{N+M}^{(n)} and f_{ij | N+M}^{(n)}  (3.8)

of the algebra S(𝔮_{N+M})^{𝔮_{M}}, see Proposition 2.6. Fix any positive integer s.

Proposition 3.2. Take the elements f_{ij | N+M}^{(n)} where

1 ⩽ n ⩽ s, 1 ⩽ i ⩽ N , 1 ⩽ |j| ⩽ N .  (3.9)

Along with these elements, take the elements c_{N+M}^{(n)} where 1 ⩽ n ⩽ s and n is odd. For any sufficiently large number M, all these elements are algebraically independent in the supercommutative algebra S(𝔮_{N+M}).

Proof. We will use arguments from [MO, Subsection 2.11]. By the Poincaré - Birkhoff -Witt theorem for Lie superalgebras, the elements

F_{kl} = f_{kl | N+M}^{(1)}

where k = 1, …, N + M and l = ±1, …, ±N + M, are free generators of the supercommutative algebra S(𝔮_{N+M}). Let X_{s} be the quotient algebra of S(𝔮_{N+M}), defined by imposing the following relations on these free generators.

For every triple (i,j ,n) satisfying the conditions (3.9), choose a subset

O_{ij}^{(n)} ⊂ {N + 1, N + 2, … }

of cardinality n − 1 in such a way, that all these subsets are disjoint. Let M be so large that all these subsets are contained in {N + 1, …, N + M − s}. If

O_{ij}^{(n)} = {l_{1} , …, l_{n−1} } ,

then put

F_{i l_{1}} = F_{l_{1}l_{2}} = … = F_{l_{n−2} l_{n−1}} = 1.

Denote by x_{ij}^{(n)} the image of the element F_{l_{n−1} j} ∈ S(𝔮_{N+M}) in the algebra X_{s}. Having done this for every triple (i,j ,n) satisfying the conditions (3.9), for every r = 1, …, s denote by x_{r} the image in X_{s} of the element

F_{N+M−s+r , N+M−s+r} ∈ S(𝔮_{N+M}).

Finally, put F_{kl} = 0 if k > 0 and (k ,l) is not one of the pairs

(i,l_{1}) , (l_{1} ,l_{2}) , …, (l_{n−2} ,l_{n−1}) , (l_{n−1} ,j)

for any triple (i,j ,n) satisfying (3.9), and not one of the pairs

(N +M −s+r,N +M −s+r) where r = 1, …, s .

The elements x_{1} , …, x_{s} and the elements x_{ij}^{(n)} for all the triples (i,j ,n) satisfying (3.9), are free generators of the algebra X_{s}. For any of these triples, the image in X_{s} of

f_{ij | N+M}^{(n)} ∈ S(𝔮_{N+M})

equals x_{ij}^{(n)} plus a certain linear combination of products of the elements x_{kl}^{(m)} where 1 ⩽ m < n. For any odd n, the image in X_{s} of the element

c_{N+M}^{(n)} ∈ S(𝔮_{N+M})

equals

2^{n}(x_{1}^{n} + … + x_{s}^{n})

plus a linear combination of products of elements x_{kl}^{(m)} where 1 ⩽ m ⩽ n. Hence all these images are algebraically independent in the quotient X_{s} of the supercommutative algebra S(𝔮_{N+M}). ⊓ ⊔

Let us show that the associative algebra A_{N} is generated by the elements C^{(1)}, C^{(3)}, … and F_{ij}^{(1)}, F_{ij}^{(2)}, …. Take any element Z ∈ A_{N}, and consider its canonical image Z_{M} = π_{M}(Z) ∈ A_{N}^{M} for any M ⩾ 0. By Theorem 1.2, the element Z_{M} is a linear combination of the products of the elements C_{N+M}^{(n)} where n = 1,3, … and of the elements F_{ij | N+M}^{(n)} where n = 1,2, … whereas i = 1, …, N and j = ±1, …, ±N; see (1.8). Choose any linear ordering of all these elements. Applying Proposition 3.1 to the algebra U(𝔮_{N+M}) instead of U(𝔮_{N}), we will assume that any of the products in the linear combination Z_{M} is an ordered monomial in these elements. If ī+ j̄ = 1, then the element F_{ij | N+M}^{(n)} may appear in any of these monomials only with the degree 1.

We will assume that for any M ⩾ 1, the map α_{M} preserves the ordering. Then for every monomial Y_{M} appearing in the linear combination Z_{M}, the monomial α_{M}(Y_{M}) may appear in the linear combination α_{M}(Z_{M}) = Z_{M−1}.

The filtration degrees of all the elements Z_{0} ,Z_{1} ,Z_{2} , … are bounded from above. Hence for any factor C_{N+M}^{(n)} or F_{ij | N+M}^{(n)} of the monomials appearing in linear combination Z_{M}, we have n ⩽ s for a certain integer s which does not depend on M. Then for a sufficiently large number M, the coefficients of the monomials appearing in the linear combinations Z_{M} ,Z_{M+1} ,Z_{M+2} , … are determined uniquely. The uniqueness follows from Proposition 3.2.

Now fix a sufficiently large number M, as above. Let Y_{M} be any monomial appearing in the linear combination Z_{M}, say with a coefficient z ∈ ℂ. We assume that z ≠ 0. Then for any integer L > M, the linear combination Z_{L} contains the summand z Y_{L} where Y_{L} is a monomial and

(α_{M+1} ∘ … ∘ α_{L})(Y_{L}) = Y_{M} .

For any non-negative integer L ⩽ M, define Y_{L} = (α_{L} ∘ … ∘ α_{M})(Y_{M}). The sequence Y_{0} ,Y_{1} ,Y_{2} , … determines an element Y ∈ A_{N}, which is a monomial in C^{(1)}, C^{(3)}, … and F_{ij}^{(1)}, F_{ij}^{(2)}, …. Here 1 ⩽ i ⩽ N and 1 ⩽ |j| ⩽ N. The element Z ∈ A_{N} is then a sum of the products of the form z Y. This sum is finite, because any such product corresponds to a summand z Y_{M} in the linear combination Z_{M}. Thus we have proved Part (a) of Theorem 1.5.

Let us now prove Parts (b) and (c). By definition, the algebra A_{N} comes with an ascending ℤ-filtration, such that the generators C^{(n)} and F_{ij}^{(n)} of A_{N} have the degree n. Denote the corresponding ℤ-graded algebra by gr A_{N}. Let c^{(n)} and f_{ij}^{(n)} be the generators of the algebra gr A_{N} corresponding to C^{(n)} and F_{ij}^{(n)}. We always assume that the index n in C^{(n)} and c^{(n)} is odd. We also assume that |i|,|j| ⩽ N in F_{ij}^{(n)} and f_{ij}^{(n)}. It follows from (1.17) that for any n = 1,2, …

f_{−i,−j}^{(n)} = (−1)^{n−1} f_{ij}^{(n)}.

The algebra gr A_{N} also inherits from A_{N} a ℤ_{2} -gradation, such that

deg c^{(n)} = 0 and deg f_{ij}^{(n)} = ī + j̄,

see (1.16). The relation (1.18) demonstrates that the ℤ_{2} -graded algebra gr A_{N} is supercommutative. To complete the proof of Theorem 1.5, it suffices to show that the elements c^{(1)}, c^{(3)}, … together with the elements f_{ij}^{(1)}, f_{ij}^{(2)}, … where i = 1, …, N and j = ±1, …, ±N, are algebraically independent in the supercommutative algebra gr A_{N}.

The algebra gr A_{N} can also be obtained as an inverse limit of the sequence of the supercommutative algebras S(𝔮_{N+M})^{𝔮_{M}} where M = 0,1,2, …. The limit is taken in the category of ℤ-graded algebras. We assume that if M = 0, then S(𝔮_{N+M})^{𝔮_{M}} = S(𝔮_{N}). The definition of the surjective homomorphism

S(𝔮_{N+M})^{𝔮_{M}} → S(𝔮_{N+M−1})^{𝔮_{M−1}}

for any M ⩾ 1 is similar to the definition of the surjective homomorphism α_{M} : A_{N}^{M} → A_{N}^{M−1}, see Lemma 1.3. Here we omit the details, but notice that the elements c^{(n)} and f_{ij}^{(n)} of gr A_{N} correspond to the sequences of elements (3.8) of the algebras S(𝔮_{N+M})^{𝔮_{M}} where M = 0,1,2, …. Proposition 3.2 now guarantees the algebraic independence of the elements c^{(1)}, c^{(3)}, … together with the elements f_{ij}^{(1)}, f_{ij}^{(2)}, … where i = 1, …, N and j = ±1, …, ±N.

Proof of Proposition 1.6. Under the correspondence F_{ij}^{(n)} ↦ (−1)^{ī} T_{ji}^{(n)}, the collection of relations (1.17) in the algebra B_{N} for the indices n = 1,2, … corresponds to the equality (1.19) in Y(𝔮_{N})[[u^{−1}]]. Put T_{ij}^{(0)} = δ_{ij} for any i,j = 1, …, ±N. Using (1.4), the relation (1.18) in B_{N} then corresponds to

(−1)^{īj̄ + īk̄ + j̄k̄} [ T_{ji}^{(m)}, T_{lk}^{(n)} ] =

∑_{r=0}^{m−1} ( T_{jk}^{(m+n−r−1)} T_{li}^{(r)} − T_{jk}^{(r)} T_{li}^{(m+n−r−1)} ) + (−1)^{ī + j̄ + 1} ×

∑_{r=0}^{m−1}(−1)^{m + r} ( T_{−j,k}^{(m+n−r−1)} T_{−l,i}^{(r)} − T_{j,−k}^{(r)} T_{l,−i}^{(m+n−r−1)} ) . (3.10)

Here we also used a remark on the summation over r = 1, …, m − 1 similar to that made immediately after the proof of Proposition 3.1.

Put T_{ij}^{(−1)} = 0 for any i,j = 1, …, ± N. The collection of relations (3.10) for m,n = 1,2, … is equivalent to the collection of relations

(−1)^{īj̄ + īk̄ + j̄k̄} ( [ T_{ji}^{(m+1)}, T_{lk}^{(n−1)} ] − [ T_{ji}^{(m−1)}, T_{lk}^{(n+1)} ] ) =

T_{jk}^{(n−1)} T_{li}^{(m)} − T_{jk}^{(m)} T_{li}^{(n−1)} + T_{jk}^{(n)} T_{li}^{(m−1)} − T_{jk}^{(m−1)} T_{li}^{(n)} + (−1)^{ī + j̄} ×

( T_{−j,k}^{(n−1)} T_{−l,i}^{(m)} − T_{j,−k}^{(m)} T_{l,−i}^{(n−1)} − T_{−j,k}^{(n)} T_{−l,i}^{(m−1)} + T_{j,−k}^{(m−1)} T_{l,−i}^{(n)} )  (3.11)

for m,n = 0,1,2, …. Multiplying the relation (3.11) by x^{1−m} y^{1−n} and taking the sum of resulting relations over m,n = 0,1,2, … we get the relation

(x^{2} − y^{2} ) · [ T_{ji}(x), T_{lk}(y)] · (−1)^{īj̄ + īk̄ + j̄k̄} =
(x + y) · (T_{jk}(y) T_{li}(x) − T_{jk}(x) T_{li}(y)) +
(x − y) · (T_{−j, k}(y) T_{−l, i}(x) − T_{j, −k}(x) T_{l, −i}(y)) · (−1)^{ī + j̄}.  (3.12)

Using (1.5), we can rewrite the left hand side of the relation (3.12) as

(y^{2} − x^{2} ) · [ T_{lk}(y), T_{ji}(x)] · (−1)^{īj̄ + īl̄ + j̄l̄}.

Replacing in the resulting relation the indices i,j ,k ,l and the parameters x,y by l,k ,j ,i and y ,x respectively, we obtain exactly the relation (1.20). Thus the defining relations of the subalgebra B_{N} ⊂ A_{N} correspond to the defining relations of the algebra Y(𝔮_{N}), see Theorem 1.5. ⊓ ⊔

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