Knizhnik-Zamolodchikov type equations for the root system $B$ and Capelli central elements

The construction of the well-known Knizhnik-Zamolodchikov equations uses the central element of the second order in the universal enveloping algebra for some Lie algebra. But in the universal enveloping algebra there are central elements of higher orders. It seems desirable to use these elements for the construction of Knizhnik-Zamolodchikov type equations. In the present paper we give a construction of such Knizhnik-Zamolodchikov type equations for the root system $B$ associated with Capelli central elements in the universal enveloping algebra for the orthogonal algebra.


Introduction
The Knizhnik-Zamolodchikov equations are a system of differential equations which is satisfied by correlation functions in the WZW theory [1]. Later it turned out that these equation are related with many other areas of mathematics (quantum algebra, isomonodromic deformation).
The Kniznik-Zamolodchikov equations are also interesting as a nontrivial example of an integrable Pfaffian system of Fuchsian type. Mention that the monodromy representation of this system is known explicitly. Thus we get a solution for the Riemann-Hilbert problem in a very particular case.
Let us write the Kniznik-Zamolodchikov equations. They have the form.
where λ is some complex parameter, y(z 1 , ..., z n ) is a vector function that takes values in a tensor power V ⊗n of a representation space V of a Lie algebra g, and τ ij is defined by the formula Here I s is base of a finite-dimensional Lie algebra, {ω(I s )} is a base dual to {I s } with respect to the Killing form. The elements I s occur on tensor factors i and j, and ρ : g → End(V ) is a representation of the Lie algebra g.
Thus τ ij is a matrix of the operator on V ⊗n . Denote a reflection corresponding to the root α as σ α .
The Matsuo's Knizhnik-Zamolodchikov type equations contain arbitrary parameters λ |α| depending on the length of the roots α and one additional parameter λ. The system has the following form.  [5].
In [6] we have constructed Knizhnik-Zamolodchikov type equations for the root system A, but our construction was based on special centra elements in the universal enveloping algebra of the orthogoanl algebra, namely the Capelli elements. In the present paper we generalize our construction to the root system B and show that the construction from [6] can be in fact interpreted in some sence as a very special case of the general Leibman's construction. This is a system of type Here τ ij is defined as in 2 where as before I s is a base of a Lie algebra, {ω(I s )} is a base dual to {I s } with respect to the Killing form.
the coefficients µ ij and ν i are defined as follows. Denote as σ an involution in the considered Lie algebra g. Then one has the elements I s occur on places i, j, the elements I s occur on the place i.
Although the Leibman's proof in [4] is done for the case of a simple Lie algebra and is based on calculations in the root base in [5] these is a proof that is valid for an arbitrary finite-dimensional Lie algebra with a fixed central element in U(g) of the second order.
3 The Lie algebra T.
In this section we introduce a Lie algebra T which playes the crucial role in our construction.
Consider the space of skew-symmetric tensors with 2n indices. Let each index of the skew symmetric tensor take values in the set −n, ..., n.
There exist a structure of an associative algebra algebra on this space As a corollary we have a structure of a Lie algebra. Denote this algebra as T.
This algebra has a representation on the space of skew-symmetric tensors with n indices defined by the formula The algebra has an involution ω defined as follows In U(T) there is a central element of the second order, namely the element where • denotes the multiplication in U(T). The proof of this fact is essentialy contained in [7].
Using general constructions described in Section 5 one can construct Knizhnik-Zamolodchikov type equations associated with the root system B based with coefficients in the Lie algebra T.
In the next section we give an interpretation of these equations as Knizhnik-Zamolodchikov type equations whose construction is based on some certain higher order central elements in U(o 2n+1 ).

Capelli elements and noncommutative pfaffians
Let us define some certain central elements in the universal enveloping algebra of the orthogonal algebra.

The split realization of the orthogonal algebra
We use the split realization of the orthogonal algebra. This means that we define the orthogonal algebra as the algebra that preserves the quadratic The row and columns are indexed by i, j = −n, −n + 1, ..., n − 1, n. The zero is skipped in the case N = 2n and is included in the case N = 2n + 1.
The algebra o N is generated by matrices The commutation relations between these generators are the following

Noncommutative pfaffians and Capelli elements
Now let us describe some special higher order central elements in the universal enveloping for the orthogoanl algebra.
Let Φ = (Φ ij ) be a k × k matrix, where k is even, whose elements belong to some noncommutative ring. The noncommutative pfaffian is defined as follows: For a subset I ⊂ {−n, ..., n} define a submatrix F I = (F ij ) i,j∈I . For this subset put The elements C k are the Capelli elements.
Tеорема 1. [7]For odd N the elements C k are algebraically independent and generate the center, for even N the same is true if one takes instead the highest Capelli element C N = (P f F ) 2 the central element P f F .
Below we need two formulas. There proofs can be found in [?].
is a sign of a permutation of the set I = {i 1 , ..., i k } that places first the subset I ′ ⊂ I and then the subset I ′′ ⊂ I.
The numbers p, q are even fixed numbers, they satisfy p + q = k = |I|.
Let ∆ be the standard comultiplication in the universal enveloping algebra.
is a sign of a permutation of the set I = {i 1 , ..., i k } that places first the subset I ′ ⊂ I and then places the subset I ′′ ⊂ I.

The action of Pfaffians on tensors
Let us decribe the action of pfaffians in th tensor representations. Proof. The proposition is proved by direct calculation using the formulaes, Prove an analog of the previous statement in an arbitrary dimension Предложение 2. On the base vectors e −n , ..., e n of the standard representation of o N the pfaffians P f F I for |I| > 2 act as zero operators.
Put q = 4, p = k − 4 in Lemma 1. One has If j / ∈ I ′′ , then obviously P f F I ′′ e j = 0. If j ∈ I ′′ , then using Proposition 1 one also obtains P f F I ′′ e j = 0.
Let us find an action of a pfaffian of the order k on a tensor product of < k 2 vectors, that is on a tensor product e r 2 ⊗ e r 4 ⊗ ... ⊗ e rt , where t < k.
Предложение 3. P f F I e r 2 ⊗ e r 4 ... ⊗ e rt = 0 where t < k Proof. The following formulae takes place By definition one has P f F I e r 2 ⊗e r 4 ⊗...⊗e r k = (∆ k P f F I )e r 2 ⊗e r 4 ⊗...⊗e r k .
Since t < k, the comultiplication ∆ k P f F I contains only summands in which on some place the pfaffian stands whose indexing set I satisfies |I| ≥ 4 (Lemma 2). From Proposition 1 it follows that every such a summand acts as a zero operator.
Find an action of a pfaffian of the order k on a tensor product of k 2 vector, that is on the tensor product e r 2 ⊗ e r 4 ⊗ ... ⊗ e r k . Otherwise take a permutation γ of I, such that (γ(i 1 ), γ(i 2 ), ..., γ(i k )) = (r 1 , r 2 , r 3 , ..., r k−1 , r k ). Then Proof. By definition one has Applying many times the formulae for comultiplication one obtains Using Proposition 3 one gets that, only the summands for which |I j | = 2, j = 1, ..., k are nonzero operators.
Hence the summation over divisions can be written in the following manner.
Show that one can consider only the permutations σ such that σ(i 2t ) = r 2t , that is the permutations of type (σ(i 1 ), r 2 , σ(i 2 ), r 3 ..., σ(i k−1 ), r k ). But when only summands corresponding to such permutations are considered one must multiply the resulting sum on 2 k 2 .
Remind that the input for σ is One has from one hand that F −σ(i 1 )σ(i 2 ) e r 2 = e −σ(i 1 ) and from the other hand , Also one has (−1) σ = −(−1) σ ′ . Thus the inputs corresponding to σ and σ ′ are the same.
The equality (−1) Taking the summation over all permutations δ, one gets Finally from the formula P f F I e r 2 ⊗ e r 4 ⊗ ... ⊗ e rt = (∆ t P f F I )e r 2 ⊗ e r 4 ⊗ ... ⊗ e rt , as in the proof of Proposition 4, one gets the formulae of the action on an arbitrary tensor e r 2 ⊗ ... ⊗ e rt .

Pfaffians and representation of the algebra T
Let us give a relation between the representation of the Lie algebra T defined by the formula 5 through the action of noncommutative pfaffians.
As a corollary of Proposition 5 we get the following proposition Then P f F P f F I ′′ J = 0 for I ′′ > 2.
The sign (−1) (I ′ I ′′ ) is defined as follows. For an index s denote as s either s for s < i, оr s − 1 for s > i.
Using the theorem 6 one obtains that for a vector v from a representation of o 2n+1 with the highest weight (1, ..., 1) the following holds.
where C is some constant.
The following theorem is proved.
where the pfaffians occur on places i, j, and ρ is a representation of o 2n+1 on the space of skew-symmetric tensors with n indices, As a corollary we get Tеорема 4. The elements 7 -9 satisfy the commutation relation for the coefficients of the Kniznik-Zamolodchikov type equations associated with the root system B