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This article is cited in 2 scientific papers (total in 2 papers) New approaches to Zhuoke Yang International Laboratory of Cluster Geometry, National Research University ``Higher School of Economics'' (HSE), Moscow
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2023, Volume 87, Issue 6, DOI: https://doi.org/10.4213/im9379 
Bibliographic databases:
    § 1. List of Symbols$\mathfrak{g}$, $\langle\,{\cdot}\,,{\cdot}\,\rangle$ – a Lie algebra endowed with a non-degenerate invariant bilinear product; $\mathfrak{gl}_N$ – the general linear Lie algebra; consists of all $N\times N$ matrices with the commutator serving as the Lie bracket; $\mathfrak{sl}_N$ – the special linear Lie algebra; consists of all $N\times N$ trace-free matrices with the commutator serving as the Lie bracket; $d$ – the dimension of Lie algebra; specifically, for $\mathfrak{gl}_N$, $d=N^2$; $D$ – a chord diagram; $n$ – the number of chords in a chord diagram; $K_n$ – the chord diagram with $n$ chords any two of which intersect one another; $\pi$ – the projection to the subspace of primitive elements in the Hopf algebra of chord diagrams whose kernel is the subspace of decomposable elements; $C_1,\dots,C_N$ – Casimir elements in $U(\mathfrak{gl}_N)$; $w$ – a weight system; $w_\mathfrak{g}$ – the Lie algebra weight system associated with a Lie algebra $\mathfrak{g}$; $\overline{w}_\mathfrak{g}$ – $w_\mathfrak{g}(\pi(\,{\cdot}\,))$; the composition of the Lie algebra weight system $w_\mathfrak{g}$ with the projection $\pi$ to the subspace of primitives; $\sigma$ – a permutation; $m$ – the number of permutated elements; for example, for the permutation determined by a chord diagram, $m=2n$; $G(\sigma)$ – the digraph of the permutation $\sigma$; $\Lambda^*(N)$ – the algebra of shifted symmetric polynomials in $N$ variables; $\phi$ – the Harish-Chandra projection; $p_1,\dots,p_N$ – shifted power sum polynomials. § 2. IntroductionIn V. A. Vassiliev’s theory of finite type knot invariants, a weight system can be associated with each such invariant. A weight system is a function on chord diagrams satisfying so-called $4$-term relations. In the opposite direction, according to a Kontsevich theorem, to each weight system taking values in a field of characteristic $0$, a finite type knot invariant can be associated in a canonical way. This makes the study of weight systems an important part of knot theory. There is a number of approaches to constructing weight systems. In particular, a huge class of weight systems can be constructed from metrized finite dimensional Lie algebras. The present paper has been motivated by an aspiration for understanding the weight system corresponding to the Lie algebra $\mathfrak{gl}_N$. The straightforward approach to computing the values of a Lie algebra weight system on a general chord diagram amounts to elaborating calculations in the non-commutative universal enveloping algebra, in spite of the fact that the result belongs to the centre of the latter. This approach is rather inefficient even for the simplest non-commutative Lie algebra $\mathfrak{sl}_2$, whose weight system is associated with the knot invariant known as the colored Jones polynomial. For this Lie algebra, however, there is a recurrence relation due to Chmutov and Varchenko [1], and numerous computations have been done using it, see, for example, [2]–[4]. In particular, recently, values of the $\mathfrak{sl}_2$-weight system were computed on certain non-trivial infinite families of chord diagrams. Much less is known about other Lie algebras; for them, explicit answers have been computed only for chord diagrams of very small order or for simple families of chord diagrams, see [5]. In particular, no recurrence similar to the Chmutov–Varchenko one exists (with the exception of the Lie superalgebra $\mathfrak{gl}_{1|1}$, see [6], [7]). The goal of the present paper is to provide two new ways to compute the values of the $\mathfrak{gl}_N$ weight system. The first approach is based on M. Kazarian’s proposal to define an invariant of permutations taking values in the centre of the universal enveloping algebra of $\mathfrak{gl}_N$. The restriction of this invariant to involutions without fixed points (such an involution determines a chord diagram) coincides with the value of the $\mathfrak{gl}_N$ weight system on this chord diagram. We describe the recursion allowing one to compute the $\mathfrak{gl}_N$ invariant of permutations and demonstrate how it works in a number of examples. For $N'<N$, the centre of the universal enveloping algebra of $\mathfrak{gl}_{N'}$ is naturally embedded into that of $\mathfrak{gl}_N$, and the $\mathfrak{gl}_N$ weight system is stable: its value on a permutation is a universal polynomial. The recursion we describe allows one to compute this polynomial simultaneously for all $N$. The calculations of the higher homogeneous part of the universal $\mathfrak{gl}_N$ weight system for some special primitive elements given by open Jacobi diagrams were the central part of the lower estimate for the dimension of the Vassiliev knot invariants in [8], [9] (see also § 14.5.4 in [10]). The second approach is based on the Harish-Chandra isomorphism for the Lie algebras $\mathfrak{gl}_N$. This isomorphism identifies the centre of the universal enveloping algebra $\mathfrak{gl}_N$ with the ring $\Lambda^*(N)$ of shifted symmetric polynomials in $N$ variables. The Harish-Chandra projection can be applied separately for each monomial in the defining polynomial of the weight system; as a result, the main body of computations can be done in a commutative algebra, rather than non-commutative one. The paper is organized as follows. In § 3, we recall the construction of Lie algebra weight systems. In § 4, we describe an extension of the $\mathfrak{gl}_N$ weight system to arbitrary permutations and a recursion to computing its values on permutations. In § 5, we apply, for the Lie algebras $\mathfrak{gl}_N$, the Harish-Chandra isomorphism to develop one more algorithm for computing the corresponding weight system. We compare the results with those obtained by the previous method. In § 6, we recall the Hopf algebra structure on the space of chord diagrams modulo $4$-term relations, and discuss the behaviour of the $\mathfrak{gl}_N$ weight system with respect to this structure. The author is grateful to M. Kazarian and G. Olshanskii for valuable suggestions, and to S. Lando for permanent attention. After the present paper has been submitted, the paper [11] appeared, where the constructions of the present paper are included in the general context of weight systems and graph invariants. In [12], we extend the results of the present paper to Lie superalgebras $\mathfrak{gl}(m|n)$. § 3. Definition of $\mathfrak{gl}_N$ weight systemBelow, we use standard notions from the theory of finite order knot invariants; see, for example, [10]. A chord diagram of order $n$ is an oriented circle (called the Wilson loop) endowed with $2n$ pairwise distinct points split into $n$ disjoint pairs, considered up to orientation-preserving diffeomorphisms of the circle. A weight system is a function $w$ on chord diagrams satisfying the $4$-term relation; see Figure 1. In figures, the outer circle of the chord diagram is always assumed to be oriented counterclockwise. Dashed arcs may contain ends of arbitrary sets of chords, same for all the four terms in the picture. Definition 3.1. The product of two chord diagrams $D_1$ and $D_2$ is defined by cutting and gluing the two circles as shown Modulo $4$-term relations, the product is well-defined, see § 4.4.3 in [10]. Given a Lie algebra $\mathfrak{g}$ equipped with a non-degenerate invariant bilinear form, one can construct a weight system with values in the centre of its universal enveloping algebra $U(\mathfrak{g})$. This is the form Kontsevich [13] gave to a construction due to Bar-Natan [14]. Kontsevich’s construction proceeds as follows. Definition 3.2 (universal Lie algebra weight system). Let $\mathfrak{g}$ be a metrized Lie algebra over $\mathbb{R}$ or $\mathbb{C}$, that is, a Lie algebra with an ad-invariant non-degenerate bilinear form $\langle\,{\cdot}\,,{\cdot}\,\rangle$. Let $d$ denote the dimension of $\mathfrak{g}$. Choose a basis $e_1,\dots,e_d$ of $\mathfrak{g}$ and let $e_1^*,\dots,e_d^*$ be the dual basis with respect to the form $\langle\,{\cdot}\,,{\cdot}\,\rangle$, $\langle e_i,e_j^*\rangle=\delta_{ij}$, where $\delta$ is the Kronecker delta. Given a chord diagram $D$ with $n$ chords, we first choose a base point on the circle, away from the ends of the chords of $D$. This gives a linear order on the endpoints of the chords, increasing in the positive direction of the Wilson loop. With each chord $a$ we associate an index, that is, an integer-valued variable, $i_a$. The values of $i_a$ will range from $1$ to $d$, the dimension of the Lie algebra. We mark the first endpoint of the chord $a$ with the symbol $e_{i_a}$ and the second endpoint with $e_{i_a}^*$. Now, write the product of all the $e_{i_a}$ and all the $e_{i_a}^*$, in the order in which they appear on the Wilson loop of $D$, and take the sum of the $d^n$ elements of the universal enveloping algebra $U(\mathfrak{g})$ obtained by substituting all possible values of the indices $i_a$ into this product. Denote by $w_\mathfrak{g}(D)$ the resulting element of $U(\mathfrak{g})$. Claim 3.3 (see [13]). The function $w_\mathfrak{g}\colon D\mapsto w_\mathfrak{g} (D)$ on chord diagrams has the following properties: 1) the element $w_\mathfrak{g} (D)$ does not depend on the choice of the base point on the diagram; 2) it does not depend on the choice of the basis ${e_i }$ of the Lie algebra $\mathfrak{g}$; 3) its image belongs to the ad-invariant subspace
$$
\begin{equation*}
U(\mathfrak{g})^{\mathfrak{g}}=\{x\in U(\mathfrak{g})\mid xy=yx \text{ for all } y\in \mathfrak{g}\}=ZU(\mathfrak{g});
\end{equation*}
\notag
$$
4) it is multiplicative, $w_\mathfrak{g} (D_1D_2)=w_\mathfrak{g} (D_1)w_\mathfrak{g} (D_2)$ for any pair of chord diagrams $D_1$, $D_2$; 5) this map from chord diagrams to $ZU(\mathfrak{g})$ satisfies the 4-term relations. Consider the Lie algebra $\mathfrak{gl}_N $ of all $N \times N$ matrices. Fix the trace of the product of matrices as the preferred ad-invariant form: $\langle x,y\rangle = \operatorname{Tr}(xy)$. The Lie algebra $\mathfrak{gl}_N $ is linearly spanned by matrix units $E_{ij}$ having $1$ on the intersection of $i$th row with $j$th column and $0$ elsewhere, $i,j=1,\dots,N$. We have $\langle E_{ij},E_{kl}\rangle = \delta_{il}\delta_{jk}$. Therefore, the duality between $\mathfrak{gl}_N $ and $\mathfrak{gl}_N^*$ defined by $\langle \,{\cdot}\,,{\cdot}\,\rangle$ is given by the formula $E_{ij}^*= E_{ji}$. The commutation relations for $\mathfrak{gl}_N$ have the form
$$
\begin{equation}
[E_{kl},E_{ji}]=E_{kl}E_{ji}-E_{ji}E_{kl}=\delta_{lj}E_{ki}-\delta_{ik}E_{jl}.
\end{equation}
\tag{1}
$$
Now, the straightforward computation of the value of the $\mathfrak{gl}_N$ weight system is as follows. Example 3.4. For a chord diagram $K_1$ with single chord, we have We denote this element by $C_2\in ZU(\mathfrak{gl}_N)$ and call the second Casimir element. Similarly, $\sum_{i=1}^{N}E_{ii}=C_1$, and, more generally, is the $k$th Casimir element in $ZU(\mathfrak{gl}_N)$. The centre $ZU(\mathfrak{gl}_N)$ is isomorphic to the ring of polynomials in the Casimir elements $C_1,\dots,C_N$: $ZU(\mathfrak{gl}_N)=\mathbb{C}[C_1,\dots,C_N]$, see [15]. The higher Casimir elements $C_{N+1},C_{N+2},\dots$ can be represented as polynomials in $C_1,\dots,C_N$. The value $w_{\mathfrak{gl}_N}(D)$ of the $\mathfrak{gl}_N$ weight system on a chord diagram $D$ with $n$ chords is a polynomial in $C_1,\dots,C_n$. Example 3.5. For the chord diagram, which we denote by $K_2$, since its intersection graph is $K_2$, the complete graph on $2$ vertices, we have Using the commutation relations (1), we obtain Even in this simple example, the straightforward computation involves a lot of steps. A much more efficient algorithm is suggested in the next section. § 4. The $\mathfrak{gl}$ weight system for permutationsThere is no recurrence relation for the weight system $w_{\mathfrak{gl}_N}$ we know about. Instead, following the suggestion by M. Kazarian, we interpret an arc diagram as an involution without fixed points on the set of its ends and extend the function $w_{\mathfrak{gl}_N}$ to arbitrary permutations of any number of permutated elements. For permutations, in contrast to chord diagrams, such a recurrence relation could be given. For any permutation $\sigma\in S_m$, we set
$$
\begin{equation*}
w_{\mathfrak{gl}_N}(\sigma)=\sum_{i_1,\dots,i_m=1}^N E_{i_1i_{\sigma(1)}}E_{i_2i_{\sigma(2)}}\cdots E_{i_mi_{\sigma(m)}}\in U(\mathfrak{gl}_N).
\end{equation*}
\notag
$$
We claim that
For example, the standard generator
$$
\begin{equation*}
C_m=\sum^N_{i_1,\dots,i_m=1}E_{i_1i_2}E_{i_2i_3}\cdots E_{i_{m-1}i_m}E_{i_mi_1}
\end{equation*}
\notag
$$
corresponds to the cyclic permutation $1\mapsto2\mapsto\cdots\mapsto m\mapsto1\in S_m$. On the other hand, a chord diagram with $n$ chords can be considered as an involution without fixed points on a set of $m=2n$ elements. The value of $w_{\mathfrak{gl}_N}$ on the corresponding involution is equal to the value of the $\mathfrak{gl}_N$ weight system on the corresponding chord diagram. Definition 4.2 (digraph of the permutation). Let us represent a permutation as an oriented graph. The $m$ vertices of the graph correspond to the permuted elements. They are ordered cyclically and are placed on a real line, subsequently connected with horizontal arrows looking right and numbered from left to right. The arc arrows show the action of the permutation (so that each vertex is incident with exactly one incoming and one outgoing arc edge). The digraph $G(\sigma)$ of a permutation $\sigma\in S_m$ consists of these $m$ vertices and $m$ oriented edges, for example: Example 4.3. The digraph of the Casimir element $C_m$, which corresponds to the cyclic permutation $1\mapsto2\mapsto\cdots\mapsto m\mapsto1\in S_m$, is the following one: Theorem 4.4. The value of the $w_{\mathfrak{gl}_N}$ invariant of permutations possesses the following properties: Proof. We only need to prove the recurrence rule, which is just the graphical explanation of the Lie bracket in $\mathfrak{gl}_N$,
$$
\begin{equation*}
\begin{aligned} \, &E_{i_ki_{\sigma(k)}}E_{i_{k+1}i_{\sigma({k+1})}}-E_{i_{k+1}i_{\sigma(k+1)}}E_{i_ki_{\sigma(k)}} \\ &\qquad =[E_{i_ki_{\sigma(k)}},E_{i_{k+1}i_{\sigma(k+1)}}] =\delta_{i_{\sigma(k)}i_{k+1}}E_{i_ki_{\sigma(k+1)}} -\delta_{i_{\sigma(k+1)}i_k}E_{i_{k+1}i_{\sigma(k)}}. \end{aligned}
\end{equation*}
\notag
$$
In the special case, when $\sigma(k+1)=k$, we have
$$
\begin{equation*}
E_{i_ki_{\sigma(k)}}E_{i_{k+1}i_k}-E_{i_{k+1}i_k}E_{i_ki_{\sigma(k)}} {=}\,[E_{i_ki_{\sigma(k)}},E_{i_{k+1}i_k}] \,{=}\,\delta_{i_{\sigma(k)}i_{k+1}}E_{i_ki_k}-\delta_{i_ki_k}E_{i_{k+1}i_{\sigma(k)}}.
\end{equation*}
\notag
$$
Summing from $i_1,\dots,i_m=1$ to $N$, we get that $\sum \delta_{i_{\sigma(k)}i_l}E_{i_ki_k}= C_1\sum\delta_{i_{\sigma(k)}i_l}$ and $\sum\delta_{i_ki_k}E_{i_li_{\sigma(k)}}=N\sum E_{i_li_{\sigma(k)}}$.
The second graph on the left-hand side corresponds to a permutation obtained from the first one by a conjugation with a transposition of two neighbouring vertices. Both graphs on the right-hand side have smaller number of vertices. Applying these relations, every graph can be reduced to a monomial in the variables $C_k$ (a concatenation of cyclic permutations) modulo terms of smaller degrees. This provides an inductive computation of the invariant $w_{\mathfrak{gl}_N}$. Theorem is proved. Remark 4.5. In the situation of permutations corresponding to chord diagrams, the difference at the right-hand side of the recurrence relation represents a Jacobi diagram with a triple vertex according to the STU relation from [14], [10]. This gives a way to calculate the weight system $w_{\mathfrak{gl}_N}$ on primitive elements given by Jacobi diagrams. For some special elements, the calculations of this sort were given in [8], [9]. Corollary 4.6. The value of $w_{\mathfrak{gl}_N}$ on a permutation is well defined, it can be represented as a polynomial in $N,C_1,C_2,\dots$, and this polynomial is universal. Definition 4.7 (universal $\mathfrak{gl}$ weight system on permutations). The universal $\mathfrak{gl}$ weight system on permutation $w_\mathfrak{gl}$ is the weight system taking values in the polynomial ring $\mathbb{C}[N,C_1,C_2,\dots]$, which satisfies $w_\mathfrak{gl}(\sigma)=w_{\mathfrak{gl}_N}(\sigma)$, for all permutations $\sigma$, and is obtained by the above recurrence relations. Example 4.8. Let us compute the value of $w_{\mathfrak{gl}}$ on the cyclic permutation $(1\ 3\ 2)$ by switching the places of nodes $2$ and $3$: The reader will find below a table of values of the $\mathfrak{gl}$ weight system on chord diagrams $K_n$, which have $n$ chords and each chord crosses each other; these results were obtained by computer calculation. These diagrams are chosen because computation of Lie algebra weight systems on them is extremely non-trivial, even for the Lie algebra $\mathfrak{sl}_2$, where we know the Chmutov–Varchenko recurrence relation. In addition, these diagrams generate a Hopf subalgebra of the Hopf algebra of chord diagrams, see § 6, which allows us to compute the $\mathfrak{gl}$ weight system on the projection of $K_n$ to the primitive space. Result 4.9. Remark 4.10. The Lie algebra $\mathfrak{gl}_N$ is not simple. Instead, it is a direct sum of a commutative one-dimensional Lie algebra and a simple Lie algebra $\mathfrak{sl}_N$. The one-dimensional commutative Lie subalgebra in $\mathfrak{gl}_N$ consists of scalar matrices, which are $\mathbb{C}$-multiples of the identity matrix. Therefore, the centre $ZU(\mathfrak{gl}_N)$ of the universal enveloping algebra of $\mathfrak{gl}_N$ is the tensor product of the centre of the universal enveloping algebra of $\mathbb{C}$ and that of $\mathfrak{sl}_N$, whence the ring of polynomials in the first Casimir $C_1$ with coefficients in $ZU(\mathfrak{sl}_N)$. Therefore, the values of the weight system $w_{\mathfrak{sl}_N}$ can be computed from that of $w_{\mathfrak{gl}_N}$ by setting $C_1=0$. In the result, $C_2,C_3,\dots$ denote the projections of the corresponding Casimir elements in $ZU(\mathfrak{gl}_N)$ to $ZU(\mathfrak{sl}_N)$. § 5. Symmetric functions and Harish-Chandra isomorphismIn this section, we make use of the Harish-Chandra isomorphism for the Lie algebras $\mathfrak{gl}_N$ to compute the corresponding weight systems. Definition 5.1 (algebra of shifted symmetric polynomials). For a positive integer $N$, the algebra $\Lambda^*(N)$ of shifted symmetric polynomials in $N$ variables $x_1, x_2,\dots,x_N$ consists of polynomials that are invariant under changes of variables for all $i=1,\dots,N-1$. Equivalently, this is the algebra of symmetric polynomials in the shifted variables $(x_1-1,x_2-2,\dots,x_N-N)$. The universal enveloping algebra $U(\mathfrak{gl}_N)$ of the Lie algebra $\mathfrak{gl}_N$ admits the direct sum decomposition:
$$
\begin{equation}
U(\mathfrak{gl}_N)=(\mathfrak{n}_-U(\mathfrak{gl}_N)+U(\mathfrak{gl}_N)\mathfrak{n}_+)\oplus U(\mathfrak{h}),
\end{equation}
\tag{2}
$$
where $\mathfrak{n}_-$ and $\mathfrak{n}_+$ are the nilpotent subalgebras of, respectively, upper and lower triangular matrices in $\mathfrak{gl}_N$, and $\mathfrak{h}$ is the subalgebra of diagonal matrices. Definition 5.2 (Harish-Chandra projection in $U(\mathfrak{gl}_N)$). The Harish-Chandra projection for $U(\mathfrak{gl}_N)$ is the projection to the second summand in (2): where $E_{11},\dots,E_{NN}$ are the diagonal matrix units in $\mathfrak{gl}_N$; they commute with one another. Theorem 5.3 (Harish-Chandra isomorphism [15], [16]). The Harish-Chandra projection, when restricted to the centre $ZU(\mathfrak{gl}_N)$, is an algebra isomorphism to the algebra $\Lambda^*(N)\subset U(\mathfrak{h})$ of shifted symmetric polynomials in $E_{11},\dots,E_{NN}$. Thus, the computation of the value of the $\mathfrak{gl}_N$ weight system on a chord diagram can be elaborated by applying the Harish-Chandra projection to each monomial of the polynomial. For such a monomial, the projection can be computed by moving variables $E_{ij}$ with $i>j$ to the left, and/or variables $E_{ij}$ with $i<j$ to the right by means of applying the commutator relations. If, in the process, we obtain monomials in $\mathfrak{n}_-U(\mathfrak{gl}_N)$ or $U(\mathfrak{gl}_N)\mathfrak{n}_+$, then we replace such a monomial with $0$. A monomial in the (mutually commuting) variables $E_{ii}$ cannot be simplified, and its projection to $U(\mathfrak{h})$ coincides with itself. The resulting polynomial in $E_{11},\dots,E_{NN}$ will be automatically shifted symmetric. Example 5.4. Let us compute the projection of the quadratic Casimir element to $U(\mathfrak{h})$. We have In this expression, the first and the third summand depend on the diagonal unit elements $E_{ii}$ only, while the second summand is in $\mathfrak{n}_-U(\mathfrak{gl}_N)+U(\mathfrak{gl}_N)\mathfrak{n}_+$, whence the image under the projection is Similarly to the ring of ordinary symmetric functions, the ring $\Lambda^*(N)$ of shifted symmetric functions in $N$ variables is isomorphic to a polynomial ring in $N$ variables. There is a variety of convenient $N$-tuples of generators in $\Lambda^*(N)$. One of them is the tuple of shifted power sum polynomials
$$
\begin{equation*}
p_k=\sum_{i}\biggl(\biggl(E_{ii}+\frac{N+1}2-i\biggr)^k-\biggl(\frac{N+1}2-i\biggr)^k\biggr).
\end{equation*}
\notag
$$
Representing $\phi(C_2)$ in the form
$$
\begin{equation*}
\phi(C_2)=\sum_{i}\biggl(\biggl(E_{ii}+\frac{N+1}{2}-i\biggr)^2 -\biggl(\frac{N+1}{2}-i\biggr)^2\biggr),
\end{equation*}
\notag
$$
we see that it is just $p_2$. Remark 5.5. Since the Harish-Chandra isomorphism can be applied to arbitrary elements of $ZU(\mathfrak{gl}_N)$, we can also apply it to the values of $w_\mathfrak{gl}$ on permutations. For $k>2$, the expression for $\phi(C_k)$ is not reduced to just linear combinations of power sums. If fact, we have the following explicit formula, which follows from [17], § 60 in [15], and Remark 2.1.20 in [18],
$$
\begin{equation*}
\begin{aligned} \, &1-Nu-\sum_{k=1}^\infty \phi(C_{k})u^{k+1}=\prod_{i=1}^N\frac{1-(E_{ii}+N-i+1)u}{1-(E_{ii}+N-i)u} \\ &\qquad=(1-Nu)\exp\biggl\{\sum_{k=1}^\infty\frac1{k}\biggl(\biggl(1-\frac{N-1}{2}u\biggr)^{-k} -\biggl(1-\frac{N+1}{2}u\biggr)^{-k}\biggr)u^kp_k \biggr\}. \end{aligned}
\end{equation*}
\notag
$$
This provides an expression for the image $\phi(C_k)$ of $C_k$ as a polynomial in $p_1,p_2,\dots$, which is valid for all $N$. The projections of the Casimir elements $C_1,\dots,C_N$ to $U(\mathfrak{h})$ can be expressed in shifted power sums $p_1,\dots,p_N$ in the following way:
$$
\begin{equation*}
\begin{aligned} \, \phi(C_1)&=p_1, \\ \phi(C_2)&=p_2, \\ \phi(C_3)&=-\frac{1}{4} N^2 p_1+\frac{N p_2}{2}+\frac{p_1}{4}+p_3-\frac{p_1^2}{2}, \\ \phi(C_4)&=-\frac{1}{4} N^3 p_1+N \biggl(-\frac{p_1^2}{2}+\frac{p_1}{4}+p_3\biggr)-p_1 p_2+\frac{p_2}{2}+p_4, \\ &\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots \end{aligned}
\end{equation*}
\notag
$$
Computations using the Harish-Chandra isomorphism are also elaborative, and the results they produce are not universal, they depend on $N$. It is more efficient, therefore, to substitute the known values $\phi(C_k)$ into the answers obtained by the previous method. For the values of the $\mathfrak{gl}$ weight system on the chord diagrams $K_n$, this yields the following result. Result 5.6. § 6. Hopf algebra structure and projection to primitivesMultiplicative weight systems often become simpler when restricted to primitive elements in the Hopf algebra of chord diagrams. This is true, in particular, for the weight systems associated with metrized Lie algebras. The degree of the value of such a weight system $w_\mathfrak{g}$ on a chord diagram with $n$ chords is $2n$, while for the projection of the chord diagram to primitives, it is at most $n$, see [1]. In many cases, knowing the value of a weight system on projections to primitives allows one to understand its structure. In this section, we recall the Hopf algebra structure on the algebra of chord diagrams modulo $4$-term relations, and discuss the values of $w_\mathfrak{gl}$ on projections of chord diagrams to primitives. Definition 6.1. The coproduct $\Delta$ of a chord diagram $D$ is defined by where the summation is taken over all subsets $J$ of the set $[D]$ of chords of $D$. Here $D_J$ is the chord subdiagram of $D$ consisting of the chords that belong to $J$ and $\overline{J} = [D] \setminus J$ is the complementary subset of chords. Claim 6.2. The algebra of chord diagrams modulo $4$-term relations endowed with the above coproduct is a graded commutative, cocommutative and connected Hopf algebra. Definition 6.3. An element $p$ of a Hopf algebra is called primitive if $\Delta(p) = 1 \otimes p + p \otimes 1$. The Milnor–Moore theorem, when applied to the Hopf algebra of chord diagrams, asserts that this Hopf algebra admits a decomposition into the direct sum of the subspace of primitive elements and the subspace of decomposable elements (polynomials in primitive elements of smaller degree). There exists, therefore, a natural projection from the space of chord diagrams to the subspace of primitive elements, whose kernel is the subspace of decomposable elements. We denote this projection by $\pi$. Theorem 6.4 (see [19], [20]). The projection $\pi(D)$ of a chord diagram $D$ to the subspace of primitive elements is given by the formula In particular, the chord diagrams $K_n$ generate a graded Hopf subalgebra in the Hopf algebra of chord diagrams (since any chord subdiagram of $K_n$ is $K_k$, for some $k$). Rewriting the formula for the projection for the exponential generating series $1+\sum_{n=1}^\infty K_nx^n/n!$ we obtain the following corollary. Corollary 6.5. The generating series for the projections $\pi(K_n)$ to the subspace of primitive elements is given by the formula Now, knowing the values of the $\mathfrak{gl}$ weight system on the diagrams $K_n$ for $n=1,2,\dots,7$, we easily obtain the values $\overline{w}_\mathfrak{gl}=w_\mathfrak{gl}\circ\pi$ on their projections to primitives. Result 6.6. In the basis $p_1,p_2,\dots$ of shifted power series, these formulas look simpler. Result 6.7.
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