September 3, 2024

Projectors on invariant subspaces of representations
$\operatorname{ad}^{\otimes 2}$ of Lie algebras $so(N)$ and $sp(2r)$ and Vogel
parametrization

A. P. Isaeva,b, A. A. Provorova,c

a Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear
Research, Dubna, Moscow region, Russia

b Faculty of Physics, M. V. Lomonosov Moscow State University, Moscow, Russia

c Moscow Institute of Physics and Technology (National Research University),
Dolgoprudny, Moscow Region, Russia

<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Explicit formulae for the projectors onto invariant subspaces in the space of
$\operatorname{ad}^{\otimes 2}$ representation of the Lie algebras $so(N)$ and
$sp(2r)$ have been found by means of the split Casimir operator. These
projectors and characteristic identities for the split Casimir operator were
considered from the viewpoint of the universal description of simple Lie
algebras by using the Vogel parametrisation.

Keywords: invariant subspace, projector, simple Lie algebra, split Casimir
operator, Vogel parameters.

##  1 Introduction

In modern theoretical physics the Yang-Baxter equations are considered to be
one of the most important objects to examine. This equation initially appeared
in works of J. B. McGuire [1] and C. N. Yang [2] and plays a key role in the
study of quantum integrable systems [3], [4]. In particular, within the
quantum inverse scattering method [5] new structures appeared, which later led
to creation of the theory of quantum groups [6], [7] (see also Refs. therein)
that describe symmetries of quantum integrable systems (see e.g. [8]–[10]).
Note that the Yang-Baxter equations are used to formulate quantum groups, as
well as to reveal their properties [11]–[13]. An important class of solutions
of the Yang-Baxter equations consists of those solutions that are invariant
under the action of a Lie group $G$ (or its Lie algebra $\mathcal{A}$) in a
particular representation $T$. Let $V$ be the representation space of $T$. In
this case a solution of the Yang-Baxter equation (the $R$-matrix) is an
operator in the representation space $V\otimes V$ of $T$. Assume that the
representation $T\otimes T$ is completely reducible and is decomposed into the
irreducible representations $T_{\lambda}$ as follows: $T\otimes
T=\sum_{\lambda}T_{\lambda}$, where the index $\lambda$ numerates the
irreducible representations. Then the required solutions of the Yang-Baxter
equation can be expanded as a sum of the projectors onto the invariant
subspaces $V_{\lambda}\subset V\otimes V$ of the representations $T_{\lambda}$
with some coefficients being functions of spectral parameters. In order to
find explicitly such solutions it is useful to have exact expressions for the
projectors onto the subspaces $V_{\lambda}$.

In the case when $T=\operatorname{ad}$ is the adjoint representation, we
construct the projectors onto the invariant subspaces of $T\otimes
T=\operatorname{ad}\otimes\operatorname{ad}$ (our approach is closely related
to the approach of [15], where the diagram technique is used). The
construction of the projectors onto the invariant subspaces of the
representation $T\otimes T=\operatorname{ad}\otimes\operatorname{ad}$ has one
more significance. It is related to the notion of the universal Lie algebra,
which was introduced by P. Vogel in [16] (see also [17], [18]; see also
historical remarks in [15], Sect.21.2). The universal Lie algebra was supposed
to be a model of all complex simple Lie algebras $\mathcal{A}$. For example,
many quantities that characterise an algebra $\mathcal{A}$ in different
representations $T_{\lambda}$ (in this case possibly reducible), that
participate in the decomposition, $\operatorname{ad}^{\otimes
k}=\sum_{\lambda}T_{\lambda}$, where $k\geq 1$, can be expressed analytically
as functions of the three Vogel parameters (see their definition in Section
5). These parameters take specific values for each of the complex simple Lie
algebras $\mathcal{A}$ (see e.g. [19], and Section 5 below). In particular it
was shown, that using the Vogel parameters one can express the dimensions of
the representations $T_{\lambda}$ in the cases of $k=2,3$ [16], the dimension
of an arbitrary representation $T_{\lambda^{\prime}}$ with the highest weight
$\lambda^{\prime}=k\lambda_{\operatorname{ad}}$, where
$\lambda_{\operatorname{ad}}$ is the highest root of a Lie algebra
$\mathcal{A}$ [20], as well as values of the higher Casimir operators in the
adjoint represenation of a Lie algebra $\mathcal{A}$ [19]. Besides, in paper
[21] it was shown, that the universal description of complex simple Lie
algebras allows to formulate some types of knot polynomials as a single
function for all the simple Lie algebras simultaneously. In our paper we also
demonstrate a particular example of the universal Lie algebra description.
Namely, the projectors onto invariant subspaces of the tensor product of two
adjoint representations of the Lie algebra $so(N,\mathbb{C})$ for $N\geq 3$
and $sp(N,\mathbb{C})$ for $N=2r\geq 2$ are written in a unified form that
illustrates a correspondence between some structures related to
$so(N,\mathbb{C})$ and $sp(N,\mathbb{C})$ Lie algebras. The correspondence is
given by substitution $N\to-N$ (see [15] and references therein). These
results having been expressed in terms of the Vogel parameters, completely
agree with consideration in papers [16], [19], [20].

##  2 A definition of the algebras $so(N,\mathbb{C})$ and $sp(N,\mathbb{C})$

In this Section, to fix notation, we give the well known definition of the Lie
algebras $so(N)$ and $sp(2n)$ which one can find in many monographs and
textbooks (see e.g. [14], [15] and [22]). We prefer to give a natural unified
definition of these algebras since it will be usefull for us in next Sections.

In order to describe the Lie algebras $so(N,\mathbb{C})$ and
$sp(N,\mathbb{C})$ uniformly, let us introduce the space $V_{N}$ of their
defining representations. The metric $||c_{ij}||_{i,j=1,\dots,N}$ on $V_{N}$
is defined to be the identity $(N\times N)$ matrix $I_{N}$ in the case of
algebra $so(N,\mathbb{C})$ and the antisymmetric $(N\times N)$ matrix

$\|c_{ij}\|=\begin{pmatrix}\hphantom{-}0&I_{r}\\\ -I_{r}&0\end{pmatrix}$ (2.1)

in the case of algebra $sp(N,\mathbb{C})$, where $N=2r$ is an even number.

We have thus $c_{ij}=\epsilon c_{ji}$, where the parameter $\epsilon$ takes
the value $+1$ in the case of algebra $so(N,\mathbb{C})$, and $-1$ in the case
of algebra $sp(N,\mathbb{C})$ (from where we have $\epsilon^{2}=1$). The
inverse matrix $\bar{c}\,^{ij}$ of the metric $c$ is defined as follows:
$\bar{c}\,^{ik}c_{kj}=\delta^{i}_{j}$. With the help of the matrices $c$ and
$\bar{c}\,$ one can raise and lower indices:
$z_{i_{1}}{}^{j_{2}j_{3}\dots}=c_{i_{1}j_{1}}z^{j_{1}j_{2}j_{3}\dots}$ and
$z^{j_{1}}{}_{i_{2}i_{3}\dots}=\bar{c}\,^{j_{1}i_{1}}z_{i_{1}i_{2}i_{3}\dots}$.

Using the matrix identities
$(e_{s}{}^{r})^{i}{}_{k}=\delta^{r}_{k}\delta^{i}_{s}$, which form a basis of
the algebra of linear operators on $V_{N}$, one can write bases of both
$so(N,\mathbb{C})$ and $sp(N,\mathbb{C})$ algebras in the defining
representation. Lowering the index $r$ yields
$(e_{sr})^{i}{}_{k}=c_{rk}\delta^{i}_{s}$. In these terms the generators of
the Lie algebras $so(N,\mathbb{C})$ and $sp(N,\mathbb{C})$ can be expressed as
follows:

$\displaystyle M_{ij}=e_{ij}-\epsilon e_{ji},$ (2.2)
$\displaystyle(M_{ij})^{k}{}_{l}=c_{jl}\delta^{k}_{i}-\epsilon
c_{il}\delta^{k}_{j}=2\delta_{[i}^{k}c_{j)l},$ (2.3)

where the notation $[ij)$ implies antisymmetrization in the case of algebra
$so(N,\mathbb{C})$ and symmetrization in the case of algebra
$sp(N,\mathbb{C})$. The commutation relation of both algebras are given by the
following formula:

$[M_{ij},M_{kl}]=c_{jk}M_{il}-\epsilon c_{ik}M_{jl}-\epsilon
c_{jl}M_{ik}+c_{il}M_{jk}=X_{ij,kl}{}^{mn}M_{mn},$ (2.4)

from where we can get the structure constants of the Lie algebras under
consideration in the basis (2.2):

$X_{ij,kl}{}^{mn}=c_{jk}\delta_{i}^{[m}\delta_{l}^{n)}-\epsilon
c_{ik}\delta_{j}^{[m}\delta_{l}^{n)}-\epsilon
c_{jl}\delta_{i}^{[m}\delta_{k}^{n)}+c_{il}\delta_{j}^{[m}\delta_{k}^{n)}.$
(2.5)

They can also be rewritten in a more concise manner:

$X_{i_{1}i_{2},j_{1}j_{2}}{}^{k_{1}k_{2}}=4\operatorname{Sym}^{\epsilon}_{1\leftrightarrow
2}(c_{i_{2}j_{1}}\delta_{i_{1}}^{k_{1}}\delta_{j_{2}}^{k_{2}}),$ (2.6)

where $\operatorname{Sym}^{\epsilon}_{1\leftrightarrow 2}$ denotes
(anti)symmetrization over the following pairs of indices $(i_{1},i_{2})$,
$(j_{1},j_{2})$, $(k_{1},k_{2})$. For example,
$\operatorname{Sym}^{\epsilon}_{1\leftrightarrow 2}(x_{i_{1}i_{2}})\equiv
x_{i_{1}i_{2}}-\epsilon x_{i_{2}i_{1}}$. Hereinafter we will be using the
designation $\mathfrak{g}_{N}^{\epsilon}$ to mean the algebra
$so(N,\mathbb{C})$ whenever $\epsilon=+1$ and $sp(N,\mathbb{C})$ if
$\epsilon=-1$.

Let us note that (anti)symmetrized pairs of indices $(i_{1},i_{2})$,
$(j_{1},j_{2})$ and $(k_{1},k_{2})$, being the indices of the basis vectors
$M_{i_{1}i_{2}}$ of the algebra $\mathfrak{g}_{N}^{\epsilon}$, can also be
viewed as coordinate indices in the space of the adjoint representation of the
algebra $\mathfrak{g}_{N}^{\epsilon}$.

##  3 The split Casimir operator

In this section we will describe a general procedure of building the
projectors onto invariant subspaces of the representation
$\operatorname{ad}^{\otimes 2}(\mathcal{A})$ of a complex simple Lie algebra
$\mathcal{A}$ by using the algebra’s split Casimir operator $\widehat{C}$. We
will also find explicit formulae for the operator $\widehat{C}$ in the
representation $\operatorname{ad}^{\otimes 2}(\mathcal{A})$ when
$\mathcal{A}=so(N,\mathbb{C})$ and $\mathcal{A}=sp(N,\mathbb{C})$.

###  3.1 The split Casimir operator in highest weight representations

Let $\mathcal{A}$ be a simple Lie algebra with the basis elements $X_{a}$ and
the structure relations

$[X_{a},X_{b}]=C^{d}_{ab}X_{d},$ (3.1)

where $C^{d}_{ab}$ are the structure constants. The Cartan-Killing metric is
defined in the standard fashion:

${\sf g}_{ab}\equiv
C^{d}_{ac}\,C^{c}_{bd}=\operatorname{tr}(\operatorname{ad}(X_{a})\cdot\operatorname{ad}(X_{b})),$
(3.2)

and the structure constants $C_{abc}\equiv C^{d}_{ab}\,{\sf g}_{dc}$ are
antisymmetric in the indices $(a,b,c)$. $\mathcal{U}(\mathcal{A})$ denotes the
universal enveloping algebra of $\mathcal{A}$. Consider the operator

$\widehat{C}={\sf g}^{ab}X_{a}\otimes
X_{b}\in\mathcal{A}\otimes\mathcal{A}\subset\mathcal{U}(\mathcal{A})\otimes\mathcal{U}(\mathcal{A}),$
(3.3)

where the matrix $\|{\sf g}^{ab}\|$ is the inverse matrix of $\|{\sf
g}_{ab}\|$, which, in turn, defines the Cartan-Killing metric (3.2):

${\sf g}^{ab}{\sf g}_{bc}=\delta^{a}_{c}.$ (3.4)

The operator $\widehat{C}$ is called the split (or polarised) Casimir operator
of the Lie algebra $\mathcal{A}$. This operator is related to the usual
quadratic Casimir operator

$C_{(2)}={\sf g}^{ab}X_{a}\cdot X_{b}$ (3.5)

according to the formula

$\Delta(C_{(2)})=C_{(2)}\otimes I+I\otimes C_{(2)}+2\,\widehat{C},$ (3.6)

where $\Delta$ is the comultiplication:

$\Delta(X_{a})=(X_{a}\otimes I+I\otimes X_{a}).$ (3.7)

Let $T_{1}$ and $T_{2}$ be two irreducible representations the highest weights
of which are $\lambda_{1}$ and $\lambda_{2}$, and which act on the spaces
$\mathcal{V}_{1}$ and $\mathcal{V}_{2}$. Then, using the formula (3.6) and the
decomposition

$\mathcal{V}_{1}\otimes\mathcal{V}_{2}=\sum_{\lambda}\mathcal{V}_{\lambda}$
(3.8)

where $\mathcal{V}_{\lambda}$ are the subspaces of the irreducible
representations $T_{\lambda}$ with the highest weights $\lambda$, we can
deduce the following relations:

$\displaystyle T(\widehat{C})\cdot(\mathcal{V}_{1}\otimes\mathcal{V}_{2})={}$
$\displaystyle\frac{1}{2}\sum_{\lambda}(c_{2}^{(\lambda)}-c_{2}^{(\lambda_{1})}-c_{2}^{(\lambda_{2})})\mathcal{V}_{\lambda}\quad\Longleftrightarrow$
(3.9) $\displaystyle\Longleftrightarrow\quad
T(\widehat{C})\cdot\mathcal{V}_{\lambda}=\frac{1}{2}(c_{2}^{(\lambda)}-c_{2}^{(\lambda_{1})}-c_{2}^{(\lambda_{2})})\mathcal{V}_{\lambda}.$

Here we used the following notation $T(\widehat{C}):=(T_{1}\otimes
T_{2})\widehat{C}$ and utilized the equalities111The first of those equalities
is a consequence of the fact, that for each $U\in\mathcal{U}(\mathcal{A})$
$(T_{1}\otimes T_{2})\Delta(U)$ commutes with the projectors $P_{\lambda}$,
which distinguish the irreducible subrepresentations of $(T_{1}\otimes T_{2})$
and hence are $\operatorname{Ad}$-invariant operators.

$\Delta(C_{(2)})\sum_{\lambda}\mathcal{V}_{\lambda}=\sum_{\lambda}c_{2}^{(\lambda)}\mathcal{V}_{\lambda},\qquad(C_{(2)}\otimes
I+I\otimes
C_{(2)})\mathcal{V}_{1}\otimes\mathcal{V}_{2}=(c_{2}^{(\lambda_{1})}+c_{2}^{(\lambda_{2})})\mathcal{V}_{1}\otimes\mathcal{V}_{2}.$

Here $c_{2}^{(\lambda)}$ is the value of the quadratic Casimir operator in the
representation with the highest weight $\lambda$,

$c_{2}^{(\lambda)}=(\lambda,\lambda+2\,\delta),\quad\quad\delta=\sum_{f=1}^{r}\lambda_{(f)}=\frac{1}{2}\sum_{\alpha>0}\alpha,$
(3.10)

$\lambda_{(f)}$ being the fundamental weights of the rank-$r$ Lie algebra
$\mathcal{A}$, $\alpha$ being the roots of $\mathcal{A}$, and the summation
being carried out over the positive roots $(\alpha>0)$. The metric in the root
space is given by the matrix (3.4). Note also, that the formula (3.9) can be
used to derive the characteristic identity

${\prod_{\lambda}}^{\,\prime}\biggl{(}T(\widehat{C})-\frac{1}{2}(c_{2}^{(\lambda)}-c_{2}^{(\lambda_{1})}-c_{2}^{(\lambda_{2})})\biggr{)}=0,$
(3.11)

where the product ${\prod}^{\,\prime}$ is performed only over those weights
$\lambda$ in (3.8), which correspond to different eigenvalues
$c_{2}^{\lambda}$.

In the next sections we will find explicit expressions for the split Casimir
operator $T(\widehat{C})=(T_{1}\otimes T_{2})(\widehat{C})$ for orthogonal and
symplectic Lie algebras in the case of $T_{1}$ and $T_{2}$ being the adjoint
representations: $T_{1}=T_{2}=\operatorname{ad}$. Then the characteristic
identity (3.11) can be rewritten in the form

${\prod_{\lambda}}^{\,\prime}\biggl{(}\operatorname{ad}^{\otimes
2}(\widehat{C})-\frac{1}{2}(c_{2}^{(\lambda)}-2c_{2}^{(\lambda_{\operatorname{ad}})})\biggr{)}=0,$
(3.12)

where $\lambda_{\operatorname{ad}}$ is the highest weight of the adjoint
representation of $\mathcal{A}$. Then, using the characteristic identity
(3.12), we will explicitly build (for orthogonal and symplectic Lie algebras)
the projectors on the invariant subspaces $\mathcal{V}_{\lambda}$ of the
irreducible representations $T_{\lambda}$, comprising the decomposition of
$\operatorname{ad}^{\otimes 2}$.

###  3.2 The split Casimir operator of the algebras $so(N,\mathbb{C})$ and
$sp(N,\mathbb{C})$ in the tensor product of adjoint representations

Let us show that the components of the split Casimir operator (3.3) in the
adjoint representation $\operatorname{ad}$ of the Lie algebras
$so(N,\mathbb{C})$ and $sp(N,\mathbb{C})$ can be written via the components of
the same operator in the defining representation.

The Cartan-Killing metric of the algebra (2.4) with the structure constants
(2.5) is given by the following relations [22]:

${\sf
g}_{i_{1}i_{2},j_{1}j_{2}}=X_{i_{1}i_{2},\ell_{1}\ell_{2}}{}^{k_{1}k_{2}}X_{j_{1}j_{2},k_{1}k_{2}}{}^{\ell_{1}\ell_{2}}=2(N-2\epsilon)(c_{i_{2}j_{1}}c_{j_{2}i_{1}}-\epsilon
c_{i_{1}j_{1}}c_{j_{2}i_{2}}).$ (3.13)

Thus, the inverse Cartan-Killing metric is of the form:

${\sf
g}^{i_{1}i_{2},j_{1}j_{2}}=\frac{1}{8(N-2\epsilon)}(\epsilon\bar{c}\,^{i_{1}j_{2}}\bar{c}\,^{i_{2}j_{1}}-\bar{c}\,^{i_{1}j_{1}}\bar{c}\,^{i_{2}j_{2}}).$
(3.14)

In what follows we will utilize the following notation for the operator
$\widehat{C}$ (that was defined in (3.3)) in the adjoint representations of
the algebras $so(N,\mathbb{C})$ and $sp(N,\mathbb{C})$:
$\operatorname{ad}^{\otimes
2}(\widehat{C})\equiv\widehat{C}_{\operatorname{ad}}$. In the basis (2.4) with
the structure constants (2.5) we have:

$\displaystyle(\widehat{C}_{\operatorname{ad}})^{k_{1}k_{2}k_{3}k_{4}}_{j_{1}j_{2}j_{3}j_{4}}$
$\displaystyle={\sf
g}^{i_{1}i_{2}i_{3}i_{4}}\,X_{i_{1}i_{2},j_{1}j_{2}}{}^{k_{1}k_{2}}X_{i_{3}i_{4},j_{3}j_{4}}{}^{k_{3}k_{4}}=$
$\displaystyle=16{\sf
g}^{i_{1}i_{2}i_{3}i_{4}}\operatorname{Sym}^{\epsilon}_{1\leftrightarrow
2}(c_{i_{2}j_{1}}\delta^{k_{1}}_{i_{1}}\delta^{k_{2}}_{j_{2}})\operatorname{Sym}^{\epsilon}_{3\leftrightarrow
4}(c_{i_{4}j_{3}}\delta^{k_{3}}_{i_{3}}\delta^{k_{4}}_{j_{4}}),$ (3.15)

where $\operatorname{Sym}_{\alpha\leftrightarrow\beta}$, as earlier, denotes
the (anti)symmetrizator over the pairs of indices $(i_{\alpha},i_{\beta})$,
$(j_{\alpha},j_{\beta})$ and $(k_{\alpha},k_{\beta})$. In these relations
those indices, that were marked with the subindices $1$ and $2$ correspond to
the first copy of the algebra $\mathfrak{g}_{N}^{\epsilon}$, while those
marked with $3$ and $4$ relate to the second copy of
$\mathfrak{g}_{N}^{\epsilon}$ (in accordance with the fact that the basis
elements of the algebras $so(N,\mathbb{C})$ and $sp(N,\mathbb{C})$ have two
indices each and can be embedded into $V_{N}\otimes V_{N}$ as vector spaces).

The split Casimir operator $\widehat{C}$ in the defining representation
$T_{f}^{\otimes 2}(\widehat{C})\equiv\widehat{C}_{f}\colon V_{N}\otimes
V_{N}\to V_{N}\otimes V_{N}$ in the basis (2.2) with the structure constants
(2.5) has the following form:

$\displaystyle(\widehat{C}_{f})^{k_{1}k_{3}}_{j_{1}j_{3}}$ $\displaystyle={\sf
g}^{i_{1}i_{2}i_{3}i_{4}}(M_{i_{1}i_{2}})^{k_{1}}{}_{j_{1}}(M_{i_{3}i_{4}})^{k_{3}}{}_{j_{3}}=4{\sf
g}^{i_{1}i_{2}i_{3}i_{4}}(\delta^{k_{1}}_{[i_{1}}c_{i_{2})j_{1}})(\delta^{k_{3}}_{[i_{3}}c_{i_{4})j_{3}}).$
(3.16)

From the expressions (3.15) and (3.16) one can conclude that

$(\widehat{C}_{\operatorname{ad}})^{k_{1}k_{2}k_{3}k_{4}}_{j_{1}j_{2}j_{3}j_{4}}=4\operatorname{Sym}^{\epsilon}_{1\leftrightarrow
2,3\leftrightarrow
4}\left((\widehat{C}_{f})^{k_{1}k_{3}}_{j_{1}j_{3}}\delta^{k_{2}}_{j_{2}}\delta^{k_{4}}_{j_{4}}\right),$
(3.17)

where $\operatorname{Sym}^{\epsilon}_{1\leftrightarrow 2,3\leftrightarrow 4}$
means (anti)symmetrization over the four pairs of indices: $(j_{1},j_{2})$,
$(k_{1},k_{2})$, $(j_{3},j_{4})$, $(k_{3},k_{4})$. Here the indices are
interpreted as the indices in the space $V_{N}$ of the defining representation
of $so(N,\mathbb{C})$ or $sp(N,\mathbb{C})$. Correspondingly, the split
Casimir operator acts on the space $V_{N}^{\otimes 2}$ in the defining
representation, and on the space $V_{N}^{\otimes 4}$ in the adjoint
representation. Here the first and the last pair of spaces $V_{N}$ in
$V_{N}^{\otimes 4}$ are antisymmetrized in the case of $so(N,\mathbb{C})$ and
symmetrized in the case of $sp(N,\mathbb{C})$. Let us introduce the operator
of (anti)symmetrization of the $a$’th and $b$’th spaces:

$\mathcal{P}_{ab}^{\epsilon}=\frac{1}{2}(I-\epsilon P_{ab}),$ (3.18)

where $a\neq b$, $a,b=1,\dots,4$, and $P_{ab}\colon V_{N}^{\otimes 4}\to
V_{N}^{\otimes 4}$ is the permutation operator, that acts on the $a$’th and
the $b$’th spaces (e.g. $P_{13}$ has the components
$(P_{13})^{i_{1}i_{2}i_{3}i_{4}}{}_{j_{1}j_{2}j_{3}j_{4}}=\delta^{i_{1}}_{j_{3}}\delta^{i_{3}}_{j_{1}}\delta^{i_{2}}_{j_{2}}\delta^{i_{4}}_{j_{4}}$).
Then for the space $V_{\operatorname{ad}}$ of the adjoint representation, as
well as for $V_{\operatorname{ad}}^{\otimes 2}$ we have

$V_{\operatorname{ad}}=\mathcal{P}^{\epsilon}_{12}V_{N}^{\otimes 2},\qquad
V_{\operatorname{ad}}\otimes
V_{\operatorname{ad}}=\mathcal{P}^{\epsilon}_{12}\mathcal{P}^{\epsilon}_{34}V_{N}^{\otimes
4}.$

Moreover, we can express the relation (3.17) between the Casimir operators in
the defining and the ajoint representations in the following fashion:

${\widehat{C}_{\operatorname{ad}}}=4\;\mathcal{P}_{12}^{\epsilon}\mathcal{P}_{34}^{\epsilon}(\widehat{C}_{f})_{13}\mathcal{P}_{12}^{\epsilon}\mathcal{P}_{34}^{\epsilon}.$
(3.19)

From (3.16) one can derive the following expression for the split Casimir
operator in the defining representation (see e.g. [22])

$(\widehat{C}_{f})_{13}=\frac{1}{2(N-2\epsilon)}(P_{13}-\epsilon K_{13}),$
(3.20)

which is to be inserted into (3.19). Here the operators
$(\widehat{C}_{f})_{13}$, $K_{13}$ and $P_{13}$ act nontrivially only on the
first and third spaces $V_{N}$ in $V_{N}^{\otimes 4}$, the operator $K_{13}$
has the following components:
$K^{i_{1}i_{2}i_{3}i_{4}}{}_{j_{1}j_{2}j_{3}j_{4}}=\bar{c}\,^{i_{1}i_{3}}c_{j_{1}j_{3}}\delta^{i_{2}}_{j_{2}}\delta^{i_{4}}_{j_{4}}$,
and the operator $P_{13}$, as it has already been noted, permutes the first
and the third spaces in $V_{N}^{\otimes 4}$. Finally we have for
$\widehat{C}_{\operatorname{ad}}$:

$\widehat{C}_{\operatorname{ad}}=\frac{2}{N-2\epsilon}\,\mathcal{P}_{12}^{\epsilon}\mathcal{P}_{34}^{\epsilon}\;(P_{13}-\epsilon
K_{13})\;\mathcal{P}_{12}^{\epsilon}\mathcal{P}_{34}^{\epsilon}.$ (3.21)

##  4 The decomposition of $\operatorname{ad}^{\otimes
2}(\mathfrak{g}_{N}^{\epsilon})$ into the irreducible representations

In this section we will use the split Casimir operator in the adjoint
representation to build projectors onto invariant subspaces of the
representation $\operatorname{ad}^{\otimes 2}(\mathfrak{g}_{N}^{\epsilon})$.
Taking into account existence of the ‘arbitrary’ isomorphisms $so(3)\cong
sl(2)\cong sp(2)$, $so(4)\cong sl(2)+sl(2)$, $so(5)\cong sp(4)$, $so(6)\cong
sl(4)$, we will consider the cases of the aforementioned algebras, apart from
$so(5)\cong sp(4)$, at the end of the chapter (the algebras $so(5)\cong sp(4)$
fit into the general picture and do not require a separate examination).

###  4.1 The characteristic identity for the operator $\widehat{C}$ in the
adjoint representation of the algebras $so(N,\mathbb{C})$ and
$sp(N,\mathbb{C})$

Let us define the spaces $V_{\operatorname{ad}}^{\epsilon}$ of the adjoint
representation of the algebras $so(N)$ and $sp(N)$ as
$V_{\operatorname{ad}}^{\epsilon}=\mathcal{P}^{(\epsilon)}_{12}V_{N}^{\otimes
2}$. The algebra $\mathfrak{g}_{N}^{\epsilon}$ coincides with
$V_{\operatorname{ad}}^{\epsilon}$ as a vector space. We introduce the
following operators that act in the space
$V_{\operatorname{ad}}^{\epsilon}\otimes
V_{\operatorname{ad}}^{\epsilon}\subset V_{N}^{\otimes 4}$:

$\displaystyle\mathbf{I}=\mathcal{P}^{(\epsilon)}_{12}\mathcal{P}^{(\epsilon)}_{34}\equiv\mathcal{P}_{12,34}^{(\epsilon)},\qquad\mathbf{P}=\mathcal{P}_{12,34}^{(\epsilon)}P_{13}P_{24}\mathcal{P}_{12,34}^{(\epsilon)},$
(4.1)
$\displaystyle\mathbf{K}=\mathcal{P}_{12,34}^{(\epsilon)}K_{13}K_{24}\mathcal{P}_{12,34}^{(\epsilon)},$

for which the following relations hold:

$\displaystyle\mathbf{I}=\mathbf{I}P_{12}P_{34}=P_{12}P_{34}\mathbf{I},\qquad\qquad\mathbf{P}=P_{12}P_{34}\mathbf{I}=\mathbf{I}P_{13}P_{24},$
(4.2)
$\displaystyle\mathbf{P}^{2}=\mathbf{I},\qquad\quad\mathbf{K}\mathbf{P}=\mathbf{P}\mathbf{K}=\mathbf{K},\qquad\quad\mathbf{K}^{2}=\frac{N(N-\epsilon)}{2}\mathbf{K}$

and

$\widehat{C}_{\operatorname{ad}}\mathbf{P}=\mathbf{P}\widehat{C}_{\operatorname{ad}},\qquad\qquad\qquad\widehat{C}_{\operatorname{ad}}\mathbf{K}=\mathbf{K}\widehat{C}_{\operatorname{ad}}=-\mathbf{K}.$
(4.3)

The operators (4.1) are obviously $\operatorname{ad}$-invariant with respect
to the adjoint action of the algebra $\mathfrak{g}_{N}^{\epsilon}$.
Hereinafter we will use the following notation $M\equiv\epsilon N$, which
allows the last relation in (4.2) to be rewritten in the form:

$\mathbf{K}^{2}=\frac{M(M-1)}{2}\mathbf{K}.$ (4.4)

In order to find the characteristic identity, it is convenient to introduce
the symmetric $\widehat{C}_{+}$ and the antisymmetric $\widehat{C}_{-}$
projectors of the operator $\widehat{C}_{\operatorname{ad}}$:

$\widehat{C}_{\pm}=\frac{1}{2}(\mathbf{I}\pm\mathbf{P})\widehat{C}_{\operatorname{ad}},$
(4.5)

which satisfy the following relations:

$\displaystyle\widehat{C}_{\pm}\widehat{C}_{\mp}=0,\qquad\mathbf{P}\widehat{C}_{\pm}=\pm\widehat{C}_{\pm},\qquad\mathbf{K}\widehat{C}_{+}=\widehat{C}_{+}\mathbf{K}=-\mathbf{K}.$
(4.6)

Substitution of (3.21) into (4.5) yields explicit formulae for the
antisymmetric and the symmetric parts of the operator
$\widehat{C}_{\operatorname{ad}}$:

$\displaystyle(\widehat{C}_{-})_{12,34}=\frac{1}{N-2\epsilon}\mathcal{P}^{(\epsilon)}_{12,34}(P_{24}-\epsilon)K_{13}\mathcal{P}^{(\epsilon)}_{12,34},$
(4.7)
$\displaystyle(\widehat{C}_{+})_{12,34}=\frac{1}{N-2\epsilon}\mathcal{P}^{(\epsilon)}_{12,34}(2P_{24}-P_{24}K_{13}-\epsilon
K_{13})\mathcal{P}^{(\epsilon)}_{12,34}.$ (4.8)

The characteristic identity for $\widehat{C}_{-}$ can be found by direct
calculations with the use of the formula (4.7):

$\widehat{C}_{-}^{2}=-\frac{1}{2}\widehat{C}_{-}\quad\iff\quad\widehat{C}_{-}\biggl{(}\widehat{C}_{-}+\frac{1}{2}\biggr{)}=0.$
(4.9)

Note also a useful consequence of the relation (4.9):

$\widehat{C}_{-}^{k}=\biggl{(}-\frac{1}{2}\biggr{)}^{k-1}\widehat{C}_{-},\qquad
k\geq 1.$ (4.10)

Analogously, one can use the explicit formula (4.8) for $\widehat{C}_{+}$, to
find an expression for $\widehat{C}_{+}^{2}$:

$\widehat{C}_{+}^{2}=\frac{1}{(M-2)^{2}}(\mathbf{I}+\mathbf{P}+\mathbf{K})-\frac{1}{M-2}\,\widehat{C}_{+}+\frac{M-8}{2(M-2)^{2}}\,\mathcal{P}^{(\epsilon)}_{12,34}K_{13}(1+\epsilon
P_{24})\mathcal{P}^{(\epsilon)}_{12,34}.$ (4.11)

If $M=8$, then the last summand in (4.11) equals zero and the identity for
$\widehat{C}_{+}$ takes the following form222The universal form (for all
exceptional Lie algebras ${\cal A}$) of this relation was obtained in [15],
eq. (17.10), in the assumption that there exists no primitive quartic ${\cal
A}$-invariant. Our definition of $\widehat{C}_{+}$ differs from definition of
${\bf Q}$ in [15] by the sign: $\widehat{C}_{+}\to-{\bf Q}$.:

$\widehat{C}_{+}^{2}=-\frac{1}{6}\widehat{C}_{+}+\frac{1}{36}(\mathbf{I}+\mathbf{P}+\mathbf{K}).$
(4.12)

If $M\neq 8$, then by multiplying (4.11) by $\widehat{C}_{+}$, we arrive at an
identity of degree three:

$\widehat{C}_{+}^{3}=-\frac{1}{2}\,\widehat{C}_{+}^{2}-\frac{M-8}{2(M-2)^{2}}\,\widehat{C}_{+}+\frac{M-4}{2(M-2)^{3}}(\mathbf{I}+\mathbf{P}-2\mathbf{K}).$
(4.13)

Since for $M=8$ the identity (4.12) for $\widehat{C}_{+}$ has the degree $2$
rather than $3$, the case $M=8$ is exceptional and will be considered later in
the subsection 4.2. Note also, that when $M=4$ the last summand in (4.13)
vanishes, hence the identity (4.13) in the case of algebra $so(4,\mathbb{C})$
is characteristic for the operator $\widehat{C}_{+}$ and has the following
explicit form:

$\widehat{C}_{+}^{3}=-\frac{1}{2}\widehat{C}_{+}^{2}+\frac{1}{2}\widehat{C}_{+}.$
(4.14)

For the purpose of getting characteristic identities for $\widehat{C}_{+}$
when $M\neq 4,8$ we need to get rid of the operators $\mathbf{P}$ and
$\mathbf{K}$ in the formula (4.13). Multiplying it by $C_{+}$ and using (4.6),
we can express $\mathbf{K}$ in terms of $\widehat{C}_{+}$:

$\frac{M-4}{(M-2)^{3}}\mathbf{K}=\widehat{C}_{+}^{4}+\frac{1}{2}\widehat{C}_{+}^{3}+\frac{M-8}{2(M-2)^{2}}\widehat{C}_{+}^{2}-\frac{M-4}{(M-2)^{3}}\widehat{C}_{+}.$
(4.15)

We then substitute $\mathbf{K}$ from (4.15) into (4.13) and find the following
identity (which also holds in the case $M=4$ as one can check by direct
calculations):

$\displaystyle\widehat{C}_{+}^{4}+\frac{3}{2}\widehat{C}_{+}^{3}+\frac{(M+1)(M-4)}{2(M-2)^{2}}\widehat{C}_{+}^{2}$
$\displaystyle+\frac{M^{2}-12M+24}{2(M-2)^{3}}\widehat{C}_{+}-\frac{M-4}{2(M-2)^{3}}(\mathbf{I}+\mathbf{P})=0.$
(4.16)

Multiplying one more time (4.16) by $\widehat{C}_{+}$ we achieve the following
characteristic identity for $\widehat{C}_{+}$:

$\displaystyle\widehat{C}_{+}^{5}+\frac{3}{2}\,\widehat{C}_{+}^{4}+\frac{(M+1)(M-4)}{2(M-2)^{2}}\,\widehat{C}_{+}^{3}$
$\displaystyle+\frac{M^{2}-12M+24}{2(M-2)^{3}}\,\widehat{C}_{+}^{2}-\frac{M-4}{(M-2)^{3}}\,\widehat{C}_{+}=0,$
(4.17)

which could be also rewritten in the form:

$\displaystyle\widehat{C}_{+}^{5}=-\frac{3}{2}\,\widehat{C}_{+}^{4}-\frac{(M+1)(M-4)}{2(M-2)^{2}}\,\widehat{C}_{+}^{3}$
$\displaystyle-\frac{M^{2}-12M+24}{2(M-2)^{3}}\,\widehat{C}_{+}^{2}+\frac{M-4}{(M-2)^{3}}\,\widehat{C}_{+}.$
(4.18)

For the upcoming calculations we will also need an expression for
$\widehat{C}_{+}^{6}$, which could be gained by multiplying the identity
(4.18) by $\widehat{C}_{+}$ followed by substitution of the known polynomial
for $\widehat{C}_{+}^{5}$ from (4.18):

$\displaystyle\widehat{C}_{+}^{6}={}$
$\displaystyle\frac{7M^{2}-30M+44}{4(M-2)^{2}}\,\widehat{C}_{+}^{4}+\frac{3M^{3}-17M^{2}+30M-24}{4(M-2)^{3}}\,\widehat{C}_{+}^{3}+{}$
$\displaystyle+\frac{3M^{2}-32M+56}{4(M-2)^{3}}\,\widehat{C}_{+}^{2}-\frac{3(M-4)}{2(M-2)^{3}}\,\widehat{C}_{+}.$
(4.19)

Now we will use the acquired expressions to find the characteristic identity
for the split Casimir operator
$\widehat{C}_{\operatorname{ad}}=\widehat{C}_{+}+\widehat{C}_{-}$. We will be
looking for such an expression in the form of a $6$-degree polynomial of
$\widehat{C}_{\operatorname{ad}}$ with the coefficients $\alpha_{i}$:

$\widehat{C}_{\operatorname{ad}}^{6}+\alpha_{5}\widehat{C}_{\operatorname{ad}}^{5}+\alpha_{4}\widehat{C}_{\operatorname{ad}}^{4}+\alpha_{3}\widehat{C}_{\operatorname{ad}}^{3}+\alpha_{2}\widehat{C}_{\operatorname{ad}}^{2}+\alpha_{1}\widehat{C}_{\operatorname{ad}}+\alpha_{0}.$
(4.20)

We need to find such $\alpha_{i}$, for which the expression (4.20) vanishes.
From the formula (4.6) it is clear that
$\widehat{C}^{k}=\widehat{C}_{+}^{k}+\widehat{C}_{-}^{k}$, hence vanishing of
the polynomial (4.20) yields the equation

$\displaystyle\widehat{C}_{+}^{6}$
$\displaystyle+\alpha_{5}\widehat{C}_{+}^{5}+\alpha_{4}\widehat{C}_{+}^{4}+\alpha_{3}\widehat{C}_{+}^{3}+\alpha_{2}\widehat{C}_{+}^{2}+\alpha_{1}\widehat{C}_{+}+{}$
$\displaystyle+\widehat{C}_{-}^{6}+\alpha_{5}\widehat{C}_{-}^{5}+\alpha_{4}\widehat{C}_{-}^{4}+\alpha_{3}\widehat{C}_{-}^{3}+\alpha_{2}\widehat{C}_{-}^{2}+\alpha_{1}\widehat{C}_{-}+\alpha_{0}=0.$

Substituting then the expressions for $\widehat{C}_{+}^{5,6}$ in terms of
$\widehat{C}_{+}^{4,3,2,1}$ according to the formulae (4.18), (4.1) and the
expressions for $\widehat{C}_{-}^{6,5,4,3,2}$ in terms of $\widehat{C}_{-}$ by
the formula (4.10) and setting the coefficients of those operators to zero, we
get the values of $\alpha_{i}$:

$\displaystyle\alpha_{0}=0,$
$\displaystyle\alpha_{1}=-\frac{M-4}{2(M-2)^{3}},$
$\displaystyle\alpha_{2}=\frac{M^{2}-16M+40}{4(M-2)^{3}},$
$\displaystyle\alpha_{3}=\frac{M^{3}-3M^{2}-22M+56}{4(M-2)^{3}},$
$\displaystyle\alpha_{4}=\frac{5M^{2}-18M+4}{4(M-2)^{2}},$
$\displaystyle\alpha_{5}=2.$

The characteristic identity for $\widehat{C}_{\operatorname{ad}}$ thus takes
the form:

$\displaystyle\widehat{C}_{\operatorname{ad}}^{6}+2\widehat{C}_{\operatorname{ad}}^{5}$
$\displaystyle+\frac{5M^{2}-18M+4}{4(M-2)^{2}}\widehat{C}_{\operatorname{ad}}^{4}+\frac{M^{3}-3M^{2}-22M+56}{4(M-2)^{3}}\widehat{C}_{\operatorname{ad}}^{3}+{}$
$\displaystyle+\frac{M^{2}-16M+40}{4(M-2)^{3}}\widehat{C}_{\operatorname{ad}}^{2}-\frac{M-4}{2(M-2)^{3}}\widehat{C}_{\operatorname{ad}}=0.$
(4.21)

The roots of the equation (4.1) can be found explicitly:

$\displaystyle a_{1}=0,\qquad a_{2}=-\frac{1}{2},\qquad a_{3}=-1,\qquad
a_{4}=\frac{1}{M-2},$ (4.22) $\displaystyle a_{5}=-\frac{2}{M-2},\qquad
a_{6}=\frac{4-M}{2(M-2)}.$

Let us note that if $M$ takes one of the following values, some of the roots
are degenerate (we discard here the values $M=0,1,2$ as they do not correspond
to semisimple Lie algebras):

$\displaystyle M=4\quad\implies\quad a_{1}=a_{6}=0,\quad a_{3}=a_{5}=-1,$
(4.23) $\displaystyle M=6\quad\implies\quad a_{2}=a_{5}=-\frac{1}{2},$ (4.24)
$\displaystyle M=8\quad\implies\quad a_{5}=a_{6}=-\frac{1}{3},$ (4.25)

from where we see that if $M=4,6,8$ then the characteristic polynomials do not
have the degree $6$, as it is in the general case, but the degrees $4,5,5$
respectively (this also manifests itself in the differences between the
identities (4.16) and (4.13) when $M=4$ and the identities (4.12) and (4.13)
when $M=8$). The cases $M=4,6$ will be considered in more details at the end
of this section, while the subsection 4.2, as it was pointed out before, will
be devoted to the case $M=8$.

Taking into account the roots (4.22) of the polynomial in the left hand side
of (4.1), the characteristic identity (4.1) can be factorized:

$\widehat{C}_{\operatorname{ad}}\biggl{(}\widehat{C}_{\operatorname{ad}}+\frac{1}{2}\biggr{)}(\widehat{C}_{\operatorname{ad}}+1)\biggl{(}\widehat{C}_{\operatorname{ad}}-\frac{1}{M\kern-1.0pt-2}\biggr{)}\biggl{(}\widehat{C}_{\operatorname{ad}}+\frac{2}{M\kern-1.0pt-2}\biggr{)}\biggl{(}\widehat{C}_{\operatorname{ad}}+\frac{M\kern-1.0pt-4}{2(M\kern-1.0pt-2)}\biggr{)}=0.$
(4.26)

The form (4.26) of the characteristic identity allows for construction of
projectors onto invariant subspaces in $V_{\operatorname{ad}}\otimes
V_{\operatorname{ad}}$. Those projectors are given by the following standard
formula (see e.g. Sect. 3.5 in [15] and Sect. 4.6.4 in [23]):

$\operatorname{P}_{j}\equiv\operatorname{P}_{a_{j}}=\prod_{\begin{subarray}{c}i=1,\\\
i\neq
j\hfill\end{subarray}}^{6}\frac{\widehat{C}-a_{i}\mathbf{I}}{a_{j}-a_{i}},$
(4.27)

where $a_{i}$ are the roots (4.22) of the characteristic equation (4.26).
Using the identities (4.6), as well as the formulas (4.10), (4.13), (4.16),
(4.18), we arrive at the following expressions for the projectors (4.27) in
terms of the operators $\mathbf{I}$, $\mathbf{P}$, $\mathbf{K}$,
$\widehat{C}_{+}$, $\widehat{C}_{-}$ (cf. projectors in [15], Table 10.3):

$\displaystyle\operatorname{P}_{1}=\frac{1}{2}(\mathbf{I}-\mathbf{P})+2\widehat{C}_{-},$
(4.28) $\displaystyle\operatorname{P}_{2}=-2\widehat{C}_{-},$
$\displaystyle\operatorname{P}_{3}=\frac{2\mathbf{K}}{(M-1)M}\,,$
$\displaystyle\operatorname{P}_{4}=\frac{2}{3}(M-2)\widehat{C}_{+}^{2}+\frac{M}{3}\,\widehat{C}_{+}+\frac{(M-4)(\mathbf{I}+\mathbf{P})}{3(M-2)}-\frac{2(M-4)\mathbf{K}}{3(M-2)(M-1)}\,,$
$\displaystyle\operatorname{P}_{5}=-\frac{2(M-2)^{2}}{3(M-8)}\,\widehat{C}_{+}^{2}-\frac{(M-2)(M-6)}{3(M-8)}\,\widehat{C}_{+}+\frac{(M-4)(\mathbf{I}+\mathbf{P})}{6(M-8)}+\frac{2\mathbf{K}}{3(M-8)}\,,$
$\displaystyle\operatorname{P}_{6}=\frac{4(M-2)}{M-8}\,\widehat{C}_{+}^{2}+\frac{4}{M-8}\,\widehat{C}_{+}-\frac{4(\mathbf{I}+\mathbf{P})}{(M-2)(M-8)}-\frac{8(M-4)\mathbf{K}}{M(M-2)(M-8)}\,.$

Note that if $M=8$ then the projectors $\operatorname{P}_{5}$ and
$\operatorname{P}_{6}$ are not formally defined. It is a consequence of the
fact, that there is a factor $a_{5}-a_{6}={(M-8)}/{2(M-2)}$ in the
denominators of those projectors in (4.27). Nevertheless, if one substitutes
the formula (4.11) for $\widehat{C}_{+}^{2}$ into the expressions for
$\operatorname{P}_{5}$ and $\operatorname{P}_{6}$, then the pole at $M=8$
reduces. As a result, we have for $\operatorname{P}_{5}$ the following
expression

$\operatorname{P}_{5}=\frac{1}{6}(1-\epsilon(P_{14}+P_{23}+P_{13}+P_{24})+P_{13}P_{24})\mathcal{P}^{(\epsilon)}_{12,34},$
(4.29)

that does not depend on $M$ and coincides with the complete antisymmetrizer on
$V_{N}^{\otimes 4}$ in the case of algebras $so(N,\mathbb{C})$ ($\epsilon=+1$)
or with the complete symmetrizer on $V_{N}^{\otimes 4}$ in the case of
algebras $sp(N,\mathbb{C})$ ($\epsilon=-1$). For the projector
$\operatorname{P}_{6}$ we have:

$\operatorname{P}_{6}=\frac{4}{M-2}\,\mathcal{P}^{(\epsilon)}_{12,34}K_{13}\biggl{[}\frac{1}{2}(1+\epsilon
P_{24})-\frac{1}{M}\leavevmode\nobreak\
K_{24}\biggr{]}\mathcal{P}^{(\epsilon)}_{12,34}.$ (4.30)

Both operators $\operatorname{P}_{5}$ and $\operatorname{P}_{6}$ are well-
defined at $M=8$. However, due to the identity (4.12) being different in
comparison to (4.13), the case of algebra $so(8,\mathbb{C})$ will be
considered separately (see Subsection 4.2).

The dimensions of the eigenspaces $V_{a_{i}}$ of
$\widehat{C}_{\operatorname{ad}}$ in the space ${V_{\operatorname{ad}}\otimes
V_{\operatorname{ad}}}$ equal to the traces of the corresponding projectors.
In order to find them we will firstly compute the following auxiliary traces:

$\displaystyle\operatorname{tr}\mathbf{I}=\frac{M^{2}}{4}(M-1)^{2},$
$\displaystyle\operatorname{tr}\mathbf{P}=\frac{M}{2}(M-1),$
$\displaystyle\operatorname{tr}\mathbf{K}=\frac{M}{2}(M-1),$ (4.31)
$\displaystyle\operatorname{tr}\widehat{C}_{+}=\frac{M}{4}(M-1),$
$\displaystyle\operatorname{tr}\widehat{C}_{+}^{2}=\frac{3M}{8}(M-1),$
$\displaystyle\operatorname{tr}\widehat{C}_{-}=-\frac{M}{4}(M-1),$

from where we have (see Table 10.3 in [15])

$\displaystyle\dim(V_{a_{1}})=\operatorname{tr}\operatorname{P}_{1}=\frac{1}{8}M(M-1)(M+2)(M-3),$
(4.32)
$\displaystyle\dim(V_{a_{2}})=\operatorname{tr}\operatorname{P}_{2}=\frac{1}{2}M(M-1),$
$\displaystyle\dim(V_{a_{3}})=\operatorname{tr}\operatorname{P}_{3}=1,$
$\displaystyle\dim(V_{a_{4}})=\operatorname{tr}\operatorname{P}_{4}=\frac{1}{12}M(M+1)(M+2)(M-3),$
$\displaystyle\dim(V_{a_{5}})=\operatorname{tr}\operatorname{P}_{5}=\frac{1}{24}M(M-1)(M-2)(M-3),$
$\displaystyle\dim(V_{a_{6}})=\operatorname{tr}\operatorname{P}_{6}=\frac{1}{2}(M-1)(M+2).$

Note that the sum of those traces

$\sum_{i=1}^{6}\operatorname{tr}\operatorname{P}_{i}=\frac{M^{2}}{4}(M-1)^{2}=\operatorname{tr}\mathbf{I}$
(4.33)

coincides with the dimension of
$\mathfrak{g}_{N}^{\epsilon}\otimes\mathfrak{g}_{N}^{\epsilon}$, as it should
be.

If $M=4,6$, the characteristic identities have degrees $4$ and $5$
respectively, as it was noted in the discussion after (4.13), as well as after
the formulae (4.23)–(4.25). However, all the operators (4.28) are well-defined
when $M=4,6$, constructed from the $\operatorname{ad}$-invariant operators
$\mathbf{I}$, $\mathbf{P}$, $\mathbf{K}$, $\widehat{C}_{+}$,
$\widehat{C}_{+}^{2}$, $\widehat{C}_{-}$ and satisfy by construction the
following relations:

$\operatorname{P}_{i}\operatorname{P}_{j}=\delta_{ij}\operatorname{P}_{i},\qquad\sum_{i=1}^{6}\operatorname{P}_{i}=\mathbf{I},\qquad
i,j=1,\dots,6\;.$ (4.34)

Thus, they form a full system of projectors on invariant subspaces of the
representation $\operatorname{ad}^{\otimes 2}(so(4))$ and
$\operatorname{ad}^{\otimes 2}(so(6))$. A decomposition of the representation
$\operatorname{ad}^{\otimes 2}(so(4))$ into irreducible subrepresentations can
be easily found, if one uses the isomorphism $so(4)\cong sl(2)+sl(2)$ and the
known decomposition $\operatorname{ad}^{\otimes 2}(sl(2))=1+3+5$, while for a
decompsotion of $\operatorname{ad}^{\otimes 2}(so(6))$ one can use, for
example, the program LieART [24] (see also Table 10.3 in [15]):

$\displaystyle\operatorname{ad}^{\otimes 2}(so(4))\equiv 6\otimes
6=1+1+3+3+5+5+9+9,$ (4.35) $\displaystyle\operatorname{ad}^{\otimes
2}(so(6))\equiv 15\otimes 15=1+15+15+20+45+45^{\prime}+84.$ (4.36)

Comparing dimensions of the representations (4.35) and dimensions of the
invariant subspaces (4.32) we see that in the case of algebra
$so(4,\mathbb{C})$ out of all the projectors (4.28) only
$\operatorname{P}_{1}$, $\operatorname{P}_{6}$, $\operatorname{P}_{3}$ and
$\operatorname{P}_{5}$ are primitive, the first two of which project onto
$10$-dimensional, and the second two project onto $1$-dimensional subspaces.
The operator $\operatorname{P}_{2}$ projects onto the sum of two
$3$-dimensional subspaces, while the operator $\operatorname{P}_{4}$ projects
onto a sum of two $5$-dimensional subspaces. In the case of algebra
$so(6,\mathbb{C})$ comparison of the formulae (4.36) and (4.32) shows, that
all the projectors apart from $\operatorname{P}_{1}$ are primitive, while
$\operatorname{P}_{1}$ projects onto the sum of two invariant subspaces, each
of which is $45$-dimensional.

Let us consider here also the case of algebra $so(3,\mathbb{C})\cong
sl(2,\mathbb{C})\cong sp(2,\mathbb{C})$, that is when $M=3$ (the algebra
$so(3,\mathbb{C})$) or $M=-2$ (the algebra $sp(2,\mathbb{C})$). Despite all
the roots (4.22) of the characteristic polynomial on the left hand side of
(4.1) being different, the identity (4.26) is not actually characteristic for
the operator $\widehat{C}_{\operatorname{ad}}$ when $M=3$ and $M=-2$. This is
due to the fact that if $M=3$ then the projectors $\operatorname{P}_{1}$,
$\operatorname{P}_{4}$ and $\operatorname{P}_{5}$ are degenerate, while if
$M=-2$ then the projectors $\operatorname{P}_{1}$, $\operatorname{P}_{4}$ and
$\operatorname{P}_{6}$ nullify (this happens because the quantities from
(4.22), that correspond to the nullifying projectors, are not eigenvalues of
$\widehat{C}_{\operatorname{ad}}$ in the considered representation). The
remaining three projectors form a full system of projectors and, as it is seen
from the decomposition

$\operatorname{ad}^{\otimes 2}(sl(2))\equiv 3\otimes 3=1+3+5,$ (4.37)

are primitive. Traces of those projectors in the case $M=3$ and $M=-2$, equal
to dimensions of the corresponding invariant subspaces, and are in agreement
with the isomorphisms $so(3,\mathbb{C})\cong sl(2,\mathbb{C})\cong
sp(2,\mathbb{C})$.

For illustration, in Tables 2, 2 we provide dimensions (4.32) of irreducible
representations in the decomposition of
$\operatorname{ad}\otimes\operatorname{ad}$ for some of the simple Lie
algebras $so(N)$, $sp(N)$ and for several values $N$. The dimensions in those
tables are in accordance with the data presented in papers [15, 25]. Note
also, that the characteristic identites (4.26) and the dimensions (4.32) for
$so(N,\mathbb{C})$ and $sp(N,\mathbb{C})$ turn into each other if one
substitutes $N\to-N$. This manifests a duality between the algebras $so(N)$
and $sp(N)$ (see [15] for more details).

Table 1: Dimensions of irreducible representations for algebra $so(N)$.

$N$ | $\dim_{1}$ | $\dim_{2}$ | $\dim_{3}$ | $\dim_{4}$ | $\dim_{5}$ | $\dim_{6}$
---|---|---|---|---|---|---
$5$ | $35$ | $10$ | $1$ | $35$ | $5$ | $14$
$7$ | $189$ | $21$ | $1$ | $168$ | $35$ | $27$
$9$ | $594$ | $36$ | $1$ | $495$ | $126$ | $44$
$10$ | $945$ | $45$ | $1$ | $770$ | $210$ | $54$
$11$ | $1434$ | $55$ | $1$ | $1144$ | $330$ | $65$

Table 2: Dimensions of irreducible representations for algebra $sp(N)$.

$N$ | $\dim_{1}$ | $\dim_{2}$ | $\dim_{3}$ | $\dim_{4}$ | $\dim_{5}$ | $\dim_{6}$
---|---|---|---|---|---|---
$4$ | $35$ | $10$ | $1$ | $14$ | $35$ | $5$
$6$ | $189$ | $21$ | $1$ | $90$ | $126$ | $14$
$8$ | $594$ | $36$ | $1$ | $308$ | $330$ | $27$
$10$ | $1430$ | $55$ | $1$ | $780$ | $715$ | $44$
$12$ | $2925$ | $78$ | $1$ | $1650$ | $1365$ | $65$

###  4.2 The case of algebra $so(8)$

In order to deduce the characteristic identity for
$\widehat{C}_{\operatorname{ad}}$ in the case of algebra $so(8)$ we use the
identities (4.9) and (4.12) for the operators $\widehat{C}_{-}$ and
$\widehat{C}_{+}$:

$\widehat{C}_{-}^{2}=-\frac{1}{2}\,\widehat{C}_{-},\qquad\widehat{C}_{+}^{2}=-\frac{1}{6}\,\widehat{C}_{+}+\frac{1}{36}(\mathbf{I}+\mathbf{P}+\mathbf{K}).$
(4.38)

Multiplying the second relation by $\widehat{C}_{+}$, multiplying then the
obtained identity by $(\widehat{C}_{+}+1)$ and using the condition ${\bf
K}(\widehat{C}_{+}+1)=0$ yields the following formula:

$\widehat{C}_{+}(\widehat{C}_{+}+1)\biggl{(}\widehat{C}_{+}+\frac{1}{3}\biggr{)}\biggl{(}\widehat{C}_{+}-\frac{1}{6}\biggr{)}=0.$

Note that the characteristic identity for the split Casimir operator
$\widehat{C}_{\operatorname{ad}}=\widehat{C}_{+}+\widehat{C}_{-}$ can be built
analogously to the general case $so(M)$ by introducing indeterminate
coefficients:

$\widehat{C}_{\operatorname{ad}}\biggl{(}\widehat{C}_{\operatorname{ad}}+\frac{1}{2}\biggr{)}(\widehat{C}_{\operatorname{ad}}+1)\biggl{(}\widehat{C}_{\operatorname{ad}}+\frac{1}{3}\biggr{)}\biggl{(}\widehat{C}_{\operatorname{ad}}-\frac{1}{6}\biggr{)}=0.$
(4.39)

In this way, the operator $\widehat{C}_{\operatorname{ad}}$ has the following
eigenvalues $a_{1}=0$, $a_{2}=-1/2$, $a_{3}=-1$, $a_{4}=-1/3$ and $a_{5}=1/6$.
Let us write all the projectors onto the invariant subspaces of
$\widehat{C}_{\operatorname{ad}}$, that can be derived from (4.39), in terms
of $\widehat{C}_{+},\ \widehat{C}_{-},\ \mathbf{I},\ \mathbf{P},\ \mathbf{K}$:

$\displaystyle\operatorname{P}_{1}^{\prime}=\frac{1}{2}(\mathbf{I}-\mathbf{P})+2\widehat{C}_{-}\equiv\operatorname{P}_{1}\Big{|}_{M=8},$
$\displaystyle\dim=350,$ (4.40)
$\displaystyle\operatorname{P}_{2}^{\prime}=-2\widehat{C}_{-}\equiv\operatorname{P}_{2}\big{|}_{M=8},$
$\displaystyle\dim=28,$
$\displaystyle\operatorname{P}_{3}^{\prime}=\frac{1}{28}\mathbf{K}\equiv\operatorname{P}_{3}\Big{|}_{M=8},$
$\displaystyle\dim=1,$
$\displaystyle\operatorname{P}_{4}^{\prime}=\frac{1}{6}(\mathbf{I}+\mathbf{P})-2\widehat{C}_{+}-\frac{1}{12}\mathbf{K}\equiv(\operatorname{P}_{5}+\operatorname{P}_{6})\Big{|}_{M=8},$
$\displaystyle\dim=105,$
$\displaystyle\operatorname{P}_{5}^{\prime}=\frac{1}{3}(\mathbf{I}+\mathbf{P})+2\widehat{C}_{+}+\frac{1}{21}\mathbf{K}\equiv\operatorname{P}_{4}\Big{|}_{M=8},$
$\displaystyle\dim=300,$

where $\operatorname{P}_{k}^{\prime}\equiv\operatorname{P}_{a_{k}}^{\prime}$,
the projectors $\operatorname{P}_{i}$ are defined in (4.28) and written for
$M=8$ with the help of the identity (4.12). The right hand side of (4.40)
contains the dimensions of the eigenspaces calculated by the formulae (4.31).

It is known that $\operatorname{ad}^{\otimes 2}(so(8))$ is decomposed into
irreducible representations as follows (see e.g. Table 17.2 in [15], and
[24]):

$\operatorname{ad}^{\otimes
2}(so(8))=1+28+35+35^{\prime}+35^{\prime\prime}+300+350\;.$ (4.41)

The subspaces, on which each of the irreducible representations from (4.41)
act, will be denoted by $V_{1}$, $V_{28}$, $V_{35}$ etc. Comparing the
dimensions of the subrepresentations in (4.40) and (4.41) yields that
$\operatorname{P}_{4}^{\prime}$ projects $V_{\operatorname{ad}}^{\otimes 2}$
onto $V_{35}+V_{35^{\prime}}+V_{35^{\prime\prime}}$. Note that for the
projector $\operatorname{P}_{4}^{\prime}$ the following decomposition holds:

$\operatorname{P}_{4}^{\prime}=\operatorname{P}_{5}|_{M=8}+\operatorname{P}_{6}|_{M=8},$
(4.42)

where $\operatorname{P}_{5}$ and $\operatorname{P}_{6}$ are defined in (4.29)
and (4.30). Besides, according to (4.32) the dimension of the subspace
$V_{a_{5}}=\operatorname{P}_{5}|_{M=8}(V_{\operatorname{ad}}\otimes
V_{\operatorname{ad}})$ is $70$, while the dimension of the subspace
$V_{a_{6}}=\operatorname{P}_{6}|_{M=8}(V_{\operatorname{ad}}\otimes
V_{\operatorname{ad}})$ equals $35$. Hence, $V_{a_{6}}$ is the space of a
$35$-dimensional irreducible representation of the algebra $so(8)$ (say,
$V_{35^{\prime\prime}}$), and the space $V_{a_{5}}$, acquired by the action of
the complete antisymmetrizer(4.29), must decompose into the sum
$V_{35}+V_{35^{\prime}}$ of the two remaining $35$-dimensional irreducible
representations.

In order to build the projectors onto the invariant subspaces $V_{35}$ and
$V_{35^{\prime}}$ we introduce an operator $E_{8}\colon V_{8}^{\otimes 4}\to
V_{8}^{\otimes 4}$ with the following components:

$(E_{8})^{i_{i}\dots i_{4}}{}_{j_{1}\dots
j_{4}}\equiv\frac{1}{4!}\,\varepsilon^{i_{i}\dots i_{4}}{}_{j_{1}\dots
j_{4}}=\frac{1}{4!}\,c_{j_{1}i_{5}}\cdots
c_{j_{1}i_{8}}\varepsilon^{i_{1}\dots i_{8}}.$ (4.43)

Here $\varepsilon^{i_{1}\dots i_{8}}$ is the completely antisymmetric
invariant tensor, $\varepsilon^{12345678}=1$. This operator, being built from
the invariant tensors $\varepsilon^{i_{1}\dots i_{8}}$ and $c_{i_{1}i_{2}}$,
is $\operatorname{ad}$-invariant. Taking into account, that in the case of
algebra $so(8)$ we have $c_{i_{1}i_{2}}=\delta_{i_{1}i_{2}}$, we do not
distinguish between upper and lower indices.

To build the projectors onto the eigenspaces of the operator $E_{8}$ we
firstly find the characteristic identity for this operator. It is convenient
to perform all the calculations for a more general case of the algebra
$so(2r)$, that is, for the $(2r)^{r}$-dimensional space $V_{2r}^{\otimes r}$
and the completely antisymmetric tensor $\varepsilon$ of the rank $2r$. The
operator $E_{r}$ in this case is defined as:

$(E_{r})^{i_{i}\dots i_{r}}{}_{j_{1}\dots
j_{r}}=\frac{(-1)^{{r}/{2}}}{r!}\,\varepsilon^{i_{i}\dots i_{r}}{}_{j_{1}\dots
j_{r}}.$ (4.44)

Note, that the following relations hold:

$\left.\begin{array}[]{l}A_{r}\,E_{r}=E_{r}\,A_{r}=E_{r}\\\
E_{r}^{2}=A_{r}\end{array}\right\\}\implies E_{r}^{3}=E_{r},$ (4.45)

where $A_{r}$ is the complete antisymmetrizer in the space of all rank $r$
tensors. The components of $A_{r}$ are:

$(A_{r})^{i_{1}\dots i_{r}}_{\qquad k_{1}\dots
k_{r}}=\frac{1}{r!}\sum_{\sigma\in
S_{r}}(-1)^{p(\sigma)}\delta^{i_{\sigma(1)}}_{k_{1}}\dots\delta^{i_{\sigma(r)}}_{k_{r}}=\frac{1}{(r!)^{2}}\varepsilon^{i_{1}\dots
i_{r}j_{1}\dots j_{r}}\varepsilon_{k_{1}\dots k_{r}j_{1}\dots j_{r}}.$ (4.46)

Here the summation is carried over all the permutations $\sigma\in S_{r}$,
$p(\sigma)$ is the parity of a permutation $\sigma$. The characteristic
identity (4.45) for $E_{r}$ can be written in the following way:

$E_{r}(E_{r}-\overline{\mathbf{I}}\,)(E_{r}+\overline{\mathbf{I}}\,)=0,$
(4.47)

where $\overline{\mathbf{I}}$ denotes the identity operator in
$V_{2r}^{\otimes r}$.

Using (4.47), we can build the projectors $\operatorname{\overline{P}}_{1}$,
$\operatorname{\overline{P}}_{2}$ and $\operatorname{\overline{P}}_{3}$ onto
the eigenspaces of $E_{r}$:

$\operatorname{\overline{P}}_{(b_{i})}\equiv\operatorname{\overline{P}}_{j}=\prod_{\begin{subarray}{c}i=1,\\\
i\neq
j\hfill\end{subarray}}^{3}\frac{E_{r}-b_{i}\overline{\mathbf{I}}}{b_{j}-b_{i}},$
(4.48)

where $b_{1}=0$, $b_{2}=1$ and $b_{3}=(-1)$ are the eigenvalues of $E_{r}$.
The exact formulae are:

$\displaystyle\operatorname{\overline{P}}_{1}=-(E_{r}-\overline{\mathbf{I}}\,)(E_{r}+\overline{\mathbf{I}}\,)=\overline{\mathbf{I}}-A_{r},$
(4.49)
$\displaystyle\operatorname{\overline{P}}_{2}=\frac{1}{2}\,E_{r}(E_{r}+\overline{\mathbf{I}}\,)=\frac{1}{2}(A_{r}+E_{r}),$
$\displaystyle\operatorname{\overline{P}}_{3}=\frac{1}{2}\,E_{r}(E_{r}-\overline{\mathbf{I}}\,)=\frac{1}{2}(A_{r}-E_{r}).$

Here we used the second formula in the curly brackets in (4.45) to simplify
the expressions.

According to (4.49) we have
$\operatorname{\overline{P}}_{2}+\operatorname{\overline{P}}_{3}=A_{r}$, and
the images of the operators $\operatorname{\overline{P}}_{2}$ and
$\operatorname{\overline{P}}_{3}$ belong to the space $V_{2r}^{\wedge r}$ of
completely antisymmetric tensors of rank $r$. The operators
$\operatorname{\overline{P}}_{2}$ and $\operatorname{\overline{P}}_{3}$ split
the space $V_{2r}^{\wedge r}$ into two parts, which are called the spaces of
self-dual and anti-self-dual tensors. The operator
$\operatorname{\overline{P}}_{1}=\overline{\mathbf{I}}-A_{r}$ acts trivially
on the space $V_{2r}^{\wedge r}$, thus, we will not use the projector
$\operatorname{\overline{P}}_{1}$ in what follows.

In order to calculate the traces of the second and the third projectors in
(4.49) and to find the dimension of the corresponding subspaces we will use
the following auxiliary traces:

$\operatorname{tr}A_{r}=\binom{2r}{r}=\frac{(2r)!}{(r!)^{2}},\qquad\operatorname{tr}E_{r}=0.$
(4.50)

As a result, the traces of the projectors $\operatorname{\overline{P}}_{2}$
and $\operatorname{\overline{P}}_{3}$ have the form:

$\operatorname{tr}\operatorname{\overline{P}}_{2}=\frac{1}{2}\,\frac{(2r)!}{(r!)^{2}},\qquad\operatorname{tr}\operatorname{\overline{P}}_{3}=\frac{1}{2}\,\frac{(2r)!}{(r!)^{2}}.$
(4.51)

Let us return now to the algebra $so(8)$ and substitute the value $r=4$ into
the expressions for the traces (4.51):

$\operatorname{tr}\operatorname{\overline{P}}_{2}|_{r=4}=35,\qquad\operatorname{tr}\operatorname{\overline{P}}_{3}|_{r=4}=35$
(4.52)

(hereinafter we will use $\operatorname{\overline{P}}_{i}$ instead of
$\operatorname{\overline{P}}_{i}|_{r=4}$ for simplicity). Thus, the dimensions
of the subspaces $\operatorname{\overline{P}}_{2}\cdot V_{8}^{\wedge 4}$ and
$\operatorname{\overline{P}}_{3}\cdot V_{8}^{\wedge 4}$ coincide and equal
$35$. Recall that the projector $\operatorname{P}_{5}$, given in (4.29), for
every $M$ equals the complete antisymmetrizer $A_{4}$, and, in particular
$\operatorname{P}_{5}|_{M=8}=A_{4}$, hence

$\displaystyle\operatorname{P}_{5}|_{M=8}=\operatorname{\overline{P}}_{2}+\operatorname{\overline{P}}_{3},$
(4.53)

and the images of $\operatorname{\overline{P}}_{2}$ and
$\operatorname{\overline{P}}_{3}$ belong to the subspace extracted by the
projector $\operatorname{P}_{5}|_{M=8}=A_{4}$. The values of the traces (4.52)
suggest that $\operatorname{\overline{P}}_{2}$ and
$\operatorname{\overline{P}}_{3}$ are the projectors onto the two
$35$-dimensional invariant subspaces of the representation
$\operatorname{ad}^{\otimes 2}(so(8))$.

To write the projector $\operatorname{P}_{6}|_{M=8}$ in terms of the operators
$E_{4}$, $A_{4}$, introduced in (4.44) and (4.46), and in terms of the
operators $\mathbf{I}$, $\mathbf{P}$, $\mathbf{K}$ and $\widehat{C}_{+}$ we
use the formula (4.42), the explicit form of the projector
$\operatorname{P}_{4}^{\prime}$, given in (4.40), as well as the fact that
$\operatorname{P}_{5}|_{M=8}=A_{4}$. As a result, we have:

$\operatorname{P}_{6}|_{M=8}=\operatorname{P}_{4}^{\prime}-\operatorname{P}_{5}|_{M=8}=\frac{1}{6}(\mathbf{I}+\mathbf{P})-2\widehat{C}_{+}-\frac{1}{12}\,\mathbf{K}-A_{4}.$
(4.54)

Finally, we have the full system of mutually orthogonal and primitive
projectors onto the spaces of the irreducible subrepresentations in
$\operatorname{ad}^{2}(so(8))$:

$\displaystyle\operatorname{P}_{1}^{\prime}=\frac{1}{2}(\mathbf{I}-\mathbf{P})+2\widehat{C}_{-},$
$\displaystyle\dim=350,$ (4.55)
$\displaystyle\operatorname{P}_{2}^{\prime}=-2\widehat{C}_{-},$
$\displaystyle\dim=28,$
$\displaystyle\operatorname{P}_{3}^{\prime}=\frac{1}{28}\mathbf{K},$
$\displaystyle\dim=1,$
$\displaystyle\operatorname{P}_{6}|_{M=8}=\frac{1}{6}(\mathbf{I}+\mathbf{P})-2\widehat{C}_{+}-\frac{1}{12}\,\mathbf{K}-A_{4},$
$\displaystyle\dim=35,$
$\displaystyle\operatorname{\overline{P}}_{2}=\frac{1}{2}(A_{4}+E_{4}),$
$\displaystyle\dim=35,$
$\displaystyle\operatorname{\overline{P}}_{3}=\frac{1}{2}(A_{4}-E_{4}),$
$\displaystyle\dim=35,$
$\displaystyle\operatorname{P}_{5}^{\prime}=\frac{1}{3}(\mathbf{I}+\mathbf{P})+2\widehat{C}_{+}+\frac{1}{21}\,\mathbf{K},$
$\displaystyle\dim=300.$

##  5 A connection between the eigenvalues of the operator $\widehat{C}$ in
the adjoint representation of $so(N,\mathbb{C})$ and $sp(N,\mathbb{C})$ Lie
algebras and the Vogel parameters

Many results of this paper, namely, construction of the characteristic
identities and projectors onto invariant subspaces of the representation
$\operatorname{ad}^{\otimes 2}(\mathfrak{g}_{N}^{\epsilon})$ in terms of the
split Casimir operator, as well as calculation of the dimensions of these
subspaces, could be also achieved by using the so-called Vogel parameters
[16], [20]. These parameters are defined as three numbers
$(\alpha,\beta,\gamma)$ modulo a common multiplier and an arbitrary
permutation from the symmetric group $S_{3}$ (and thus can be interpreted as
coordinates in the space $\mathbb{P}^{2}/S_{3}$). It is known [16], [17], [20]
that certain values of the Vogel parameters (or what is the same, a certain
point in the space $\mathbb{P}^{2}/S_{3}$) correspond to some simple Lie
algebra. For the simple Lie algebras of classical series values of these
parameters are given in Table 3 (see [19]), where $t\equiv\alpha+\beta+\gamma$
is the dual Coxeter number.

Table 3: Vogel parameters for simple Lie algebras. Type | Lie Algebra | $\alpha$ | $\beta$ | $\gamma$ | $t$
---|---|---|---|---|---
$A_{r}$ | $sl(r+1)$ | $-2$ | $2$ | $r+1$ | $r+1$
$B_{r}$ | $so(2r+1)$ | $-2$ | $4$ | $2r-3$ | $2r-1$
$C_{r}$ | $sp(2r)$ | $-2$ | $1$ | $r+2$ | $r+1$
$D_{r}$ | $so(2r)$ | $-2$ | $4$ | $2r-4$ | $2r-2$

The representation $\operatorname{ad}^{\otimes 2}\mathcal{A}$ of any simple
Lie algebra $\mathcal{A}$ can be decomposed into symmetric and antisymmetric
subspaces. According to papers [15], [16]–[18] the symmetric part (in all the
cases of simple Lie algebras of classical series, apart from
$sl(2,\mathbb{C})\cong sp(2,\mathbb{C})\cong so(3,\mathbb{C})$,
$sl(3,\mathbb{C})$ and $so(8,\mathbb{C})$), in its turn, decomposes into four
irreducible subrepresentations – a singlet $T_{0}$ and three irreducible
representations denoted by $Y_{2}^{(\alpha)}$, $Y_{2}^{(\beta)}$ and
$Y_{2}^{(\gamma)}$, where $\alpha$, $\beta$ and $\gamma$ are the Vogel
parameters of the algebra $\mathcal{A}$ from Table 3. The dimensions of these
four representations as well as the eigenvalues of the quadratic Casimir
operator $C_{(2)}$ (which was defined in (3.5)), acting on the spaces of these
representations, are given by formulas [16], [20]

$\displaystyle\dim T_{0}(\mathcal{A})=1,$ $\displaystyle c_{2}^{0}=0,$ (5.1)
$\displaystyle\dim
Y_{2}^{(\alpha)}(\mathcal{A})=-\frac{(3\alpha-2t)(\beta-2t)(\gamma-2t)t(\beta+t)(\gamma+t)}{\alpha^{2}(\alpha-\beta)\beta(\alpha-\gamma)\gamma},$
$\displaystyle c_{2}^{\alpha}=2-\frac{\alpha}{t}\,,$ $\displaystyle\dim
Y_{2}^{(\beta)}(\mathcal{A})=-\frac{(3\beta-2t)(\alpha-2t)(\gamma-2t)t(\alpha+t)(\gamma+t)}{\beta^{2}(\beta-\alpha)\alpha(\beta-\gamma)\gamma},$
$\displaystyle c_{2}^{\beta}=2-\frac{\beta}{t}\,,$ $\displaystyle\dim
Y_{2}^{(\gamma)}(\mathcal{A})=-\frac{(3\gamma-2t)(\beta-2t)(\alpha-2t)t(\beta+t)(\alpha+t)}{\gamma^{2}(\gamma-\beta)(\beta(\gamma-\alpha)\alpha},$
$\displaystyle c_{2}^{\gamma}=2-\frac{\gamma}{t}\,,$

where $c_{2}^{0}$, $c_{2}^{\alpha}$, $c_{2}^{\beta}$ and $c_{2}^{\gamma}$ are
the eigenvalues of the operator $C_{(2)}$, acting in the representations
$T_{0}$, $Y_{2}^{(\alpha)}$, $Y_{2}^{(\beta)}$, $Y_{2}^{(\gamma)}$
respectively (here the upper index of $c_{2}$ is not the highest weight, as in
the formula (3.10), but is chosen to accord with the notation of the
corresponding representation).

The antisymmetric part decomposes into the direct sum of two representations,
one of which is the adjoint representation $\operatorname{ad}$. The dimension
of this representation and the value $c_{2}^{\operatorname{ad}}$ of the
quadratic Casimir operator $C_{(2)}$ in it are expressed by the following
formulae:

$\dim(\operatorname{ad})\equiv\dim\mathcal{A}=\frac{(\alpha-2t)(\beta-2t)(\gamma-2t)}{\alpha\beta\gamma},\qquad
c_{2}^{\operatorname{ad}}=1.$ (5.2)

The second antisymmetric subrepresentation is denoted $X_{2}$. Its dimension
and the value $c_{2}^{X}$ of the operator $C_{(2)}$ in it are:

$\dim(X_{2})=\frac{1}{2}\dim\mathcal{A}(\dim\mathcal{A}-3),\qquad
c_{2}^{X}=2.$ (5.3)

Comparing the dimensions [24] of the irreducible representations in
$\operatorname{ad}^{\otimes 2}$ with the dimensions (4.32) indicates that the
representation $X_{2}$ for the Lie algebras of the types $B_{r},C_{r}$ and
$D_{r}$ is irreducible. Note that the representation $X_{2}$ for Lie algebras
of the type $A_{r}$, $r>1$, is reducible and decomposes into the sum of two
irreducible representations with the same dimensions.

The final decomposition of the representation $\operatorname{ad}^{\otimes
2}(\mathcal{A})$, where $\mathcal{A}$ is a classical simple Lie algebra
(except for the algebras $sl(2,\mathbb{C})\cong sp(2,\mathbb{C})\cong
so(3,\mathbb{C})$, $sl(3,\mathbb{C})$ and $so(8,\mathbb{C})$), takes the form

$\operatorname{ad}^{\otimes
2}(\mathcal{A})=T_{0}(\mathcal{A})+Y_{2}^{(\alpha)}(\mathcal{A})+Y_{2}^{(\beta)}(\mathcal{A})+Y_{2}^{(\gamma)}(\mathcal{A})+\operatorname{ad}(\mathcal{A})+X_{2}(\mathcal{A}).$
(5.4)

If we know the values (5.1)–(5.3) of the quadratic Casimir operator $C_{(2)}$
in the six representations (in the case of those representations being
irreducible), we can calculate the values of the split Casimir operator
$\widehat{C}$ (which was defined in (3.3)) in these representations by using
the formula (3.9). A connection between the eigenvalues of $\widehat{C}$ and
$C_{(2)}$ in any irreducible subrepresentation $T_{\lambda}$ in the
decomposition $\operatorname{ad}\otimes\operatorname{ad}$ with the help of the
formula (3.9), where one needs to set the values $\lambda_{1}$ and
$\lambda_{2}$ equal to the highest weight of the irreducible representation of
the algebra $\mathcal{A}$. As the values of the quadratic Casimir operator of
any simple lie Algebra $\mathcal{A}$ in the adjoint representation equal $1$,
(see e.g. [22]), then $c_{2}^{(\lambda_{1})}=c_{2}^{(\lambda_{2})}=1$ and
according to (3.9) we have

$\hat{c}^{(\lambda)}=\frac{1}{2}c_{2}^{(\lambda)}-1.$ (5.5)

In what follows we will only consider the cases of the algebras
$so(N,\mathbb{C})$ and $sp(N,\mathbb{C})$, which our paper is devoted to.
Substituting the values $\alpha$, $\beta$, $\gamma$ and $t$ for the algebras
$so(N,\mathbb{C})$ and $sp(N,\mathbb{C})$ from Table 3 into the formulas
(5.1)–(5.3), yields that the dimensions of the six representations $T_{0}$,
$Y_{2}^{(\alpha)}$, $Y_{2}^{(\beta)}$, $Y_{2}^{(\gamma)}$,
$\operatorname{ad}(\mathfrak{g}_{N}^{\epsilon})$ and $X_{2}$ coincide with the
dimensions (4.32) of the invariant subspaces of the representations
$\operatorname{ad}^{\otimes 2}(\mathfrak{g}_{N}^{\epsilon})$. Taking into
account the formula (5.5) one can show, that the values of the split Casimir
operator in each of the aforementioned representations coincide with the roots
(4.22) of the polynomial on the left hand side of (4.26), which is the
characteristic polynomial of $\widehat{C}$ in the representation
$\operatorname{ad}^{\otimes 2}(\mathfrak{g}_{N}^{\epsilon})$. It allows us to
conclude that, the spaces of the representations on the right hand side of
(5.4) coincide with the invariant subspaces, that were extracted by the
projectors (4.28).This correspondence is shown in Table 4, where in the top
row all the six projectors (4.28) are given, while in the second, third and
forth rows there are subrepresentations, subspaces of which are extracted in
$V_{\operatorname{ad}}\otimes V_{\operatorname{ad}}$ by the projectors in the
top row.

Table 4: Correspondence between the representations and the projectors. |  $\operatorname{P}_{1}$ |  $\operatorname{P}_{2}$ |  $\operatorname{P}_{3}$ |  $\operatorname{P}_{4}$ |  $\operatorname{P}_{5}$ |  $\operatorname{P}_{6}$
---|---|---|---|---|---|---
$B_{r}$ |  $X_{2}$ |  $\operatorname{ad}(so(2r+1))$ |  $T_{0}$ |  $Y_{2}^{(\alpha)}$ |  $Y_{2}^{(\beta)}$ |  $Y_{2}^{(\gamma)}$
$C_{r}$ |  $X_{2}$ |  $\operatorname{ad}(sp(2r))$ |  $T_{0}$ |  $Y_{2}^{(\beta)}$ |  $Y_{2}^{(\alpha)}$ |  $Y_{2}^{(\gamma)}$
$D_{r}$ |  $X_{2}$ |  $\operatorname{ad}(so(2r))$ |  $T_{0}$ |  $Y_{2}^{(\alpha)}$ |  $Y_{2}^{(\beta)}$ |  $Y_{2}^{(\gamma)}$

Thus, we obtain that the calculated in the previous sections values of the
split Casimir operator in the irreducible subrepresentations in
$\operatorname{ad}^{\otimes 2}$ for the cases of $so(N,\mathbb{C})$ and
$sp(N,\mathbb{C})$ Lie algebras, as well as the dimensions of the
corresponding invariant subspaces could be written in terms of the Vogel
parameters in complete accordance with papers [16]–[18].

##  6 Conclusion

In our work we obtained explicit formulae for projectors onto the spaces of
the irreducible subrepresentations comprising the tensor product of two
adjoint representation for all complex simple Lie algebras $so(N,\mathbb{C})$
for $N\geq 3$ and $sp(N,\mathbb{C})$ for $N\geq 2$. For Lie algebras
$so(N,\mathbb{C})$ ($N\geq 3$) the analogous projectors where obtained in [15]
(see Table 10.3) by using the diagram technique. We proved the natural
conjecture that the projectors for algebras $sp(N,\mathbb{C})$ $(N=2r)$ are
related to the corresponding projectors for algebras $so(N,\mathbb{C})$ by the
duality transformation333Generally speaking this duality does not work for
representations which can not be constructed from the adjoint representations.
$N\to-N$. In all considered cases, except the case of algebra $so(8)$, the
formulation was conducted by finding the characteristic identities for the
split Casimir operator $\widehat{C}$ in adjoint representation. In special
case of $so(8,\mathbb{C})$ (considered separately) an additional invariant
operator (independent of $\widehat{C}$) was used to construct the complete set
of projectors. This operator allows to decompose the completely antisymmetric
representation of rank $4$ into the selfdual and the anti-selfdual parts. It
was then shown that the dimensions of the irreducible subrepresentations in
$\operatorname{ad}^{\otimes 2}$, as well as the values of the quadratic
Casimir opertor in these subrepresentations are in complete correspondence
with the results of papers [16]–[18], where these values were written in terms
of the Vogel parameters.

### Acknowledgements

The authors would like to thank P. Cvitanović, S.O. Krivonos, and O.V.
Ogievetsky for useful discussions. The work of A.P. Isaev was supported by the
Russian Foundation for Basic Research grant 19-01-00726. The work of A.A.
Provorov was supported by the Russian Foundation for Basic Research grant
20-52-12003\20.

## References

  * [1] J. B. McGuire, Study of exactly soluble one-dimensional $N$-body problems, J. Math. Phys., 5, No. 5, 622–636 (1964).
  * [2] C. N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett., 19, No. 23, 1312–1315 (1967).
  * [3] R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, New York (1982).
  * [4] A. B. Zamolodchikov and Al. B. Zamolodchikov, Factorized $S$-matrices in two dimensions as the exact solutions of certain relativistic quantum field models, Ann. Phys. (N. Y.), 120, No. 2, 253–291 (1979).
  * [5] E. K. Sklyanin, L. A. Takhtadzhyan, and L. D. Faddeev, Quantum inverse problem method. I, Theoret. and Math. Phys.,40, No. 2, 688–706 (1979).
  * [6] V. G. Drinfeld, Quantum groups in: Proceedings of the International Congress of Mathematicians (Berkeley, California, USA, August 3–11, 1986)(A. M. Gleason, ed.), Vol 1, AMS, Providence, RI (1986), pp.798–820.
  * [7] M. Jimbo, A $q$-analogue of $U(\mathfrak{gl}(N+1))$, Hecke algebra and the Yang–Baxter equation, Lett. Math. Phys., 11, No. 3, 247–252 (1986); Introduction to the Yang–Baxter equation, Internat. J. Modern Phys. A, 4, No. 15, 3759–3777 (1989).
  * [8] V. Pasquier and H. Saleur, Common structures between finite systems and conformal field theories through quantum groups, Nucl. Phys. B, 330, No. 2–3, 523–556 (1990).
  * [9] M. Karowski and A. Zapletal, Quantum-group-invariant integrable $n$-state vertex models with periodic boundary conditions, Nucl. Phys. B, 419, No. 3, 567–588 (1994), [arXiv:hep-th/9312008].
  * [10] P. P. Kulish, Quantum groups and quantum algebras as symmetries of dynamical systems, Preprint YITP/K-959, Yukawa Institute for Theoretical Physics, Kyoto (1991).
  * [11] L. Faddeev, N. Reshetikhin, and L. Takhtajan, Quantization of Lie groups and Lie algebras, Leningrad Math. J., 1, 193–225 (1990).
  * [12] V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge Univ. Press, Cambridge (1994).
  * [13] A. P. Isaev, Quantum groups and Yang–Baxter equations, Preprint MPIM (Bonn), MPI 2004-132, (http://webdoc.sub.gwdg.de/ebook/serien/e/mpi_mathematik/2004/132.pdf)
  * [14] H. Weyl, The Classical Groups: Their Invariants and Representations, Princeton University Press (1997).
  * [15] P. Cvitanović, Group Theory: Birdtracks, Lie’s, and Exceptional Groups, Princeton Univ. Press, Oxford, (2008).
  * [16] P. Vogel, The universal Lie algebra, Preprint, Université Paris VII, UFR de mathématiques, Paris (1999).
  * [17] P. Deligne, La série exceptionnelle des groupes de Lie, C. R. Acad. Sci. Paris Sér. I Math., 322, No. 4, 321–326 (1996).
  * [18] J. M. Landsberg and L. Manivel, A universal dimension formula for complex simple Lie algebras, Adv. Math., 201, No. 2, 379–407 (2006).
  * [19] R. L. Mkrtchyan, A. N. Sergeev, and A. P. Veselov, Casimir eigenvalues for universal Lie algebra, J. Math. Phys., 53, No. 10, 102–106, (2012).
  * [20] J. M. Landsberg and L. Manivel, Triality, exceptional Lie algebras and Deligne dimension formulas, Adv. Math., 171, No. 1, 59–85 (2002).
  * [21] A. Mironov, R. Mkrtchyan, and A. Morozov, On universal knot polynomials, JHEP, 2016, No. 2, 78 (2016).
  * [22] A. P. Isaev and V. A. Rubakov, Theory of Groups and Symmetries. I. Finite Groups, Lie Groups, and Lie Algebras, World Sci., Singapore (2018).
  * [23] A. P. Isaev and V. A. Rubakov, Theory of Groups and Symmetries. II. Representations of Groups and Lie Algebras, Applications, World Sci., Singapore (2019).
  * [24] R. Feger and Th. W. Kephart, LieART – A mathematica application for Lie algebras and representation theory, Comput. Phys. Commun., 92, 166–195 (2015), [arXiv:1206.6379]
  * [25] N. Yamatsu, Finite-dimensional Lie algebras and their representations for unified model building, arXiv:1511.08771 [hep-ph]