on $\Hom_\C(V^{\ot r}, V^{\ot s})$  ($r, s\in \N$ fixed).  Any $g\in G_0$ and $X\in \fg$ act on $\varphi\in \Hom_\C(V^{\ot r}, V^{\ot s})$
($r, s\in \N$ fixed) by

\[
\baln
& (X.\varphi)({\bf v})= X.(\varphi({\bf v}))- (-1)^{[X][\varphi]} \varphi(X.{\bf v}), \\
& (g.\varphi)({\bf v})= g. (\varphi( g^{-1}.{\bf v})), \quad \forall {\bf v}\in V^{\ot r}.
\ealn
\]

This $G$-action extends to $\U(\fg)\ot \Hom_\C(V^{\ot r}, V^{\ot s})$ as follows. For any $X\in\fg$, $g\in G_0$,
$u\in \U(\fg)$, and $\varphi\in \Hom_\C(V^{\ot r}, V^{\ot s})$,

X.(uφ) = X.u φ+ (-1)^[X][u] X.φ,   g.(uφ) = g. u g.φ.
Now it is evident that $\U(\fg)\ot \Hom_\C(V^{\ot r}, V^{\ot s})\ot \Hom_\C(V^{\ot r'}, V^{\ot r'})$ also acquires a $G$-module structure.

For any $u, u'\in\U(\fg)$, $\varphi\in \Hom_\C(V^{\ot r}, V^{\ot r'})$ and $\psi \in \Hom_\C(V^{\ot r'}, V^{\ot s})$,
we have the composition $(u'\ot\psi)(u\ot \varphi)=(-1)^{[u][\psi]}u' u \ot \psi\varphi$. Then one checks easily that
the $G$-action defined by (<ref>) respects composition.

Let us write $\U=\U(\fg)$, and consider it as a left $\U$-module, the action being multiplication. The functor $\CF_M$ for $M=\U$ then becomes
_: ()_(V) ,
where $\CF_\U(\I_0)=1\in\U$ and $\CF_\U(\BH)=(\id_\U\ot \mu)(t)$. Clearly $\CF_\U(\BH)$ is $G$-invariant with respect to the action (<ref>),
and so is also $\CF(D)$ for any morphism $D$ of $\CB(\sdim)$.
As $\I_0, \BH$ and all $D$ generate morphisms of $\AB(\sdim)$,

_(_()(r, s)) ⊆_(V^r, V^s)^G.

We are now in a postion to prove the following crucial result.

The identity element together with $\{\CF_\U(Z_{2\ell})\mid \ell\ge 1\}$  generate $\U(\fg)^G$. Further,
if $\dim(V_{\bar 0})$ is odd or $0$, these elements generate $Z(\U(\fg))$.

We begin by laying some groundwork for the proof.
For any $G$-module $M$,  let $T(M)=\oplus_{r\ge 0} M^{\ot r}$ be the tensor algebra on $M$.
For each $r$, there is the usual semi-simple $\Sym_r$ action on $M^{\ot r}$, which commutes with the $G$-action.
Let $\Sigma_{\pm 1}=\sum_{\sigma\in\Sym_r}(\mp 1)^{\ell(\sigma)}\sigma$, where $\ell(\sigma)$ is the length of $\sigma$.
Then $\Sigma_{-1}$ (resp. $\Sigma_{+1}$) is the Young symmetriser associated with the standard Young tableau of shape
$(r)$ (resp. $(1^r)$). Write $S_{gr}^k (M)=\Sigma_{-1}M^{\ot r}$ and $\wedge_{gr}^k M= \Sigma_{+1} M^{\ot r}$;
these are $G$-module direct summands of $M^{\ot r}$.  Write $S_{gr}(M)=\oplus_r S_{gr}^k M$ and $\wedge_{gr} (M)= \oplus_r \wedge_{gr}^k M$.
These are our $G$-module direct summands of the tensor algebra $T(M)=\oplus_r M^{\ot r}$.

Now consider the natural $G$-module $V$, and the following $G$-module  isomorphism

\[
\begin{aligned}
j:\   &  \wedge^2_{gr}V\stackrel{\sim}\lra \fg, & e_a\wedge e_b \mapsto X_{a b} \quad \forall a, b,
\end{aligned}
\]

where $e_a\wedge e_b =e_a\ot e_b- (-1)^{[a][b]}e_b\ot e_a$, and $X_{a b}$ are the basis elements of $\fg$ defined in Section <ref>.
We extend the map $j$ to a $\U(\fg)$-module isomorphism
\Upsilon:  S_{gr}(\wedge^2_{gr} V) \stackrel{\sim}\lra \U(\fg)
in the following way.

For any $r\ge 2$, the symmetric group
$\Sym_r$ acts on $\U(\fg)^{\ot r}$ by permuting the factors with appropriate sign.
Explicitly, the Chevalley generators $s_i$ of $\Sym_r$ act by
$\id_{\U}^{\ot(i-1)}\ot\tau_{\U, \U}\ot \id_{\U}^{\ot(r-i-1)}$,
where $\tau_{\U, \U}$ is defined by (<ref>).
As above, let $\Sigma_{-1}\in \C\Sym_r$ be the Young symmetriser associated with the shape $(r)$.
Further, let $\mathfrak{M}_r: \U(\fg)^{\ot r}\lra \U(\fg)$ be the multiplication defined by
$u_1\ot u_2\ot \dots \ot u_r\mapsto u_1u_2\dots u_r$, for any $u_1, \dots, u_r\in\U(\fg)$.
Now we define $\Upsilon$ by requiring that it map any element $Y_1 Y_2\dots Y_r\in S_{gr}^r(\wedge^2_{gr} V)$
to $\frac{1}{r!}{\mathfrak M}_r\Sigma_{-1}\left(j(Y_1)\ot j(Y_2)\ot \dots\ot  j(Y_r)\right)$ in $\U(\fg)$ for any $r$.

We now prove the theorem.  Clearly, $(S_{gr}(\wedge^2_{gr} V))^G=\sum_{k\ge 0}(S^k_{gr}(\wedge^2_{gr} V))^G$ is $\Z_+$-graded with
(S_{gr}^0(\wedge^2_{gr} V))^G =\C$ and $(S_{gr}^1(\wedge^2_{gr} V))^G=0.
It follows from the first fundamental theorem of invariant theory for $G$ [6, 24] that the elements

\[
\wh{T}[\ell]= \sum_{a_1,\dots, a_{\ell-1},  b} (-1)^{\sum_{i=1}^{\ell-1} [a_i]} j^{-1}(X^{a_1}_b)  j^{-1}( X^{a_2}_{a_1})  j^{-1}(X^{a_3}_{a_2})\dots j^{-1}(X^b_{a_{\ell-1}}), \quad \ell\ge 2,
\]

together with the identity generate $(S_{gr}(\wedge^2_{gr} V))^G$.
Equation  (<ref>) states that

\[
\CF_\U(Z_\ell)= \sum_{a_1,\dots, a_{\ell-1},  b=1}^{m+2n} (-1)^{\sum_{i=1}^{\ell-1} [a_i]} X^{a_1}_b  X^{a_2}_{a_1} X^{a_3}_{a_2}\dots X^b_{a_{\ell-1}}.
\]

Thus for any $\ell$, we have

Υ^-1(_(Z_ℓ))=T[ℓ](S_gr^ℓ-1(∧^2_gr V))^G.
In particular, $\Upsilon^{-1}(\CF_\U(Z_2))=\wh{T}[2]$,  and by Lemma <ref>,
It is also evident that

Υ^-1(_(Z_k Z_ℓ))=T[k] T[ℓ](S_gr^k+ℓ-1(∧^2_gr V))^G.
Using the relations (<ref>) and (<ref>),  we can show by induction on the degree of $(S_{gr}(\wedge^2_{gr} V))^G$ that  the identity and
$\Upsilon^{-1}(\CF_\U(Z_\ell))$ for all $\ell\ge 2$ generate  $(S_{gr}(\wedge^2_{gr} V))^G$.
Since $\Upsilon^{-1}(\U(\fg)^G)=(S_{gr}(\wedge^2_{gr} V))^G$, the identity and the elements   $\CF_\U(Z_\ell)$ for $\ell\ge 2$ together generate $\U(\fg)^G$.
As the elements $Z_{2\ell}$ for all $\ell\ge 1$ generate $\Hom_{\AB(\sdim)}(0, 0)$ by Theorem <ref>(2),
we obtain the first statement of the theorem.

The second statement of the theorem now follows from Lemma <ref>.  This completes the proof.

In summary, we have

For any $r, s$, the functor $\CF_\U$ sends $\Hom_{\AB(\sdim)}(r, s)$ to the subspace $\left(\U\ot_\C \Hom_\C\left(V^{\ot r}, V^{\ot s}\right)\right)^G$ of  $G$-invariants. In particular,

_(_()(0, 0))=()^G.
Further,  if $\dim(V_{\bar 0})$ is odd or $0$, then
$Z(\U(\fg))=\CF_\U(\Hom_{\AB(\sdim)}(0, 0))$.

The first statement follows (<ref>), and (<ref>) follows from Theorem <ref>.
If $\dim(V_{\bar 0})$ is odd or $0$, then $\U(\fg)^\fg=\U(\fg)^G$ by Lemma <ref>.
This completes the proof of the Corollary.

To close this section, we complete the proof of Theorem <ref>. Let us recall some well known facts about central elements of the universal enveloping algebra $\U(\fso_m)$ of the orthogonal Lie algebra $\fso_m$.

By taking $V=\C^m$ to be purely even and equipped with the bilinear form $\omega(e^i, e^j)=\delta_{i j}$ for all $i, j$ in  Appendix <ref>, we have the orthogonal Lie algebra $\fso_m$, which is spanned by $J_{i j} = E^i_j-E^j_i$ for $i, j=1, 2, \dots, m$, where $E_j^i$ are the $m\times m$ matrix units.
Denote by  $X_{i j}$ the image  of $J_{i j}$ in $\U(\fso_m)$. Then it follows immediately from the first fundamental theorem of invariant theory for the orthogonal group
$O_m(\C)$ that as a subalgebra of $\U(\fso_m)$, $\U(\fso_m)^{O_m(\C)}$ is generated by

\[
I(r)=\sum_{i_1, i_2, \dots, i_r} X_{i_1 i_2} X_{i_2 i_3} \dots X_{i_r, i_1},  \quad r\ge 2.
\]

It is well-known that the central elements $I(2j)$ for $j=1, 2, \dots, \begin{bmatrix}\frac{m}{2}\end{bmatrix}$ are algebraically independent  and generate $\U(\fso_m)^{O_m(\C)}$.
These facts are easily proved following the line of reasoning suggested in <cit.>.

It is also straightforward to verify that $I(2k+1)$ may be explicitly written as a polynomial in the $I(2j)$.
This also follows from the more general statement in Theorem <ref>(2) (which is proved independently) by using  the functor $\CF_\U$ for $\fso_m$.

Retain the notation in Remark <ref>.
Let $\fg=\fso_m$ for any $m\ge 3$, and thus $\U(\fg)=\U(\fso_m)$. By using (<ref>),  we obtain

\[
\CF_{\U}(Z_{2j})=I(2j), \quad \forall j\ge 1.
\]

Now the $I(2j)$ for $j=1, 2, \dots, \left[\frac{m}{2}\right]$ are algebraically independent. Thus the elements
$Z_{2j}$ for all $j=1, 2, \dots, \left[\frac{m}{2}\right]$ must be algebraically independent, and this is true for any $m\ge 3$.
This proves Theorem  <ref>(4).

§.§ Categorification of characteristic identities for $\mathfrak{so}_m$ and $\mathfrak{sp}_{2n}$

In general, $\ker(\CF_\U)\ne 0$, and non-zero elements of $\ker(\CF_\U|_r)$ lead to interesting identities in $\U\ot\End_\C(V^{\ot r})$. Bracken and Green <cit.>,
[4, 15] discovered certain `characteristic identities' for Lie algebras in $\U\ot\End_\C(V)$, which may be interpreted this way.
Such characteristic identities have been widely studied in the mathematical physics literature (see [17] for a brief review and further references).

Throughout this section, $V$ is either $\C^{m|0}$ or $\C^{0|2n}$, and thus $\fg$ is $\mathfrak{so}_m$ or $\mathfrak{sp}_{2n}$.  We denote $d=\dim(V)$.

§.§.§ Characteristic identities for the orthogonal and symplectic Lie algebras

We begin by stating precisely what is meant by the term “characteristic identity” [3, 4, 15] for $\fg$ (also see [10]). Let $E$ be the element of  $\U(\fg)\ot\End_\C(V)$
given by $E=(\id_{\U(\fg)}\ot\mu)(t)$, where, as usual, $\mu$ denotes the $\fg$-action on $V$ and $t$ is the tempered Casimir element..
We say that $E$ satisfies a characteristic identity if there are explicitly known central elements $Q_i\in\Z(\U(\fg))$ such that

E^d + Q_1E^d-1 + Q_2E^d-2+…+Q_d=0.
Here the $Q_i$ in (<ref>) are known explicitly  in the sense that their images under the Harish-Chandra isomorphism are explicitly given, as explained in the proof of Theorem <ref>.

The central elements $Q_i$ appearing in any characteristic identity (<ref>) all belong to $\U(\fg)^G$.

By Lemma <ref>, it is always true that $Q_i\in \U(\fg)^G$ for all $i$ if $\fg=\mathfrak{sp}_{2n}$ or $\mathfrak{so}_m$ for odd $m$.
If $\fg=\mathfrak{so}_m$ with $m$ even, we let $\eta_0\in {\rm O}_m$ be such that $\det(\eta_0)=-1$. We can express $Q_i$ as $Q_i=Q_{i, +}+Q_{i, -}$
with $Q_{i, \pm}\in  Z(\U(\fg))$ such that $\eta_0.Q_{i, \pm}=\pm Q_{i, \pm}$, that is, $Q_{i, +}\in \U(\fg)^G$, and $Q_{i, -} \in \U(\fg)^{G, \sdet}$.
Since $\eta_0. E= E$ and $\eta_0.Q_i=Q_{i, +}-Q_{i, -}$, by combining (<ref>) with

\[
E^d + \eta_0.Q_1 E^{d-1} + \eta_0.Q_2 E^{d-2}+\dots+\eta_0.Q_d =0,
\]

we obtain
E^d + Q_{1, +}E^{d-1} + Q_{2, +}E^{d-2}+\dots+Q_{d, +}=0.
This proves the lemma.

§.§.§ Categorification of characteristic identities

The following theorem gives a categorical derivation of characteristic identities for  $\mathfrak{so}_m$ and $\mathfrak{sp}_{2n}$.

Maintain the above notation, and write $d=\dim(V)$. There exist elements $\wh{Q}_i\in \Hom_{\AB(\sdim)}(0, 0)$, for $i=1, 2, \dots, d=\dim(V)$, such that the element

\[
\wh\Q:=\BH^d + \wh{\Q}_1\BH^{d-1} + \wh{\Q}_2\BH^{d-2} +\dots + \wh{\Q}_{d}
\]

belongs to $\ker(\CF_\U|_1)$, where $\wh{\Q}_i = \wh{Q}_i\ot I$.
Furthermore, the characteristic identity (<ref>) may simply be stated in the present context as
\CF_\U(\wh\Q)=0.

This proof is partly based on [10].
Denote by $M_\lambda$ the Verma module with highest weight $\lambda$, and by $L_\lambda$ the simple quotient module. Then $M_\lambda\ot V= \sum_{\lambda_i}M_{\lambda+\lambda_i}$,
where the $\lambda_i$ for $i=1, ,2, \dots, d$ are the weights of a basis of weight vectors of $V$. Now
$E$ takes the eigenvalue

\[
e_i(\lambda)=\frac{1}{2}\left((\lambda+\lambda_i+2\rho, \lambda+\lambda_i)- (\lambda+2\rho, \lambda)-\chi_V(C)\right),
\]

in each submodule $M_{\lambda+\lambda_i}$, where $\chi_V(C)$ is the eigenvalue of $C$ in $V$.  Let

\[
R_\lambda=\prod_{i=1}^{d}\left(E- e_i(\lambda)\right).
\]

Then $(\mu_{M_\lambda}\ot\id)(R_\lambda)=0$ as an element of $\End_{\U}(M_\lambda\ot V)$. This implies

(μ_L_λ𝕀)(R_λ)=0,   ∀λ.

We expand $R_\lambda$ into a polynomial in $E$, obtaining
R_λ=E^d + P_1(λ) E^d-1 + P_2 (λ) E^d-2+…+ P_d (λ),
where $P_i(\lambda)$ are polynomials in the functions $e_j(\lambda)$ of $\lambda$,
which are essentially the elementary symmetric functions in the $e_i(\lambda)$, with suitable signs; e.g., $P_1(\lambda)=-\sum_i e_i(\lambda)$,
$P_2(\lambda)=\sum_{i\ne j} e_i(\lambda)e_j(\lambda)$,  and etc..
Denote $E(\lambda)=(\mu_{L_\lambda}\ot\id)(E)$. Then (<ref>) is equivalent to

E(λ)^d + P_1(λ) E(λ)^d-1 + P_2 (λ) E(λ)^d-2+…+ P_d (λ) =0,   ∀λ.

Using the fact that the set of weights of $V$ is invariant
with respect to the usual action of the Weyl group $W$ of $\fg$, one easily shows, as in [10] that the
$P_i(\lambda)$ are all invariant under the dot-action of $W$. Hence by Harish-Chandra's isomorphism, there exist elements $Q_i\in Z(\U)$ such that
$\mu_{M_\lambda}(Q_i)=P_i(\lambda)$ for all $i$.  We define
R=E^d + Q_1 E^d-1 + Q_2 E^d-2+…+ Q_d,
which is equal to the left hand side of (<ref>).  Now equation (<ref>) becomes
(μ_L_λ𝕀)(R)=0,   ∀λ.

Note that $R$ is a square matrix with entries in $\U(\fg)$. The above equation states that all entries of $R$ vanish in the irreducible representations $\mu_{L_\lambda}$ for all $\lambda$. This implies that $R=0$, which is equation (<ref>).

By Lemma <ref> and Corollary <ref>, there exist $\wh{Q}_i\in \Hom_{\AB(\sdim)}(0, 0)$ for $1\le i\le d$ such that $Q_i= \CF_\U(\wh{Q}_i)$,  and hence $Q_i\ot \id_V= \CF_\U(\wh{\Q}_i)$. Since $E=\CF_\U(\BH)$, the left hand side of  (<ref>) is the image of the element $\wh Q$ given in Theorem <ref> under $\CF_\U$. This proves the theorem.

When $V=\C^{0|2}$, we are dealing with the Lie algebra $\mathfrak{sp}_2$.

\[
\wh\Q=\BH^2 - 2\BH + \frac{1}{2}Z_2\ot I
\]

belongs to the kernel of $\CF_\U|_1$, and
the characteristic identity in this case is given by $\CF_\U(\wh\Q)=0$.

To be more explicit, let $X, Y, T$ be the standard generators of $\mathfrak{sp}_2=\fsl_2$ with

\[
[T, X]=2X,  \quad [T, Y]=-2Y, \quad [X, Y]=T.
\]

Then  the quadratic Casimir operator (<ref>) is given by

C= - 2 (H^2/2+ X Y + Y X)
in the current notation. As $V=\C^{0|2}$ has highest weight $1$, the eigenvalue of $C$ in $V$ is equal to $-2\left(\frac{1}{2}+1\right)= - 3= \sdim-1$.
The standard generators $X,Y,T$ act on $V$ with respect to the standard basis as the matrices $\begin{pmatrix} 0 & 1\\0 & 0 \end{pmatrix}$,
$\begin{pmatrix} 0 & 0\\1 & 0 \end{pmatrix}$ and $\begin{pmatrix} 1 & 0\\0 & -1 \end{pmatrix}$ respectively.

The corresponding tempered Casimir element is  $t=- 2\left( \frac{T\ot T}{2}+ X\ot Y + Y\ot X\right)$. Hence

\[
\begin{aligned}
\CF_\U(\BH)&=(\id_\U\ot\mu)(t)\\
&=-2\left( \frac{1}{2}T\ot  \begin{pmatrix} 1 & 0\\0 & -1 \end{pmatrix}    + X\ot \begin{pmatrix} 0 & 0\\1 & 0 \end{pmatrix}
+Y\ot \begin{pmatrix} 0 & 1\\0 & 0 \end{pmatrix}\right)\\
&=-2 \begin{pmatrix} \frac{T^2}{2} & Y\\  X & -  \frac{T^2}{2} \end{pmatrix}, \text{ and } \\
\CF_\U(Z_2)&=\text{str} (\CF_\U(\BH)^2)=2C.
\end{aligned}
\]

§.§.§ Comments on characteristic identities for Lie superalgebras and quantum groups

There are discussions of characteristic identities for Lie superalgebras in the literature (see, e.g., [18]), but largely in the form of (<ref>)
for any finite dimensional simple module $L_\lambda$ (that is, replacing $M_\lambda$ by $L_\lambda$).
It appears to be difficul to generalise the method of [10] to lift (<ref>) for all $\lambda$ to the identity (<ref>)
in $\U\ot\End_\C(V)$ for $\osp_{m|2n}$ and $\gl_{m|n}$ of general $m, n$.
The problem lies in the fact that the $P_i(\lambda)$ for atypical $\lambda$ are not invariant
under translation by scalar multiples of atypical roots $\gamma$, that is,
$P_i(\lambda)\ne P_i(\lambda+c\gamma)$ in general for nonzero $c\in\C$ and isotropic root $\gamma$ such that $(\lambda+\rho, \gamma)=0$. By the generalised Harish-Chandra isomorphism for Lie superalgebras,  there exist no central elements
of $\U$ whose eigenvalues are $P_i(\lambda)$ in $M_\lambda$ for all $\lambda$.  This problem does not exist for  $\osp_{1|2n}$, as this Lie superalgebra has no atypical weights.

However, any non-zero element $\wh\Q\in \ker(\CF_\U|_1)$ would lead to an identity
$\CF_\U(\wh\Q)=0$  in $\U\ot\End_\C(V)$, which may be considered a generalisation of the characteristic identities above.
For the purpose of constructing such identities, we need only consider the generators of $\ker(\CF_\U|_1)$ as a $2$-sided ideal in $\Hom_{\AB(\sdim)}(1, 1)$.

Characteristic identities for quantum groups were investigated in [11] (also see [33]).
It is quite straightforward to use the restricted coloured tangle category (with one pole only) introduced
in <cit.> to give a categorical treatment of the quantum characteristic identities analogous to what has been presented here.

§ APPLICATION TO CATEGORY $\MATHCAL O$ OF $\FSP_2$

Throughout this section, we assume that $V=\C^{0|2}$ and $\omega=\begin{pmatrix}0 &1\\ -1 & 0\end{pmatrix}$. Thus $\osp(V; \omega)$ is the Lie algebra $\fsp_2(\C)$.

We investigate the functor of Theorem <ref> in some depth in this special case. Clearly $\sdim=-2$. The following result shows that the functor
$\CF_M$ factors through the Temperley-Lieb category in this case.

For any  $\fsp_2$-module $M$,  the functor $
\CF_M: \AB(-2)\lra \CT_M(V)$ given in Theorem <ref> factors through $\ATL(-2)$, that is,
there exists a unique functor $\CF^{TL}_M: \ATL(-2) \lra \CT_M(V)$ such that the following diagram commutes,

\[
\begin{tikzcd}
\AB(-2) \arrow[d] \arrow[r, "\CF_M"] & \CT_M(V)\\
\ATL(-2) \arrow[ur, "\CF^{TL}_M"'pos=0.43].
\end{tikzcd}
\]

Note that $V\ot V=\C^{0|2}\ot\C^{0|2}=L_s \oplus L_a$ as $\fsp_2$-module, where $L_s$ and $L_a$ are simple submodules of dimensions $1$ and $3$ respectively. Thus the permutation $\tau: V\ot V\lra V\ot V$ is given by
$\tau = -P_a + P_s = 1 + 2 P_s$.  As $\sdim(V)=-2$, the map $e$ in (<ref>) is given by $e=-2 P_s$ in this case, and hence
\tau = 1- e.
This shows that $\CF(\Theta)=0$ because $\delta=\sdim(V)=-2$, where $\Theta$ is the element of $\AB(-2)(2,2)$ depicted in Fig. 14. By definition, this shows
that the functor $\CF_M$ factors through $\ATL(-2)$.

Maintain the notation above and fix $\lambda\in\C$. Assume that the $\fsp_2$-module $M$ is either the Verma module $M_\lambda$ with highest weight $\lambda$ or the simple module $L_\lambda$. Then the functor
$\CF^{TL}_M: \ATL(-2) \lra \CT_M(V)$ factors through $\TLBC(-2, \lambda)$, that is there exists a unique functor
$\CF^{TLB}_\lambda: \TLBC(-2, \lambda)\lra \CT_M(V)$
such that the following diagram commutes,

\[
\begin{tikzcd}
\ATL(-2) \arrow[d] \arrow[r, "\CF^{TL}_M"] & \CT_M(V)\\
\TLBC(-2, \lambda) \arrow[ur, "\CF^{TLB}_\lambda"'pos=0.43],
\end{tikzcd}
\]

where the vertical functor is the quotient functor (see Definition <ref>).

We have $\CF^{TL}_M(Z_2)=2C: M\lra M$, where $C$ is the quadratic Casimir given in (<ref>), which acts on $M=M_\lambda$ and a fortiori on $L_\lambda$
as multiplication by the scalar $\chi_\lambda(C)= - \lambda(\lambda +2)$. Hence for $\delta=-2$,

\[
\CF^{TL}_M\left(\frac{Z_2}{\delta}\right)= \lambda(\lambda +2) = -\lambda\left(\frac{\delta-2}{2}-\lambda\right).
\]

This shows that $\CF^{TL}_M$ factors through $\TLBC(-2, \lambda)$.

Let us denote $\Z_{<-1}=\{-2, -3, -4, \dots\}$.  For any given $\lambda\in\C\backslash \Z_{<-1}$, the Verma module $M_\lambda$ is projective in category ${\CO}$ of $\fsp_2$.

The next theorem is an analogue in our context of <cit.>.

Maintain the above notation and let $M_\lambda$ be the Verma module for $\fsp_2$ with highest weight $\lambda\in\C\backslash \Z_{<-1}$. Then
the functor $\CF^{TLB}_\lambda: \TLBC(-2, \lambda)\lra \CT_{M_\lambda}(V)$
is an isomorphism of categories.

It is evident that the restriction of $\CF^{TLB}_\lambda$  to objects is an isomorphism. Since $M_\lambda$ with $\lambda\in\C\backslash \Z_{<-1}$ is projective in category ${\CO}$, we have

\[
\begin{aligned}
\dim \Hom_{\fsp_2}(M_\lambda \ot V^{\ot r}, M_\lambda \ot V^{\ot s})
&\cong \dim \Hom_{\fsp_2}(M_\lambda, M_\lambda \ot V^{\ot (r+s)}) \\
&\cong \dim (V^{\ot (r+s)})_0, \quad \forall r, s,
\end{aligned}
\]

where $(V^{\ot (r+s)})_0$ is the $0$-weight subspace of $V^{\ot (r+s)}$, which has dimension $\begin{pmatrix}2N \\ N\end{pmatrix}$ if $r+s=2N$ is even, and is $0$ otherwise.
It follows  from Theorem <ref> that for all $r, s$,

\[
\dim \Hom_{\TLBC(-2, \lambda)}(r, s) =\dim \Hom_{\fsp_2}(M_\lambda \ot V^{\ot r}, M_\lambda \ot V^{\ot s}).
\]

Hence it suffices to show that the restriction of $\CF^{TLB}_\lambda$  to morphisms is injective.
Only the case with $r+s=2N$ requires proof, and we will adapt the main idea in the treatment of $\U_q(\fsl_2)$ representations in [16] to the present context.

Write $\CW(r, 2N-r)=\Hom_{\TLBC(-2, \lambda)}(r, 2N-r)$ for all $r$, and $\CW(2N)=\CW(0, 2N)$.
Let $E^0_{2N}$ be the subalgebra of $\CW(2N, 2N)$ spanned by diagrams without connectors, which is isomorphic to the Temperley-Lieb algebra $TL_{2N}(-2)$. Note that
$\CF^{TLB}_\lambda(E^0_{2N})=\CF(E^0_{2N}) \cong E^0_{2N}$ as algebra. Furthermore, $\CW(2N)$ is a module for this algebra.
We want to show that $\CF^{TLB}_\lambda(\CW(2N))\cong \CW(2N)$ as $TL_{2N}(-2)$-module.

Consider images of the standard affine Temperley-Lieb diagrams as shown in Figure <ref>, which will be called standard
affine Temperley-Lieb diagrams of type $B$. They form a basis of $\CW(2N)$. As in the situation in the proof of Theorem <ref>,
the standard $(0, 2N)$-diagrams with at most $i\le N$ connectors span a $TL_{2N}(-2)$-submodule $F_i\CW(2N)$ of $\CW(2N)$, and
$\CW_{2i}(2N) = \frac{F_i \CW(2N)}{F_{i-1}\CW(2N)}$ is a simple $TL_{2N}(-2)$-module.
Hence $\CF^{TLB}_\lambda(\CW_{2i}(2N)) = \frac{\CF^{TLB}_\lambda(F_i \CW(2N))}{\CF^{TLB}_\lambda(F_{i-1}\CW(2N))}\cong \CW_{2i}(2N)$ or $0$.  We show that it is non-zero for all $i$.

Let $\BD_t$ be the standard $(0, 2N)$-diagram with $t$ connectors as depicted below.

\[
\begin{picture}(180, 80)(-40, 20)
\put(-40, 60){$\BD_t\  =$}
\linethickness{1mm}
\put(0, 20){\line(0, 1){80}}
\put(0, 90){\uwave{\hspace{5mm}}}

\put(15, 100){\line(0, -1){20}}
\put(25, 100){\line(0, -1){20}}
\qbezier(15, 80)(20, 75)(25, 80)

\put(0, 70){\uwave{\hspace{12mm}}}

\put(35, 100){\line(0, -1){40}}
\put(45, 100){\line(0, -1){40}}
\qbezier(35, 60)(40, 55)(45, 60)

\put(0, 40){\uwave{\hspace{23mm}}}
\put(65, 100){\line(0, -1){70}}
\put(75, 100){\line(0, -1){70}}
\qbezier(65, 30)(70, 25)(75, 30)

\put(48, 80){$\dots$}
\put(10, 50){$\vdots$}

\put(85, 100){\line(0, -1){20}}
\put(95, 100){\line(0, -1){20}}
\qbezier(85, 80)(90, 75)(95, 80)

\put(100, 90){$\dots$}

\put(120, 100){\line(0, -1){20}}
\put(130, 100){\line(0, -1){20}}
\qbezier(120, 80)(125, 75)(130, 80)

%\qbezier(55, 28)(70, 38)(75, 0)
%\qbezier(55, 8)(70, -2)(75, 40)

\put(135, 25){.}
\end{picture}
\]

Then $\BD_t$ belongs to $F_t\CW(2N)$, but is not in $F_{t-1}\CW(2N)$.
Postmultiplying $\I_{2t}\ot \underbrace{\cap \dots \cap}_{N-t}$ with $\BD_t$, we obtain the standard $(0, 2t)$-diagram $\wt\BD_t$ shown below
up to a scalar factor.

\[
\begin{picture}(150, 80)(-40, 20)
\put(-40, 60){$\wt\BD_t\  =$}
\linethickness{1mm}
\put(0, 20){\line(0, 1){80}}
\put(0, 90){\uwave{\hspace{5mm}}}

\put(15, 100){\line(0, -1){20}}
\put(25, 100){\line(0, -1){20}}
\qbezier(15, 80)(20, 75)(25, 80)

\put(0, 70){\uwave{\hspace{12mm}}}

\put(35, 100){\line(0, -1){40}}
\put(45, 100){\line(0, -1){40}}
\qbezier(35, 60)(40, 55)(45, 60)

\put(0, 40){\uwave{\hspace{23mm}}}
\put(65, 100){\line(0, -1){70}}
\put(75, 100){\line(0, -1){70}}
\qbezier(65, 30)(70, 25)(75, 30)

\put(48, 80){$\dots$}
\put(10, 50){$\vdots$}

\put(85, 25){;}
\end{picture}
\begin{picture}(150, 80)(-40, 20)
\put(-40, 60){$\wt\BD_t^*\  =$}
\linethickness{1mm}
\put(0, 20){\line(0, 1){80}}

\put(0, 90){\uwave{\hspace{23mm}}}
\put(0, 70){\uwave{\hspace{12mm}}}
\put(0, 40){\uwave{\hspace{5mm}}}

\put(15, 20){\line(0, 1){25}}
\put(25, 20){\line(0, 1){25}}
\qbezier(15, 45)(20, 50)(25, 45)

\put(35, 20){\line(0, 1){55}}
\put(45, 20){\line(0, 1){55}}
\qbezier(35, 75)(40, 80)(45, 75)

\put(65, 20){\line(0, 1){75}}
\put(75, 20){\line(0, 1){75}}
\qbezier(65, 95)(70, 100)(75, 95)

\put(48, 50){$\dots$}
\put(5, 50){$\vdots$}

\put(85, 25){.}
\end{picture}
\]

Now premultiply the tensor product $\wt{\BD}_t\ot I_{2t}$ with the $(0, 4t)$-diagram

\[
\begin{picture}(150, 40)(-40, -5)
\put(-40, 10){$\A_{4t}\ =$}
\linethickness{1mm}
\put(0, 0){\line(0, 1){30}}

\qbezier(10,0)(50, 50)(90, 0)
\qbezier(40,0)(50, 40)(60, 0)

\put(25, 10){$\dots$}
\put(25, 0){\tiny$2t$}

\put(95, 0){.}
\end{picture}
\]

One obtains essentially (up to a non-zero scalar) the $(2t, 0)$-diagram $\wt{\BD}_t^*$ shown above.
Denote by $m_+$ the highest weight vector of $M_\lambda$.  It is easy to show that
^TLB_λ(_t^*)(m_+ v_-1…v_-1_2t)= (-2 Y)^t m_+0.

Let $\BD_{<t}$ be an element in $F_{t-1}\CW(2N)$.  We perform the same operations as above on $\BD_{<t}$ to obtain an element $\wt\BD_{<t}^* \in \Hom_{\TLBC(-2, \lambda)}(2t, 0)$, which can be expressed as a linear combination of $(2t, 0)$-diagrams with no more than $t-1$ connectors. Thus each $(2t, 0)$-diagram in $\wt\BD_{<t}^*$ with non-zero coefficient must have a
component of the form $\{\cap\}$ which is not connected to the pole via a connector.
Note that for all $i+j=2t-2$, we have
$\id_{M_\lambda}\ot \id_V^{\ot i} \ot \CF(\cap)\ot \id_V^{\ot j} (m_+\ot \underbrace{ v_{-1}\ot\dots\ot v_{-1}}_{2t})=0$ because $\omega(v_{-1}, v_{-1})=0$. Thus
^TLB_λ(_<t^*)(m_+ v_-1…v_-1_2t)=0.

Comparing the two equations above, we conclude that
\CF^{TLB}_\lambda(\CW_{2t}(2N)) \ne 0
for all $t$, and hence by cellular theory [12], $\CF^{TLB}_\lambda(\CW_{2t}(2N))$  $\cong \CW_{2t}(2N)
This shows that $\CF^{TLB}_\lambda$ is injective when restricted to morphisms, completing the proof of the theorem.

§ COMMENTS AND OUTLOOK

We point out directions for further research, and make more comments on relationship of the present paper to other work.

§.§ Relationship to other work

§.§.§ Comments on other work
In contrast to the categories in [29, 1], both of which are monoidal categories, our category $\AB(\delta)$ was designed to only admit a tensor product functor $\ot: \AB(\delta) \times \CB(\delta)\lra  \AB(\delta)$ with the usual Brauer category $\CB(\delta)$, but not a natural monoidal structure. The reason for this is that we want to use $\AB(\delta)$ to study the category $\CT_M(V)$ of $\fg$-modules with objects $M\ot V^{\ot r}$, which only has a natural tensor product  $\ot: \CT_M(V)\times \CT(V) \lra \CT_M(V)$ with the category $\CT(V)$ of the modules $V^{\ot r}$.  We do not allow for cups and caps which would close the pole into a loop as $M$ may be infinite dimensional.

In order to compare our category with that of [29], we consider the respective endomorphism algebras.
The endomorphism algebras of the category in [29] are the Nazarov–Wenzl algebra
at different degrees defined in <cit.>  with generators and relations
$(VW.1)$ – $(VW.8)$, which involve an infinite family of
arbitrary parameters $w_k$ for all non-negative integers $k$, see $(VW.3)$ and $(VW.4)$ in <cit.> .

An inspection of the structures of the algebras under consideration suggests that we consider quotient algebras of $\Hom_{\AB(\delta)}(r, r)$ defined as follows. Given any scalars $z_{2\ell}$ for $\ell=0, 1, 2, \dots$, we let $\mathfrak{M}_z$ be the maximal ideal  of $\Hom_{\AB(\delta)}(0, 0)$ generated by $Z_{2\ell}-z_{2\ell}$ for all $\ell$.
Then $\Hom_{\AB(\delta)}(0, 0)/\mathfrak{M}_z\cong K$, as all $Z_{2\ell}$ are algebraically independent, and all $Z_{2j+1}$ ($Z_1=0$) can be expressed as polynomials in the elements $Z_{2\ell}$, as shown in Theorem <ref>. Denote by $z_{2\ell+1}$ the image of  $Z_{2\ell+1}$ for all $\ell$. Then in particular, we have $z_1=0$.

The following fact is immediate by using the description of $\Hom_{\AB(\delta)}(r, r)$ given in Theorem <ref>.

Given any scalars $z_{2\ell}\in\C$ for $\ell\ge 0$ with $z_0=\delta$, and let $z_{2\ell+1}\in\C$ be defined as above. There is an algebra homomorphism from $\Hom_{\AB(\delta)}(r, r)/\mathfrak{M}_z\ot I_r$ to
the Nazarov–Wenzl algebra of degree $r$ (in the notation of <cit.>) with parameters
$w_0=\delta$ and $w_k= \sum_{i=1}^k \begin{pmatrix}k\\ i\end{pmatrix}\left(\frac{1-\delta}2\right)^i  z_i$ for $k\ge 1$, defined by
ι(s_i)↦s_i,   ι(e_i)↦e_i,   Θ_i+1-δ/2 _r↦y_i.

This indicates that the affine Brauer category of [29] is a quotient of $\AB(\delta)$.

The affine VW supercategory in [1] was built for studying
$\CT_M(V)$ for the periplectic Lie superalgebra $\mathfrak{p}(n)$.  The cups and caps in the
affine VW supercategory
(analogues of $\cup$ and $\cap$ in our $\AB(\delta)$) were designed to account for
the odd invariant non-degenerate bilinear form of the natural  module $V$.
This makes the supercategory different from $\AB(\delta)$ at a fundamental level.

However, an inspection reveals that the endomorphism algebras of the affine VW supercategory
have similar general features as those of $\AB(\delta)$ at $\delta=0$, ignoring the differences
in signs in the defining relations.
It would be interesting to follow [1] to construct a priori an affine VW category of dotted Brauer diagrams, adapted to the study of $\osp_{m|2n}(\C)$ modules.
An important question would then be whether it is equivalent to the category introduced in this paper.

§.§.§ Quantum versus classical

Heuristically [16] may be regarded as the quantum version of the present work.
The diagrammatic affine Brauer category studied here may be considered as
the “semi-classical limit” of a polar BMW category [16].
We identify over and under crossings of thin strings in polar braid diagrams,
and replace the combination of polar braid diagrams on the left hand side of the following relation

\[   %\begin{figure}[h]
\setlength{\unitlength}{0.2mm}
\begin{picture}(50, 50)(-30,40)
\linethickness{.7mm}
\put(-15, 30){\line(0, 1){18}}
\put(-15, 52){\line(0, 1){38}}
\qbezier(0, 90)(-7, 80)(-12, 72)
\qbezier(-18, 68)(-25, 60)(-15, 50)
\qbezier(-15, 50)(-15, 50)(0, 30)
\put(5, 50){$-$}
\end{picture}
\setlength{\unitlength}{0.2mm}
\begin{picture}(100, 50)(-30,40)

\linethickness{.7mm}
\put(-15, 30){\line(0, 1){60}}
\put(5, 30){\line(0, 1){60}}
\put(25, 50){$\rightsquigarrow$}
\end{picture}
\setlength{\unitlength}{0.2mm}
\begin{picture}(45, 50)(-30,40)
\put(-35, 50){$\hbar$}
\linethickness{.7mm}
\put(-15, 30){\line(0, 1){60}}
\put(-14, 60){\uwave{\hspace{4mm}}}
\put(8, 30){\line(0, 1){60}}
\put(20, 50){$+\  o(\hbar^2)$,}
\end{picture}
\]

by the combination of affine Brauer diagrams on the right side.
Then the affine Brauer diagram
(22, 20)(0,3)

(5, 0)(0, 1)20

(3, 12)

(15, 0)(0, 1)20
(18, 5),

which is the new ingredient needed for the development of our diagrammatic affine Brauer category,  can be considered as the $\hbar\to 0$ limit of $\frac{1}{\hbar}$ times the left hand side of the above relation.
It is such heuristics that led us to the investigation in this paper. We point out that heuristics of a similar nature also appeared in <cit.>.

§.§ Outlook

The present work points to several directions for further exploration.

§.§.§ Oriented affine Brauer category

Recall from <cit.> the oriented Brauer category $\OB(\delta)$, which is the natural diagrammatic category required for studying the representations of the general linear superalgebra $\gl(V)$ of the type
$V^{\ot (\varepsilon_1, \varepsilon_2, \dots, \varepsilon_r)}:=V^{\varepsilon_1}\ot V^{\varepsilon_r}\ot  \dots \ot V^{\varepsilon_r}$,  for all $r\in\N$, with $\varepsilon_i=\pm$, where $V^+=V$ is the natural module, and $V^-=V^*$ is the dual module.

The objects of $\OB(\delta)$ are sequences $(\varepsilon_1, \varepsilon_2, \dots, \varepsilon_r)$ for $r\in\N$ with $\varepsilon_i=\pm$, and morphisms are spanned by oriented Brauer diagrams, that is, Brauer diagrams with oriented strings.   An oriented Brauer $(k, \ell)$-diagram belongs to
$\Hom_{\OB(\delta)}(s(\Gamma), t(\Gamma))$ with $s(\Gamma)=(\varepsilon_1,   \varepsilon_2, ...,
\varepsilon_k)$ and target $t(\Gamma)=(\varepsilon'_1,  \varepsilon'_2, ...,
\varepsilon'_\ell)$, if for each $p=1, \dots, k$, the $p$-th bottom vertex is an end point of a string with the arrow pointing outward (resp. inward) when $\varepsilon_p=+$ (resp. $\varepsilon_p=-$), and for each $q=1, \dots, \ell$,
the $q$-th top vertex is an end point of a string with the arrow pointing inward (resp. outward)  when $\varepsilon'_q=+$ (resp. $\varepsilon'_q=-$). Composition of morphisms preserves orientation.

Define the oriented $H$-diagrams in $\OB(\delta)$ by Figure <ref>.

(120, 40)(0, 0)

(0, 0)(0, 1)40
(0, 20)(1, 0)20

(20, 0)(0, 1)40

(0, 40)(0, 1)1
(20, 40)(0, 1)1

(35, 15)$:=\ $

(60, 40)(80, 0)(80, 0)
(60, 0)(80, 40)(80, 40)

(60, 40)(-1, 2)1
(80, 40)(1, 2)1

(90, 5),

(80, 40)(0, 0)

(0, 0)(0, 1)40
(0, 20)(1, 0)20

(20, 0)(0, 1)40

(0, 0)(0, -1)1

(20, 0)(0, -1)1

(35, 15)$:=\ $

(60, 40)(80, 0)(80, 0)
(60, 0)(80, 40)(80, 40)

(80, 0)(1, -2)1
(60, 0)(-1, -2)1

(120, 60)(0, 0)

(0, 0)(0, 1)40
(0, 20)(1, 0)20

(20, 0)(0, 1)40

(0, 40)(0, 1)1

(20, 2)(0, -1)1

(35, 15)$:= -  $

(65, 40)(75, 10)(85, 40)
(65, 0)(75, 30)(85, 0)

(65, 40)(-1, 2)1
(85, 0)(1, -2)1
(95, 5),

(80, 60)(0, 0)

(0, 0)(0, 1)40
(0, 20)(1, 0)20

(20, 0)(0, 1)40

(0, 0)(0, -1)1

(20, 40)(0, 1)1

(35, 15)$:= -  $

(65, 40)(75, 10)(85, 40)
(65, 0)(75, 30)(85, 0)

(65, 0)(-1, -2)1
(85, 40)(1, 2)1

Oriented $H$-diagrams

There is a natural oriented version of the diagrammatic affine Brauer category $\AB(\delta)$.  To construct such a category, we introduce two oriented analogues of $\BH$ as depicted in Figure <ref>.

(80, 40)(0,0)

(35, 0)(0, 1)40

(35, 20)

(55, 0)(0, 1)40

(65, 5),

(80, 40)(0,0)

(35, 0)(0, 1)40

(35, 20)

(55, 40)(0, -1)40

Oriented $\BH$-diagrams

We can then build oriented affine Brauer diagrams as in Figure <ref> with oriented Brauer diagrams and oriented $\BH$-diagrams, which obey

* oriented versions of four-term-relations,  and

* skew symmetry of oriented $\BH$-diagrams.

The oriented four-term-relations involve oriented analogues of $\BH_{12}$, which are the juxtapositions of a pole with any of the oriented $H$-diagrams shown in Figure <ref>. Skew symmetry of oriented $\BH$-diagrams is shown in Figure <ref>.  Note that the oriented $H$-diagrams in Figure <ref> also have similar skew symmetry properties.

(90, 40)(30, 0)

(35, 0)(0, 1)40

(35, 20)

(55, 8)(0, 1)20

(55, 28)(70, 38)(75, 0)
(55, 8)(70, -2)(75, 40)
(75, 40)(0, 1)1

(85, 15)$= \ - $

(50, 40)(40,0)

(35, 0)(0, 1)40

(35, 20)

(55, 0)(0, 1)40

(65, 5);

(90, 40)(30, 0)

(35, 0)(0, 1)40

(35, 20)

(55, 8)(0, 1)20

(55, 28)(70, 38)(75, 0)
(55, 8)(70, -2)(75, 40)
(75, 0)(0, -1)1

(85, 15)$= \ - $

(30, 40)(40,0)

(35, 0)(0, 1)40

(35, 20)

(55, 40)(0, -1)40

Skew symmetry of oriented $\BH$-diagrams

Now the oriented diagrammatic affine Brauer category $\OAB(\delta)$ has the  same objects as $\OB(\delta)$, and morphisms being linear combinations of oriented affine Brauer diagrams.

There is a functor from $\OAB(\sdim(V))$ to the category of $\gl(V)$-modules with objects $M\ot V^{\ot (\varepsilon_1, \varepsilon_2, \dots, \varepsilon_r)}$ for an arbitrary but fixed $\gl(V)$-module $M$.  In particular, the work [30] can be cast into this framework.  The functor is full for any of the special class of typical Kac modules $M$ (which are projectiv in the category of finite dimensional $\gl(V)$-modules) studied in [30].

§.§.§ Enhanced affine Brauer category

As we pointed out in Remark <ref>,  while the Brauer category enables us to describe all the morphisms among the tensor powers of the natural module $V$
for the orthosymplectic supergroup $\OSp(V; \omega)$, it does not yield all the $\osp(V; \omega)$ morphisms among the tensor powers of the natural module.
This problem is present even for the orthogonal Lie algebra $\fso_m(\C)$. We solved the problem in the $\fso_m(\C)$ case by introducing an enhanced Brauer category
in [25], which admits a full functor to the full subcategory of $\fso_m(\C)$-modules with objects $(\C^m)^{\ot r}$ for all $r\in\N$.

It is straightforward to follow the procedure of Section <ref> to develop an enhanced affine Brauer category by replacing usual Brauer diagrams by  the enhanced Brauer diagrams of [25]. Such a category will enable us to resolve the problem discussed in Remark <ref>  in the case of $\fso_m$, that is, the map $\id_M\ot \det: M\lra M\ot (\C^m)^{\ot m}$ with
$\C \det(1)=\wedge^m(C^m)$ can be counted for by the category. One can also build such a category in the general $\osp$ case once an “enhanced Brauer category” similar to that of  [25] is developed for this case.

§.§.§ Generalised Schur-Weyl-Brauer dualities

An interesting question from the representation theoretical point of view is to extend the work of Section <ref> on $\fsp_2$ to find out for what $\U(\fg)$-modules $M$, the analogues functors of $\CF_M$ from the affine Brauer category, oriented affine Brauer category and enhanced affine Brauer category to appropriate representation categories of classical Lie superalgebras are full. Such full functors lead to
generalised Schur-Weyl-Brauer dualities.  Beside the projective Verma modules for $\fsp_2$ studied in Section <ref>,
the typical Kac modules for $\fg=\gl(m|n)$ investigated in [30], and the class of projective Verma modules for $\U_q(\fsl_2)$ studied in [16] are known examples.

In another publication, we investigate this question for many classes of projective generalised Verma modules in parabolic category $\CO$.

§ BASICS OF THE LIE SUPERALGEBRA $\OSP(V; \OMEGA)$

We retain the notation in Section <ref>. In particular, $V=\C^{m|2n}$ is equipped with the non-degenerate bilinear form $\omega$ which is even and supersymmetric, and we write $\fg=\mathfrak{osp}(V; \omega)$.

We give a complete proof of (<ref>). When $\sdim(V)\ne 0$, the formula (<ref>) can be proved
using representation theoretical arguments as in Section <ref>. However, the case $\sdim(V)=0$  is  more difficult,
and the most efficient way to prove it is by explicit calculations with convenient bases for $V$ and $\fg$.

We also present a self-contained treatment of the quadratic Casimir of $\U(\fg)$, and obtain explicit formulae for central elements which essentially generate the centre.

§.§ Tensorial notation for $\osp(V; \omega)$

Let us take the standard basis
$\{e^a\mid 1\le a\le {m+2n}\}$ for $V$, which is homogeneous with $e^i$ being even for all $i\le m$, and $e^{m+j}$ being odd for all $j\ge 1$.  We use $[v]=0, 1$ to denote the $\Z_2$-degree of a homogeneous element $v\in V$, and write $[a]=[e^a]$ for all $a$ for simplicity of notation.

The form $\omega$ can be  represented by an invertible matrix $g^{-1}=(g^{a b})$ with respect to the standard basis $\{e^a\mid a=1, 2, \dots, m+2n\}$ of $V$,  where $g^{a b}=\omega(e^a, e^b)$ for all $a, b$. We write $g=(g_{a b})$.  Note that $g$ has the following symmetry properties: $g_{a b}\ne 0$ only if $[a]=[b]$, and $g_{a b} = (-1)^{[a]} g_{b a} = (-1)^{[b]} g_{b a}$. The elements $e_b= \sum_c g_{b c} e_c$ satisfy

\[
\omega(e_b, e^a) = \sum_c g_{b c} \omega(e^c, e^a)  = \sum_c g_{b c} g^{c a}  = \delta^a_b,
\]

where $\delta^a_b$ is the usual Kronecker delta, which is equal to $1$ if $a=b$ and $0$ otherwise.  Thus $\{e_a\mid a=1, 2, \dots, m+2n\}$ is a basis of $V$ dual to the standard basis.  We have $e^a = \sum_b g^{a b} e_b$.
Note in particular that the element $j^{-1}(\id_V)$ in (<ref>) can be expressed as $j^{-1}(\id_V)=\sum_{a=1}^{m+2n} e^a\ot e_a$.

Let $E^a_b\in\End_\C(V)$ (where $a, b=1, 2, \dots, m+2n$) be the matrix units defined relative to the standard basis, that is,  $E^a_b(e^c)=\delta^c_b e^a$. Let $E_{a b}=\sum_c g_{a c} E^c _b$, then $E_{a b}(e^c)=e_a\delta_b^c$.
Introduce the following elements of $\End_\C(V)$,

J_a b = E_a b - (-1)^[a][b] E_b a,   a, b=1, 2, …, m+2n,
where clearly $J_{a b}=-(-1)^{[a][b]}J_{b a}$. We also let

\[
\begin{aligned}
J^a_b :=\sum_{a'} g^{a a'} J_{a' b}	&=E^a_b -  (-1)^{[a][b]}  \sum_{b', a'} g^{a a'} g_{b b'} E^{b'}_{a'}.
%	&= E^a_b - (-1)^{[a]([a]+[b])} \sum_{b', a'}   g_{b b'} E^{b'}_{a'} g^{a' a}
\end{aligned}
\]

The elements $J_{a b}$ satisfy the following commutation relations [18, 32]

[J_a b, J_c d]    = g_c b J_a d + (-1)^[a]([b]+[c])g_d a  J_b c

- (-1)^[c][d] g_d b J_a c - (-1)^[a][b] g_c a J_b d,

where $[X, Y]=XY - (-1)^{[X][Y]} Y X$ is the $\Z_2$-graded commutator of $X, Y$.
They span the orthosymplectic Lie superalgebra $\fg$. Clearly $J_{i j}$, $J_{i, m+k}$ and  $J_{m+k, m+\ell}$ with $1\le i< j\le m$, $1\le k\le \ell \le 2n$ are linearly independent, and hence $\dim(\fg)=\frac{1}{2}m(m-1) + \frac{1}{2}2n(2n+1) + 2m n$.

Let us verify that the bilinear form $\omega$ is indeed invariant with respect to this Lie superalgebra.
Note that

J_a b(e^c)   =e_a δ^c_b - (-1)^[a][b] e_b δ_a^c.

Using this relation we obtain

\[
\begin{aligned}
\omega(J_{a b}(e^c), e^d) &= \delta_a^d \delta^c_b - (-1)^{[a][b]} \delta_b^d\delta_a^c, \\
\omega(e^c, J_{a b}(e^d))
%&=  (e^c, e_a\delta^d_b - (-1)^{[a][b]} e_b\delta^d_a) \\
%&= (-1)^{[a]} \delta^c_a\delta^d_b - (-1)^{[a][b] +[b]} \delta^c_b\delta^d_a\\
&= -(-1)^{([a]+[b])[c]} \left(\delta^d_a \delta^c_b- (-1)^{[a][b]} \delta^d_b \delta^c_a\right),
\end{aligned}
\]

which immediately lead to

\[
\omega(J_{a b}(e^c), e^d) + (-1)^{([a]+[b])[c]} \omega(e^c, J_{a b}(e^d))=0, \quad \forall a, b, c, d.
\]

This proves the invariance of $\omega$ with respect to the
Lie superalgebra spanned by the elements $J_{a b}$.

§.§ Proof of Lemma <ref>

Denote by $X_{a b}$ (resp. $X^a_b$) the image of $J_{a b}$ (resp. $J^a_b$) in $\U(\fg)$ under the canonical embedding $\fg\hookrightarrow \U(\fg)$. Now we use “tensor contraction” to write down
the quadratic Casimir of $\U(\fg)$,

C=1/2∑_a, b=1 ^m+2n(-1)^[b] X^a_b X^b_a,
where the normalisation factor $\frac{1}{2}$ is chosen in order for Theorem <ref> to be valid.

The Casimir operator (<ref>) in the case of $\mathfrak{sp}_2=\fsl_2$ is $-2$ times the standard Casimir operator of $\fsl_2$, see Example <ref>.

Now the tempered Casimir element $t=\frac{1}{2}\left(\Delta(C)- C\ot 1 - 1 \ot C\right)$ is equal to

t=1/2∑_a, b=1 ^m+2n X^a_b (-1)^[b] X^b_a.

We have the following formula.
(𝕀_μ)(t)   =   ∑_a, b X^a_b(-1)^[b] E^b_a.

Using  $\mu(X^a_b)=J^a_b$ and $X^b_a=\sum_c g^{b c} X_{c b}$, we can express $2(\id_\U\ot \mu)(t)$ as

\[
\begin{aligned}
2(\id_\U\ot \mu)(t)& = \sum_{a, b} X^b_a\ot (-1)^{[a]}E^a_b \\
&-  \sum_{a, b, c} g^{b c} X_{c a}\ot(-1)^{[a][b]}  \sum_{b', a'} g^{a a'} g_{b b'} E^{b'}_{a'}.
\end{aligned}
\]

Note that the first term is equal to the right hand side of (<ref>), thus we want to show that the second term is the same. Using symmetry properties of $g$, we can re-write the second term as

\[
- \sum_{a, b, c} X_{c a}\ot (-1)^{[a][c] +[a]+[c]}  g^{a b}  E^{c}_{b}
- \sum_{a, b, c} g^{ba}  (-1)^{[a][c]} X_{c a}\ot (-1)^{[c]}   E^{c}_{b}.
\]

Using $X_{c a}=-(-1)^{[a][c]}X_{a c}$, we can re-write the right hand side of the above identity as

\[
\begin{aligned}
\sum_{a, b, c} g^{b a} X_{a c}\ot (-1)^{[c]}   E^{c}_{b}
= \sum_{b, c}X^b_{c}\ot (-1)^{[c]}   E^{c}_{b},
\end{aligned}
\]

which is equal to the right hand side of (<ref>). This completes the proof.

The following result is the Key Lemma (i.e., Lemma <ref>), which plays a crucial role in the main body of this paper.
The action of $t$ on $V\ot V$ is given by
(μμ)(t)=1/2∑_a, b J^a_b (-1)^[b]J^b_a = τ- e.

By expressing $J$'s in terms of $E$'s using (<ref>), we obtain from (<ref>)

\[
(\mu\ot\mu)(t)= \frac{1}{2}\sum_{a, b} J^a_b \ot (-1)^{[b]}J^b_a=\sum_{a, b} J^a_b \ot (-1)^{[b]}E^b_a  =T_1 - T_2,
\]

\[
\begin{aligned}
T_1&= \sum (-1)^{[b]} E^a_b\ot  E^b_a, \\
T_2&= \sum (-1)^{[b]} (-1)^{[a]([a]+[b])}   g_{b b''} E^{b''}_{a''} g^{a'' a} \ot E^b_a.
\end{aligned}
\]

Clearly $T_1=\tau$. For any $e^c\ot e^d$, a long handed calculation leads to

\[
\begin{aligned}
T_2(e^c\ot e^d)%&=\sum  g_{b' a} E^a_b e^c \ot  (-1)^{[a]([a]+[b]) + [c]([b]+[b'])}   E^{b'}_{a'} g^{a' b} e^d\\
%&=\sum  g_{b' a} e^a\delta_b^c \ot  (-1)^{[a]([a]+[b]) + [b]([b]+[b'])}   e^{b'}\delta_{a'}^d g^{a' b}\\
%&=\sum  g_{b' a} e^a \ot  (-1)^{[a]([a]+[c]) + [c]([c]+[b'])}   e^{b'} g^{d c}\\
%&=\sum  g_{b' a} e^a \ot  (-1)^{[a]+[c]}   e^{b'} g^{d c}\\
%&=\sum  g_{b' a} e^a \ot  (-1)^{[a]}   e^{b'} g^{c d}\\
%&=\sum  g_{b' a} e^a \ot  (-1)^{[b']}   e^{b'} g^{c d}\\
%&=\sum_b  e_b \ot  (-1)^{[b]}   e^{b} g^{c d}\\
&= g^{c d} \check{c}(1)= \check{c}\circ \hat{c}(e^c\ot e^d),
\end{aligned}
\]

that is,  $T_2=e$.  This completes the proof.

Let us compute the eigenvalue of $C$ in $V$.

The eigenvalue of $C$ in $V$ is equal to

χ_V(C) = (V)-1.

We have
\chi_V(C)\id_V=\frac{1}{2}\sum_{a, b} J^a_b (-1)^{[b]}J^b_a=\sum_{a, b} J^a_b (-1)^{[b]}E^b_a.
Expressing $J$'s in terms of $E$'s, we obtain
=Q_1 - Q_2$, with

\[
\begin{aligned}
Q_1&= \sum (-1)^{[b]} E^a_b E^b_a, \\  %= \sum (-1)^{[b]} \sum E^a_a, \\
%= \sdim(V) \id_V,\\
Q_2&= \sum (-1)^{[b]} (-1)^{[a]([a]+[b])}   g_{b b''} E^{b''}_{a''} g^{a'' a} E^b_a.
%= \id_V,\\
\end{aligned}
\]

By using properties of $g$ and carefully keeping track of the sign factors, one can show that
Q_1 = \sdim(V) \id_V$  and $Q_2 = \id_V$. This immediately leads to

§.§ Formulae for central elements of $\U(\osp(V; \omega))$

Let $T[1]_b^a=X_b^a$, and define $T[k]_b^a$ for $k>1$ recursively by

\[
T[k]_b^a=\sum_c  (-1)^{[c]} X^c_b T[k-1]_c^a.
\]

Then we easily obtain the following formula by using (<ref>).
(𝕀_μ)(t^k)   =    ∑_a, b T[k]^a_b(-1)^[b] E^b_a.

Recall the elements $Z_\ell\in \Hom_{\AB(\delta)}(0, 0)$ for $\ell\ge 1$ defined by (<ref>). It was shown that $Z_1=0$. Applying $\CF_\U$ to them and using (<ref>), we obtain
\CF_\U(Z_\ell)= \sum_c T[\ell]^c_c,
which can be expressed as

_(Z_ℓ)= ∑_a_1,…, a_ℓ-1,  b=1^m+2n (-1)^∑_i=1^ℓ-1 [a_i] X^a_1_b  X^a_2_a_1 X^a_3_a_2…X^b_a_ℓ-1.
It is shown in Theorem <ref> that these elements are central, and that in the case $\sdim(V)$ is odd, they generate the centre $Z(\U(\fg))$ of $\U(\fg)$.

§.§ Acknowledgements

We thank Tony Bracken, Mark Gould and Yang Zhang for Zoom discussions and email correspondence, and Catherina Stroppel for discussions.

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#  Quintessences Universe in $f(R,L_{m})$ gravity with special form of
deceleration parameter

B. K. Shukla<EMAIL_ADDRESS>Department of Mathematics, Govt.
College, Bandri Sagar (M.P.) India  R. K. Tiwari<EMAIL_ADDRESS>Department of Mathematics, Govt. Model Science College, Rewa 486 001 (M.P.)
India  D. Sofuo$\widehat{g}$lu<EMAIL_ADDRESS>Department of Physics,
Istanbul University Vezneciler 34134, Fatih, Istanbul, Turkey  A. Beesham
<EMAIL_ADDRESS>Department of Mathematical Sciences, University of
Zululand,P Bag X1001,Kwa-Dlangezwa 3886, South Africa  Faculty of Natural
Sciences, Mangosuthu University of Technology, P O Box 12363, Jacobs, South
Africa  National Institute for Theoretical and Computational Sciences, South
Africa

###### Abstract

In this paper We have investigated a homogeneous and isotropic FRW
cosmological model with perfect fluid in the framework of $f(R,L_{m})$
gravity. We have explored for the non linear case of $f(R,L_{m})$ model,
namely $f(R,L_{m})=\frac{R}{2}+L_{m}$ and obtained the solution by using the
condition that the deceleration parameter is a linear function of the Hubble
parameter. We employ 57 Hubble data points and 1048 Pantheon supernovae type
Ia data samples to restrict the model parameters. Additionally, we employ
Markoc Chain Monte Carlo (MCMC) simulation for our statistical analysis.
Additionally, we analyse the jerk and om diagnostic parameters for our model
using the parameter values that were obtained.

###### pacs:

2

##  I Introduction

Numerous scientific investigations have been conducted in the last several
decades to understand the mysterious behaviour of the universe. Throughout
their entire existence, gravitational waves, dark energy, early time
inflation, late time acceleration,BH ,WH and cosmological constants have been
probing the very nature of the universe. The universe was thought to be both
isotropic and homogenous on an enormous scale, just to investigate into the
cosmological principle. But in 1992, COBE convincingly asserted that the
large-scale CMB contains a minor anisotropy [1]. This was also confirmed by
observations taken by the Plank collaborations [2], CBI [3], WMAP,[4] and
BOOMERanG [5] in following years. The observational findings of the two teams
led by Perlmutter and Riess also led to exciting developments in the realm of
cosmology [6, 7]. These investigations aim to support the idea that the cosmos
is presently experiencing an accelerated expansion phase. The isotropic
character of the universe’s expansion has been a topic of debate up to this
point. The universe appears to grow at a variable rate in other directions,
according to interesting recent findings [8].Despite being the most
successful, FLRW cosmology is founded on cosmic principles.
In the current instance, the modified theoretic technique seems more effective
to deal with the examination of such problems. One of these is the $f(R)$
theory of gravity, which has created a solid framework to evaluate the present
state of the universe’s evolution [9]. In fact, the interpretations of late-
time acceleration[10, 11], the exclusion of the dark matter entity in the
examination of the dynamics of large test particles [12], and the unification
of inflation with dark energy [13] may all be satisfactorily explained by
$f(R)$ theories. Numerous arguments also suggest that higher-order theories,
such as $f(R)$ gravity, are able to account for the flatness of galaxies’
rotating curves [14]. Numerous coupling ideas were created as a result of
these reasons[15, 16, 17].The $f(R,L_{m})$ theory of gravity is one of them
[18]. Notably, this favours the occurrence of an additional, four-velocity-
orthogonal force. Additionally, the ’additional’ force explains the test
particle’s non-geodesic movements. As a result, it is possible to identify an
equivalence principle failure. The literature has produced many improvements
to this hypothesis [19, 20, 21, 22, 23, 24, 25, 26]. Jaybhaye et al. recently
investigated cosmology under $f(R,L_{m})$ gravity [27].
In this paper, we have investigated homogeneous and spatially isotropic FRW
model with a perfect fluid in this study. Within the constraints of
$f(R,L_{m})$ gravity, this is succeeded in doing. We focus on a nonlinear
$f(R,L_{m})$ model, namely $f(R,L_{m})=\frac{R}{2}+L_{m}$. Furthermore, the
deceleration parameter is a unique variation of the Hubble parameter. We
employ 57 Hubble data points and 1048 Pantheon supernovae type Ia data samples
to restrict the model parameters. Additionally, we employ Markoc Chain Monte
Carlo (MCMC) modelling for our statistical analysis. Additionally, we analyse
the jerk and om diagnostic parameters for our model using the parameter values
that were obtained.
The content of this paper is structured as follows: Section II presents the
fundamental definition of $f(R,L_{m})$ gravity. In section III, the study of
deceleration parameter is a special form of Hubble parameter. The examination
of observational conditions and a discussion of the findings are presented in
Section IV.In section V, we explain the cosmographic parameter and om
diagnostic in section VI. A few closing remarks are provided in section VII,
the last section.

##  II Review of $f(R,L_{m})$ gravity

The gravitational action for $f(R,L_{m})$ is given by

$\displaystyle\mathcal{S}=\int f(R,L_{m})\sqrt{-g}dx^{4}.$ (1)

Here $R$ denote the Ricci scalar and $L_{m}$ denotes the matter Lagrangian .
Ricci scalar $R$ is obtained by the following method

$\displaystyle R=g^{\mu\nu}R_{\mu\nu},$ (2)

where the Ricci tensor $R_{\mu\nu}$ is written in the following form as

$\displaystyle
R_{\mu\nu}=\partial_{\lambda}\Gamma^{\lambda}_{\mu\nu}-\partial_{\mu}\Gamma^{\lambda}_{\lambda\nu}+\Gamma^{\lambda}_{\mu\nu}\Gamma^{\sigma}_{\sigma\lambda}-\Gamma^{\lambda}_{\nu\sigma}\Gamma^{\sigma}_{\mu\lambda},$
(3)

where $\Gamma^{\lambda}_{\alpha\beta}$ is the components of Levi-Civita
connection.
Now the given field equation is obtained by with respect to the metric tensor
$g_{\mu\nu}$

$\displaystyle
f_{R}R_{\mu\nu}+(g_{\mu\nu}\square-\nabla_{\mu}\nabla_{\nu})f_{R}-\frac{1}{2}(f-f_{L_{m}}L_{m})g_{\mu\nu}=\frac{1}{2}f_{L_{m}}T_{\mu\nu},$
(4)

Here $f_{R}=\frac{\partial f}{\partial R},f_{L_{m}}=\frac{\partial f}{\partial
L_{m}}$ and $T_{\mu\nu}$ is called the stress-energy tensor for the cosmic
fluid, given by

$\displaystyle
T_{\mu\nu}=\frac{-2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}L_{m})}{\delta
g^{\mu\nu}}.$ (5)

Now, we can obtain the following relation by using covariant derivative in
equation (LABEL:4)

$\displaystyle\nabla^{\mu}T_{\mu\nu}=2\nabla^{\mu}ln(f_{L_{m}})\frac{\partial
L_{m}}{\partial g^{\mu\nu}}.$ (6)

To find the cosmological implications, we consider the following homogeneous
and spatially isotropic FLRW metric

$\displaystyle ds^{2}=-dt^{2}+a^{2}(t)(dx^{2}+dy^{2}+dz^{2}),$ (7)

where, $a(t)$ is the cosmic scale factor. Now , the Ricci scalar for the
metric (7) is obtained

$\displaystyle R=6(\dot{H}+2H^{2}),$ (8)

where $H$ is Hubble parameter and defined as

$\displaystyle H=\frac{\dot{a}}{a}.$ (9)

The energy-momentum tensor for energy density $\rho$ and the pressure $p$ of
the cosmic pressure is taken by,

$\displaystyle T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu}.$ (10)

The corresponding field equations for above are given by

$\displaystyle
3H^{2}f_{R}+\frac{1}{2}(f-f_{R}R-f_{L_{m}}L_{m})+3H\dot{f}_{R}=\frac{1}{2}f_{L_{m}}\rho,$
(11)
$\displaystyle\dot{H}f_{R}+3H^{2}f_{R}-\ddot{f}_{R}-3H\dot{f}_{R}+\frac{1}{2}(f_{L_{m}L_{m}}-f)=\frac{1}{2}f_{L_{m}}p.$
(12)

##  III Deceleration parameter in term of Hubble parameter

In this section, we explore a variable deceleration parameter in order to find
the exact solution of the field equations.To explain how the universe changed
from an era of decelerating expansion to one of current acceleration, a
variable deceleration parameter, q, has been suggested [28, 29, 30, 31, 32,
33, 34].
The cosmos expands either more accelerating $(q<0)$ or more decelerating
$(q>0)$, depending on the geometric parameter known as the deceleration
parameter.When $q=0$, the universe grows at a steady rate. When $q<-1$, the
accelerated growth is referred to as super-exponential expansion. A new
variable deceleration parameter has been assumed with the Hubble parameter as
[35]

$\displaystyle q=\alpha-\frac{\beta}{H^{2}},$ (13)

where the positive constants $\alpha$ and $\beta$ are present. Using the
formula for the Hubble parameter $H=\frac{\dot{a}}{a}$,and the deceleration
parameter in equation (13) is $q=-1-\frac{\dot{H}}{H^{2}}$. We can determine
the scale factor $a$

$\displaystyle a=[\sinh\tau]^{\frac{1}{1+\alpha}}.$ (14)

where $\tau=\\{(\sqrt{(1+\alpha)\beta)}t+c\\}$ and $c$ is an integration
constant. The Hubble parameter $H$ and the deceleration parameter $q$ are
rewritten by the value of (14)

$\displaystyle H=\sqrt{\frac{\beta}{1+\alpha}}\coth\tau,$ (15) $\displaystyle
q=\alpha-(1+\alpha)\tanh^{2}\tau,.$ (16)

Furthermore, we may express the Hubble parameter and deceleration parameter in
terms of redshift by using the relationship between the redshift and the scale
factor of the universe, $a(t)=(1+z)^{-1}$ .

$\displaystyle
H=\sqrt{\frac{\beta}{1+\alpha}}[1+(1+z)^{2(1+\alpha)}]^{\frac{1}{2}},$ (17)
$\displaystyle q=\alpha-\frac{(1+\alpha)}{[1+(1+z)^{2(1+\alpha)}]}.$ (18)

##  IV Evaluation to model parameters according to observational data

The Hubble parameter for redshift, which has been given by

$\displaystyle
H=\sqrt{\frac{\beta}{1+\alpha}}[1+(1+z)^{2(1+\alpha)}]^{\frac{1}{2}},$ (19)

Using the Hubble parameter’s most recent value, determine the parameter

$\displaystyle H=\frac{H_{0}}{\sqrt{2}}[1+(1+z)^{2(1+\alpha)}]^{\frac{1}{2}},$
(20)

where $H_{0}=\sqrt{\frac{2\beta}{(1+\alpha)}}$.

###  IV.1 Cosmic Chronometers datasets

We then employ 57 Hubble data sets in the redshift region of $0.07<z<2.36$
offered by Magana et al. [36] to determine the model parameter and the latest
value of the Hubble parameter $H_{0}$.
The associated chi-square function is written as such to determine the mean
value of the parameters and $H_{0}$:

$\displaystyle\chi^{2}_{H}(\alpha,H_{0})=\displaystyle\sum_{i=1}^{57}\frac{[H_{th}(\alpha,H_{0},z_{i})-H_{obs}(z_{i})]^{2}}{\sigma^{2}_{i}},$
(21)

where $H_{th}(\alpha,H_{0},z_{i})$ and $H_{obs}(z_{i})$, respectively,
represent the model-predicted and observed Hubble parameter measurements. The
standard error of the Hubble parameter measurement is denoted by the symbol
$\sigma_{i}$.

###  IV.2 Pantheon datasets

In this some cases, we use 1048 points from the Pantheon Type 1a supernova
data set (Scolnic et al.[37]) to estimate the parameters $\alpha$ and $H_{0}$.
This data collection contains 1048 apparent magnitude observations $m_{B}$
that support the redshift range of 0$0.01<z<2.3$.
In conventional cosmology, the distance modulus $\mu(z)$ is defined as

$\displaystyle\mu=m_{B}-M=5log_{10}(d_{l}(z))+25,$ (22)

where $m_{B}$ is the perceived magnitude, $M$ is the absolute magnitude, and
$d_{l}(z)$ is the luminosity distance for the flat FRLW universe

$\displaystyle
d_{l}(z)=\frac{c(1+z)}{H_{0}}\int_{0}^{z}\frac{dz^{*}}{E(z^{*})}.$ (23)

Given that all supernovae are thought to have the identical actual size, we
may use supernova 2002 cr, whose possesses a $m_{B}$ value of $13.907\pm
0.1982$ at a low redshift of $z=0.0101$, to calculate the actual size $M$
using the method described by Goswami et al. [38].

$\displaystyle M=5log_{10}\left(\frac{H_{0}}{c}\right)-1.093.$ (24)

Using equations (22) and (24), we arrive at the apparent magnitude equation
shown below.

$\displaystyle m_{B}=5log_{10}\left(\frac{d_{l}(z)H_{0}}{c}\right)+23.907.$
(25)

Now,we define chi-square function is

$\displaystyle\chi^{2}_{Pan}(\alpha,H_{0})=\displaystyle\sum_{i=1}^{1048}\frac{[\mu_{th}(\alpha,H_{0},z_{i})-\mu_{obs}(z_{i})]^{2}}{\sigma^{2}_{i}},$
(26)

where $\mu_{th}(\alpha,H_{0},z_{i}),\mu_{obs}(z_{i})$ and $\sigma_{i}^{2}$ are
theoretical distance , observational distance modulus and variance at $z_{i}$
respectively.

###  IV.3 An observational finding

We are capable to obtain limits on the parameters of my cosmological model for
all of the two distinct situations for the combined $H(z)$+Pantheon data set
by optimising the complete chi-squared function$\chi_{H}^{2}+\chi_{Pan}^{2}$.

Fig-1. The MCMC likelihood curves at $1\sigma$ and $2\sigma$ obtained from the
CC+SN datasets are shown in the above figure.

Fig-2. The figure show that the theoretical curve of $H(Z)$. Fig-3. The figure
show that the theoretical curve of $\mu(z)$.

Table 1 is an overview of the MCMC findings from the paper.

Observations data name | Parameters |  $F(R,L_{m})$ model
---|---|---
CC |  $\alpha$, $H_{0}$ |  $0.208573\pm 0.011896$, $67.166192\pm 0.401849$
SN |  $\alpha$, $H_{0}$ |  $0.208573\pm 0.026998$, $67.166192\pm 0.7202$

##  V  Cosmographic analysis

In this section of our examination of the evolution of the universe, we employ
a method known as ”cosmography”. It is predicated on the cosmological
principle, which holds that our physical universe is homogeneous and isotropic
on infinite sizes. Through this research, we are able to learn more about the
evolution of the universe independent of the cosmological principle without
supposing anything about certain cosmological theories. It makes advantage of
the Taylor series’ scale factor expansion

$\displaystyle\frac{a(t)}{a(t_{0})}=1+H_{0}(t-t_{0})-q_{0}H_{0}^{2}\frac{(t-t_{0})^{2}}{2!}+j_{0}H_{0}^{3}\frac{(t-t_{0})^{3}}{3!}-......,$
(27)

Here, subscript $0$ denotes the present value of the parameter.
We define Hubble parameter, deceleration parameter and jerk parameter in term
of redshift as

$\displaystyle H(z)=-(1+z)^{-1}\frac{dz}{dt},$ (28) $\displaystyle
q(z)=-1+\frac{(1+z)\frac{dH}{dz}}{H^{2}}.$ (29) $\displaystyle
j(z)=(1+z)\frac{dq}{dz}+q(1+2q).$ (30)

We obtained the value of jerk parameter in terms of redshift

$\displaystyle
j(z)=\frac{2(1+\alpha)^{2}(1+z)^{2(1+\alpha)}}{[1+(1+z)^{2(1+\alpha)}]^{2}}+\left[\alpha-\frac{(1+\alpha)}{[1+(1+z)^{2(1+\alpha)}]}\right]+2\left[\alpha-\frac{(1+\alpha)}{[1+(1+z)^{2(1+\alpha)}]}\right]^{2}.$
(31)

Fig-4. The above figure shows the $q$ vs $z$. Fig-5. The above figure shows
the $j$ vs $z$.

Fig-6. The figure show that the deceleration parameter vs jerk parameter.

This adequately accounts for the observations of the standard $\Lambda$CDM
model. Figures 4, 5, and 6, which express the evolution of the jerk parameter
and the deceleration parameter, respectively, also explain the characteristics
of these parameters.

###  V.1 Statefinder parameters

Current observational data cannot rule out all of the cosmological dark energy
concepts that are currently under consideration. To differentiate between
alternative dark energy models, however, two new cosmological parameters were
devised a few years ago. The scale factor and its derivatives with respect to
cosmic time are used to express these so-called statefinder parameters
completely up to the third order[38]. The statefinder’s parameters are
described below

$\displaystyle r=\frac{\dddot{a}}{aH^{3}},$ (32) $\displaystyle
s=\frac{r-1}{3(q-\frac{1}{2})}.$ (33)

The characteristics of the plane’s paths can be extremely different for
several existing models. The departure of these trajectories from the (1,0)
point determines the separation between a particular model and the LCDM model.
The statefinder pair $(r-s)$ successfully separates interacting dark energy
theories, brane models, quintessence, and Chaplygin gas from a variety of
other cosmological models. For a particular model, it is possible to determine
the pair $(r,s)$ and the trajectory in the $(r,s)$ plane.

Fig-7. The figure show that the statefinder parameter.

##  VI Om DIAGNOSTIC $Om(z)$

Om is a mathematical diagnostic which includes the redshift and the Hubble
parameter. With and without consideration of matter density, it can
distinguish between a dynamical dark energy model and the CDM. Dark energy is
a cosmological constant ($\Lambda$CDM) because $Om(z)$ behaves consistently
with regard to $z$. $Om(z)$ has a positive slope, which suggests that dark
energy is phantom, and a negative slope, which suggests that dark energy
behaves like quintessence. The definition of $Om(z)$ for a spatially flat
Universe is given by

$\displaystyle
Om(z)=\dfrac{\left[\frac{H(z)}{H_{0}}\right]^{2}-1}{(1+z)^{3}-1},$ (34)

where $H_{0}$ is initial value of Hubble parameter.

Fig-8. The figure show that the om diagnostics.

##  VII  $f(R,L_{m})$ gravity MODEL

To examine the behaviour of the universe’s, we determine this $f(R,L_{m})$
gravity.

$\displaystyle f(R,L_{m})=\frac{R}{2}+L_{m}^{n},$ (35)

where $n$ is free parameter.
Following that, for that particular functional type of $L_{m}=\rho$ [40], the
universe is given by the Friedmann equations (11) and (12)

$\displaystyle 3H^{2}=(2n-1)\rho^{n},$ (36) $\displaystyle
2\dot{H}+3H^{2}=[(n-1)\rho-np)\rho^{n-1}.$ (37)

the energy density and cosmic pressure in term of red shift

$\displaystyle\rho=\left[\frac{H_{0}^{2}[1+(1+z)^{2(1+\alpha)}]}{2(2n-2)}\right]^{\frac{1}{n}},$
(38) $\displaystyle
p=-\frac{1}{n}\dfrac{[-H_{0}^{2}(1+z)^{2(1+\alpha)}]+\frac{3}{2}H_{0}^{2}[1+(1+z)^{2(1+\alpha)}]}{\left[\frac{H_{0}^{2}[1+(1+z)^{2(1+\alpha)}]}{2(2n-1)}\right]^{\frac{n-1}{n}}}+\frac{(n-1)}{n}\left[\frac{H_{0}^{2}[1+(1+z)^{2(1+\alpha)}]}{2(2n-1)}\right]^{\frac{1}{n}},$
(39)
$\displaystyle\omega=\dfrac{-\frac{1}{n}\dfrac{[-H_{0}^{2}(1+z)^{2(1+\alpha)}]+\frac{3}{2}H_{0}^{2}[1+(1+z)^{2(1+\alpha)}]}{\left[\frac{H_{0}^{2}[1+(1+z)^{2(1+\alpha)}]}{2(2n-1)}\right]^{\frac{n-1}{n}}}+\frac{(n-1)}{n}\left[\frac{H_{0}^{2}[1+(1+z)^{2(1+\alpha)}]}{2(2n-1)}\right]^{\frac{1}{n}}}{\left[\frac{H_{0}^{2}[1+(1+z)^{2(1+\alpha)}]}{2(2n-1)}\right]^{\frac{1}{n}}}.$
(40)

Fig-9. Energy density $\rho$ vs redshift $z$. Fig-10. Cosmic pressure $p$ vs
redshift $z$.

Fig-11.The EoS $\omega$ vs redshift $z$.

This figure 9 demonstrates that the energy density is a positive function of z
and increases as cosmological redshift increases. When $z\rightarrow-1$, it
begins with a positive value and moves towards zero. In Fig.10, which depicts
the pressure behaviour as a function of redshift, we can see that the pressure
in the current model decreases as the cosmological redshift increases. It
initially has a high negative value and is currently approaching zero. Recent
observations indicate that the so-called dark energy, which has a negative
pressure, is the cause of the Universe’s accelerated expansion phase. As a
result, the pressure for our model fits recent data.
The EoS parameter is a crucial instrument for describing the cosmic epochs and
comprehending the properties of dark energy. Each different dark energy model
has a different range of values for this parameter. If the cosmological
constant ($\Lambda$CDM) characterises dark energy, then $\omega=-1$. While if
$-1<\omega<-0.33$, we claim that dark energy is quintessential, and if
$\omega<-1$, the model’s phantom character is indicated. Our model equation of
state parameter lies between -1 and -0.55.

##  VIII Conclusion

By adopting the novel relation of deceleration parameter and Hubble parameter
in the non-linear f(R,Lm) formalism for FRW cosmological model. We have
analyzed the Hubble parameter $H_{0}$ by using 57 Hubble data points in the
redshift range $0.07<z<2.36$ and we use 1048 points from the Pantheon Type 1a
supernova data to estimate the parameters $\alpha$ and $H_{0}$. This data
collection contains 1048 apparent magnitude observations $m_{B}$ that support
the redshift range of $00.01<z<2.3$.The matter density is a decreasing
function of time. The behavior of model is closed to Lambda CDM model . The
statefinder parameters $(rs)$ indicate for dark energy theories, brane models,
quintessence, and Chaplygin gas .The EoS parameter is a crucial instrument for
describing the cosmic epochs and comprehending the properties of dark energy.
Each different dark energy model has a different range of values for this
parameter. If the cosmological constant ($\Lambda$CDM) characterises dark
energy, then $\omega=-1$. While if $-1<\omega<-0.33$, we claim that dark
energy is quintessential, and if $\omega<-1$, the model’s phantom character is
indicated. Our model equation of state parameter lies between -1 and -0.55.
Abbreviations
The following abbreviations are used in this manuscript:
BH:black holes,
WH: wormholes
COBE:Cosmic Background Explorer
CMB:cosmic microwave background
CBI:Cosmic Background Imager
WMAP:Wilkinson Microwave Anisotropy Probe
BOOMERanG:Balloon Observations of Millimetric Extragalactic Radiation and
Geophysics

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