\section{Introduction}

\textit{Editor's note}. After Egorov died, I composed \cite{Eg} from scrap papers he left to me; he intended to improve them, but his sudden death intervened. Egorov was almost blind, so it was sometimes difficult to decipher intersecting lines of the notes he wrote to himself. Being guided by what I recalled of our discussions, I composed the introduction and supplied with the bibliography, but was unable to recover some of the proofs, see \cite{Eg}, published in not easily accessible collection. To the current version of Egorov's paper I updated the references trying to provide with the latest sources with more correct and reasonable definitions of the basics (e.g., Cartan matrix, Dynkin-Kac diagram, double extension) than the first definitions and more convenient notation. Ivan Penkov considerably improved the draft I composed for this Special volume in memory of Arkady Onishchik, I am very thankful to him for help. I heartily thank Osmo Pekonen who sent me the pdf of \cite{Eg} so I could work on it trying to make Egorov's findings more available. In what follows, ``I" means ``Egorov''.~\textit{D.Leites.}

\smallskip

In this paper, the ground field is $\Cee$.

\subsection{Lie algebras} There are many contenders for the role of $\fgl (n)$ in the case where $n =\infty$. Lately, some of these contenders are distinguished thanks to their applications in the theory of differential equations such as KdV and KP hierarchies, see~\cite{R} and references therein. For other applications of $\fgl (\infty )$, see \cite{B,K,KL1, KL2,FF,Lq,V}. (The best, in my opinion, short review on KdV is an MPI-1987 preprint by Date.)

Let me list some of the Lie algebras of infinite matrices that enjoy everybody's recognition. It goes without saying that we identify a~linear transformation with its matrix having chosen a~basis in the space $V$ of the tautological representation of the Lie algebra $\fgl (V)$. Thus, we replace the Lie algebra of operators $\fgl (V)$ by the Lie algebra of matrices $\fgl (\dim V)$ and for infinite-dimensional (of countable dimension) $V$, so every element of $V$ is a finite sum of basis elements, we get:
\[
\begin{array}{ll}
\fgl (\infty ) &= \{X \mid X\text{~~ is a~linear endomorphism of $\Cee^\infty=\Cee^\Zee$} \}\\
&= \{X\text{~~ is a~matrix with finitely many non-zero entries in every column} \};\\
\fgl _c(\infty ) &= \{X \in \fgl (\infty ) \mid \text{~~ nonzero entries of $X$ belong to a~band}\\ &\text{which contains the main diagonal and whose boundaries are parallel to it}\};\\
\fgl _f(\infty ) &= \{X \in \fgl (\infty ) \mid X\text{~~ has finitely many nonzero entries}\}.\\
\end{array}
\]

Each of these three Lie algebras of infinite both ways matrices has a~subalgebra of matrices whose rows and columns are only labelled with positive integers. Denote the respective subalgebras by $\fgl ^{+} (\infty )$, $\fgl_c ^{+} (\infty )$, $\fgl_f ^{+} (\infty )$.

Let us explain what is so natural about these algebras. For the definition of Lie (super)algebras with Cartan matrix and the corresponding Dynkin--Kac diagrams, see \cite{CCLL}; for realizations of finite-dimensional Lie superalgebras, see \cite{BGLLS}. Now, look at the Dynkin diagram of $\fgl (n)$; more precisely, of $\fsl(n)$ as $n\tto \infty $. There are two ways to tend $n$ to $\infty $ and that is how these algebras are obtained: they are generated by countably many Chevalley generators subject to the relations governed by the Cartan matrix corresponding to one of the following infinite Dynkin diagrams:
\[
\begin{array}{lll}
\bigcirc -\ldots -\bigcirc &\tto \bigcirc -\ldots &\text{(corresponds to $\fgl _f^{+} (\infty )$)}\\
\bigcirc -\ldots -\bigcirc &\tto \ldots -\bigcirc -\ldots & \text{(corresponds to $\fgl _f (\infty )$)}
\end{array}
\]
Observe that although seemingly different, the limit algebras are isomorphic, as Penkov proved by a simple argument: they are transformations of the space whose basis vectors are labelled by all or only positive integers; the isomorphism of the sets of labels establishes isomorphism $\fgl _f^{+} (\infty )\simeq\fgl _f (\infty )$ which does not come from any analog of the Weyl group considered in what follows.

For applications, however, these algebras turn out to be too small and it is desirable to enlarge them so that the enlarged Lie algebra has a~nontrivial central extension of degree 0. (This central extension is needed in ``quantization'', cf.~\cite{Lq, LSh}.)


The above-defined intermediate Lie algebras $\fgl _c(\infty )$ and $\fgl _c^{+}(\infty )$ prove to be just fine, although no analog of Dynkin diagram corresponds to them. (We will show that they themselves have interesting enlargements. R.~Bullough has reported that he found a~need in an enlargement of the conventional $\fgl _c(\infty )$, see \cite{BT}. Hopefully, the even parts of the Lie superalgebras introduced in this paper will satisfy these needs.) The central extension we have mentioned is given by the cocycle
\[
X,Y \mapsto \tr (J[X,Y])\quad \text{for}\quad X,Y \in \fgl _c (\infty )
\]
where $J = \diag(\ldots ,1,{-}1,\ldots )$ with the 1's occupying the slots with negative indices and the $-1$'s occupying the slots with positive indices.

\subsection{The Lie algebras studied by Penkov with co-authors} See \cite{PS1, PS2, FPS, PP1, PP2, PP3, PP4, IP, IPW, PSZ, PZ, ChP1, ChP2, HPS, CouP, GP, Pe, CP1, CP2} summarized in \cite{HP}, and an amendment in \cite{HZ}.

\subsection{Bases of Lie superalgebras} As is known, some~simple finite-dimen\-sio\-nal Lie superalgebras $\fg$ can have several bases (a.k.a. systems of simple roots) not isomorphic with respect to the Weyl group of $\fg_{\overline 0}$. The simplest way to see the reason for this phenomenon is to consider $\fgl (V)$ for a~finite-dimensional superspace $V$.

The matrix expression of elements from $\fgl (V)$ now depends not only on the basis of $V$ (which is always supposed to consist of homogeneous elements) but also on the \textit{parities} of the elements of the basis. The general linear Lie superalgebra of $n \times n$ matrices is determined by a~ set of $\sdim V=(\dim V_\ev, \dim V\od)$ basis vectors and the set \text{Par} of their parities, of which there are $\dim V_\ev$ even and $\dim V_\od$ odd ones, so that $\dim V_\ev+\dim V_\od = n$.


V. Serganova suggested several generalizations of the Weyl group all of which act by permutations of all the bases of a~simple Lie algebra $\fg$ (either finite-dimensional or a~(twisted) loop one) with Cartan matrix~\cite{S1,S2,LSS}. Let us recall them.

\subsection{Super Weyl groups}

\subsubsection{The universal super Weyl group} For simplicity, consider the finite-dimen\-sional case.

Let $\fg$ be a~simple Lie superalgebra, $\fh$ its maximal torus, $B$ a~base (system of simple roots), $\fn_B$ the nilpotent subalgebra generated by root vectors corresponding to~$B$. A subalgebra of the form $\fb_B:=n_B \oplus \fh$ is called a~\textit{Borel subalgebra}. Observe that $\fb_B$ is not the maximal nilpotent Lie superalgebra, see \cite{Shch}; Penkov showed that in the study of irreducible $\fg$-modules the non-maximal subalgebra $\fb_B$ is more convenient than the maximal one. 

Let $L(\lambda ,B)$ be the finite-dimensional irreducible $\fg$-module with $\fb_B$-highest weight $\lambda $; let $\Lambda =\Lambda _B$ be the set of $\fb_B$-highest weights of all the finite-dimensional irreducible $\fg$-modules and $\Gamma _\lambda $ the set of weights of $L(\lambda, B)$.

For any $\alpha \in R$ and $\gamma \in \Gamma _\lambda $, denote by $S_\alpha (\gamma )$ the \textit{$\alpha $-string through $\gamma $}, i.e., the set
\[
\gamma -q \alpha ,\ldots ,\gamma -\alpha ,\gamma ,\gamma +\alpha ,\ldots ,\gamma +p \alpha 
\]
such that $\gamma -(q+1)\alpha $, $\gamma +(p+1)\alpha \notin \Gamma _\lambda $. The number $l_\alpha =p+q-1$ is called the \textit{length} of the $\alpha $-string.

Set $r_\alpha (\gamma )=\gamma + (p - q)\alpha $. Since the weight $r_\alpha (\gamma )$ is defined for any $\gamma \in \Gamma _\lambda $, there exists a~map $r_\alpha \colon \Gamma _\lambda \mapsto \Gamma _\lambda $. This map will be called the \textit{reflection in $\alpha $}.

Let $F_R$ be the free group with generators $f_\alpha $ for every $\alpha \in R$. Then, for any $\lambda \in \Lambda $, there is defined an $F_R$-action on $\Gamma _\lambda $ by the formula $f_\alpha (\gamma ) := r_\alpha (\gamma )$.

Let $I_{\lambda ,R}$ be the normal subgroup of $F_R$ singled out by the formula
\[
I_{\lambda ,R}=\{f\in F_R\mid f(\gamma )=\gamma \text{ for all }\gamma \in \Gamma _\lambda \}.
\]

The group $UW_R = F_R/I_R$, where $I_R =\cap _{\lambda \in \Lambda }I_{\lambda ,R}$, will be called the \textit{universal super Weyl group} or just the univesal Weyl group of the root system $R$ (or of $\fg$ and then we denote $UW_R$ by $UW_{\fg}$).

Denote by $r_\alpha $ the image of $f_\alpha $ under the natural projection. By construction, $UW_{\fg}$ acts on $\Gamma _\lambda $ for any $\lambda \in \Lambda $, in particular, it acts on~$R$.

\parbegin{Lemma}
\textup{a)} $r_\alpha ^2 = 1$.

\textup{b)} $r_{-\alpha } = r_\alpha $.

\textup{c)} The Weyl group $W_{\fg_{\overline 0}}$ of the Lie algebra $\fg_{\overline 0}$ is naturally embedded into $UW_{\fg}$.

\textup{d)} Let $w \in W_{\fg_{\overline 0}}$. Then, $r_{w(\alpha )}=wr_\alpha w^{-1}$.
\end{Lemma}

\begin{proof} The properties a) and b) directly follow from the definition of the~$r_\alpha $, the property c) is obvious.

Since $r_\alpha $ is a~linear transformation of $\fh^*$ and maps $\Gamma _\lambda $ into itself, then
\[
S_{w(\alpha )}(w(\gamma )) = w(S_\alpha (\gamma ))\text{ for any }w\in UW_{\fg},\gamma \in \Gamma _\lambda
\]
implying d).
\end{proof}

\parbegin{Example} For $\fg = \fsl(m | n)$, select a~base $B =\{\alpha _1,\ldots ,\alpha _{m+n-1}\}$ corresponding to the Dynkin diagram of the form
\[
\bigcirc --\cdots--\bigcirc --\otimes --\bigcirc --\cdots--\bigcirc 
\]
By heading d) of Lemma we can take $r_{\alpha _1},\ldots ,r_{\alpha _{m+n-1}}$ for the generators of $UW_{\fg}$. \end{Example}

\parbegin{Proposition}[\cite{S2}] $UW_{\fg}$ is a~Coxeter group for $\fg = \fsl(m | n)$ and $\fg = \fosp(2 | 2n)$ with relations indicated in Table $3.1$. 
\end{Proposition}


\parbegin{Conjecture} $UW_{\fg}$ is a~Coxeter group for any simple Lie superalgebra with Cartan matrix with relations indicated in Table $3.1$. \end{Conjecture}


\subsubsection{The linear Weyl group}
Under the notations of the previous section, let $A$ be the Cartan matrix (see \cite{CCLL}) and define the reflections $r_\alpha $ by the formulas
\[
r_{\alpha _i}(\alpha _j)=\begin{cases}
-\alpha _j&\text{for }i = j\\
\alpha _j-a_{ij}\alpha _i&\text{for }i \ne j \text{ and }A_{ii} = 2\\
\alpha _j-2A_{ij}\alpha _i&\text{for }i \ne j \text{ and }A_{ii} = 1\\
\alpha _j+\alpha _i&\text{for }i \ne j \text{ and }A_{ii}=0,\ A_{ji}\ne 0\\
\alpha _j&\text{for }i \ne j \text{ and }A_{ii} = A_{ji} = 0
\end{cases}
\]
Let $LW_R$ be the group generated by such reflections when $\alpha $ runs over simple roots of \textit{all} bases of $\fg$. We will call $LW_R$ the \textit{linear Weyl group} of $R$ (or of $\fg$).

\subsection{Formulation of the Problem} M. Saveliev was, probably, the firt to observe that in applications of Lie superalgebras to integrable dynamical systems, the base of a~simple Lie superalgebra that consists of only odd elements plays a~distinguished role, see~\cite{LSS}. In Table~ 6 of~\cite{LSS}, there are listed Lie superalgebras with which there are associated by a~superization of a~method by Drinfeld and Sokolov, also described in~\cite{LSS}, super versions of KdV and (with infinite diagrams) KP, see a~pioneering paper~\cite{MR} and later works~\cite{KL1,KL2,R}. Note that only a~\textit{method} to recover an equation from a~superalgebra is written in~\cite{LSS}, the exceptional super KdVs themselves corresponding to exceptional diagrams from Table~ 6 of~\cite{LSS} were never written explicitly.

Not every Lie superalgebra has such a~basis. Serganova's result states that if the Lie superalgebra is finite-dimensional or loop one (perhaps, twisted, or is its double extension with Cartan matrix, see \cite{BLS}) it does not matter which base we start with as long as the superalgebra has a~distinguished base since we can reach any of the bases from any given one with the help of any of Serganova's super Weyl groups.

\textbf{Question: Is it true for $\fgl _c(\infty |\infty )$ that there is just one class of bases with respect to an analogue of the Weyl group? } Since only $\fgl _c(\infty |\infty )$ was chosen so far (\cite{R,KL1, KL2,B}) to study superized versions of KdV and KP, the answer is important for these interesting differential equations.

It turns out that the answer is ``NO''.

\subsection{Formulation of the answer} In what follows, I will give a~complete and explicit description of bases (systems of simple roots) for the so far conventional $\fgl(\infty |\infty)$; namely, 
\[
\text{the bases are indexed by the points of $[a, b]\cup [c, d]$, where $0<a \leqslant b<c \leqslant d<1$.}
\] 
I also define interesting enlargements of $\fgl _c(\infty |\infty )$. For the largest of these enlargments, there is just one class of bases \textit{and} for all of these types of Lie superalgebras, there exists a~ nontrivial cocycle of degree~0 which defines a nontrivial central extension.

I will also describe analogues of the Weyl group for these enlarged $\fgl (\infty |\infty )$'s.

I also give a~similar description for $\fgl_c(p|\infty )$ and $\fgl _c(\infty |q)$ with ${p,q < \infty }$.

In the last section, I describe various important subalgebras of $\fgl _c(\infty |\infty )$: its orthosymplectic, periplectic and queer versions; how to embed certain affine Kac-Moody (see \cite{BLS}) superalgebras into some of these Lie superalgebras of infinite matrices.




\section{Examples of infinite matrix Lie superalgebras and their ``Weyl groups''}\renewcommand\thesubsection{\arabic{section}.\arabic{subsection}}

In this section, we introduce several (super)spaces of infinite matrices. Let us start with the underlying spaces. Let ${\Mat: =\{a = (a_{i,j})_{i,j\in \Zee}\}}$ designate the space of all infinite matrices. Set
\begin{gather*}
\Mat_f := \{x \in \Mat \mid \card (\Supp x) <\infty \},\text{ where}\\
 \Supp (a_{i,j}) := \{(i,j) \in \Zee \times \Zee\mid a_{i,j}\ne 0\}.
\end{gather*}

In what follows, an arbitrary map $p \colon \Zee \tto \Zee/2$ will be called a~\textit{parity function}. The following two parity functions are very important for us:
\begin{gather*}
p_{+}(x)=\begin{cases}
\overline 1,&\text{if } x \geqslant 0\\
\overline 0,&\text{otherwise.}
\end{cases}\\
p_{\text{st}}=x \mod 2.
\end{gather*}

On $\Mat_f$, introduce a~Lie superalgebra structure by defining the supermatrices with the help of a~fixed parity function $p$ and by setting
\be\label{1}
[a,b]_{ij}=\sum\limits _k(a_{ik}b_{kj}-(-1)^{(p(i)+p(k))(p(j)+p(k))}b_{ik}a_{kj});
\ee
denote the obtained Lie superalgebra by $\fgl _{p,f}$. The Dynkin diagrams corresponding to $p_{+}$ and $p_{\text{st}}$, respectively, are:
\[
\cdots--\bigcirc --\cdots--\bigcirc --\otimes --\bigcirc --\cdots--\bigcirc --\cdots
\]
and
\[
\cdots--\otimes --\cdots--\otimes --\cdots
\]

\subsection{Several superized $\fgl (\infty )$s} We will consider the following four en\-lar\-ge\-ments of $\fgl_{p,f}$ with the bracket defined by the formula \eqref{1}:
\begin{gather*}
\fgl _{p,g}:=\{(a_{ij})_{i,j\in \Zee} \in \Mat\mid \text{for any $i_0$, $j_0$ there exists}\\
\text{some $c = c(i_0,j_0) > 0$ such that}\\
a_{ij}=a_{ji}=0 \text{ for all $i$, $j$ such that }(i - i_0)(j_0 - j) > c\}
\end{gather*}
It is an easy exercise for the reader to prove that there are finitely many entries in every row and every column as well as in the first and the  third quadrants of every matrix in~ $\fgl _{p,g}$.
\begin{gather*}
\fgl _{p,l}:=\{a=(a_{ij})_{i,j\in \Zee} \in \Mat\mid \text{ for some $\lambda =\lambda (a)>0$ the matrix $a$ satisfies}\\
\lim\limits _{\substack{(i,j) \in \Supp a\\
|i| \tto \infty }}\frac {|i-j|^\lambda }{|i+j|}=\lim\limits _{\substack{(i,j) \in \Supp a\\
|j| \tto \infty }}\frac {|i-j|^\lambda }{|i+j|}=0)\};\\
\fgl _{p,o}:=\{a=(a_{ij})_{i,j\in \Zee} \in \Mat\mid \lim\limits _{\substack{(i,j) \in \Supp a\\
|i| \tto \infty }}\frac {|i-j|^\lambda }{|i+j|}=\lim\limits _{\substack{(i,j) \in \Supp a\\
|j| \tto \infty }}\frac {|i-j|^\lambda }{|i+j|}=0 \\
 \text{for all}\quad \lambda >0\};\\
\fgl _{p,c}:=\{a=(a_{ij})_{i,j\in \Zee}\in \Mat\mid a_{ij}=0 \text{ for all $i$, $j$ such that}\\
\text{$|i -j| > c$ for some constant $c = c(a)$}\}.
\end{gather*}

\sssbegin{Proposition} $\fgl _{p,*}$, where $* = g,l,o,c$, are Lie superalgebras, i.e., for any $a,b\in \fgl _{p,*}$ we have $[a,b] \in \fgl _{p,*}$.
\end{Proposition}

\begin{proof} Every row and column of any $(a_{ij})_{i,j\in \Zee} \in \fgl _{p,g}$ is finite, so the bracket is well defined. Let $c = [a, b]$, where $a, b \in \fgl _{p,*}$; we have to find $\Suppp c$, having given $\Suppp a$ and $\Suppp b$. We have to prove that the supercommutator does not lead out of $\fgl _{p,g}$.

If $(i,j) \in \Suppp c$, then $(i, k), (k, j) \in \Suppp a~\cup \Suppp b$ for some $k \in \Zee$. For any $i_0$ and $j_0$, there exist $c_1$ and $c_2$ such that $(i - i_0)(j_0 -k) < c_1$ and $(k - i_0)(j_0 - j) < c_2$. The details are left to the reader.
\end{proof}

\sssbegin{Lemma} \textup{1)} The superalgebra $\fgl _{p,g}$ has a~unique nontrivial central extension $\widetilde {\fgl }_{p,g}$
given by the cocycle:
\[
a,b\tto\str([a,b]J)z,
\]
where $J\in \Mat$ is such that $J_{ik}=\delta _{ik}\sign(i)$.

\textup{2)} The restrictions of this cocycle to $\fgl _{p,l}$, $\fgl _{p,c}$ and $\fgl _{p,o}$ are nontrivial.
\end{Lemma}

We will denote the central extensions corresponding to heading 2 of Lemma by $\widetilde {\fgl }_{p,l}$, $\widetilde {\fgl }_{p,c}$ and $\widetilde {\fgl }_{p,o}$, respectively.

\subsection{Weyl groups for $\fgl _{p,g}$ and its subalgebras} Let $p_1$ and $p_2$ be two different parity functions. What are the conditions for these two functions to determine isomorphic Lie superalgebras of type $\widetilde {\fgl }_{p,c}$ (resp. $\widetilde {\fgl }_{p,o}$, $\widetilde {\fgl }_{p,l}$ and $\widetilde {\fgl }_{p,g}$)? To answer this question we need the following infinite permutations groups that will serve as analogues of the Weyl group of $\fgl (n)$.

\ssbegin{Convention} In what follows we will only consider isomorphisms of these superalgebras which transfer their common Lie subsuperalgebra of finite matrices $\Mat_f$ into itself.
\end{Convention}

Let $S_{\Zee}$ be the group of all permutations of $\Zee$, i.e., the group of all one-to-one maps $\sigma: \Zee \tto \Zee$. We define the groups
\begin{gather*}
S_g:=\{\sigma \in S_{\Zee} \mid \text{ for some $c_\sigma >0$ we have $\frac {\sigma (i)}i>0$}\\
\text{for all $i$ such that: $|i| >c_\sigma $}\};\\
S_l:=\{\sigma \in S_g\mid \lim\limits _{|i| \tto \infty }\frac {|\sigma (i)-i|^\lambda }{|i|}=0 \text{ for some }\lambda _\sigma =\lambda >0\};\\
S_m:=\{\sigma \in S_g\mid \limsup\limits _{|i| \tto \infty }\frac {|\sigma (i)-i|}{|i|}<\infty ,\text{ and } \liminf\limits _{|i| \tto \infty }\frac {|\sigma (i)-i|}{|i|}>0\};\\
S_n:=\{\sigma \in S_{\Zee} \mid \lim\limits _{|i| \tto \infty }\frac {|\sigma (i)-i|}{|i|}=0\};\\
S_o:=\{\sigma \in S_{\Zee} \mid \lim\limits _{|i| \tto \infty }\frac {|\sigma (i)-i|^\lambda }{|i|}=0 \text{ for all }\lambda >0\};\\
S_c:=\{\sigma \in S_{\Zee} \mid |\sigma (i)-i|<c_\sigma \text{ for some }c_\sigma >0 \text{ and }i\in \Zee\}.
\end{gather*}

It is clear that 
\[
S_c \subseteq S_o \subseteq S_n \subseteq S_m \subseteq S_l \subseteq S_g \subseteq S_{\Zee}.
\]
Define the action of $S_{\Zee}$ on parity functions by setting $(\sigma p)(i)=p(\sigma (i))$.

\ssbegin{Proposition} Let $p$ and $p'$ be parity functions, $\sigma \in S_g$ and $p'=\sigma p$. Let 
\[
(J_\phi )_{ij} := J_{\sigma ^{-1}i,\sigma ^{-1}j}-J_{ij}. 
\]
Then, the formulas 
\[
\phi _\sigma (z)=z,\phi _\sigma (a_{ij})=(a_{\sigma (i),\sigma (j)}+\str(J_\phi a)z,
\]
determine an isomorphism ${\phi _\sigma \colon \widetilde {\fgl }_{p,g} \tto \widetilde {\fgl }_{p',g}}$.

If $\sigma \in S^l$ \textup{(}or $S^o$, or $S^c$\textup{)}, then $\phi _\sigma $ induces the isomorphism of $\widetilde {\fgl }_{p,l}$ and $\widetilde {\fgl }_{p',l}$ \textup{(}of $\widetilde {\fgl }_{p,o}$ and $\widetilde {\fgl }_{p',o}$, or $\widetilde {\fgl }_{p,c}$ and $\widetilde {\fgl }_{p',c}$, respectively\textup{)}.
\end{Proposition}

\begin{proof} Let $pr \colon \widetilde {\fgl }_{p,g} \tto \fgl _{p,g}$ be the natural projection. Let us first prove that $\phi _\sigma $ is an isomorphism. It is clear that
\[
pr \phi _\sigma pr([a,b])=pr([\phi _\sigma (a),\phi _\sigma (b)]).
\]
Then, 
\begin{gather*}
\phi _\sigma ([a,b])=\phi _\sigma (pr([a,b])+\str(pr([a,b])J)z)\\{}=
pr \phi _\sigma pr([a,b])+\str(pr([a,b])J_\sigma )z+\str(pr([a,b])J)z\\{}=
pr([\phi _\sigma (a),\phi _\sigma (b)])+\sum\limits (-1)^{p(i)}[a,b]_{ii}(J_{\sigma ^{-1}i,\sigma ^{-1}i}-J_{ii}+J_{ii})z\\{}=
{}[\phi _\sigma (a),\phi _\sigma (b)].
\end{gather*}

Let $*$ be $c$, $o$ or $l$. We should prove that $\phi _\sigma (a) \in \fgl _{p,*}$ for any $a\in \fgl _{p,*}$ and $\sigma \in S_*$.

Let $*$ be $c$. Then, $a_{ij}=0$ for any $i$, $j$ such that $|i - j| > c_a $. For any $i$, $j$ such that $|i - j| >c_a+2c_\sigma $, we have
\[
(\phi _\sigma a)_{ij}=a_{\sigma ^{-1}(i),\sigma ^{-1}(j)}=0,
\]
because
\[
|\sigma ^{-1}i-\sigma ^{-1}j|>>|i-j|-|\sigma ^{-1}i-i|-|\sigma ^{-1}j-j|>c_a.
\]
Let $*$ be $o$. Then, 
\[
\lim\limits _{\substack{i,j\tto \infty \\
(i,j) \in \Supp \phi _\sigma (a)}}\frac {|i-j|^\lambda }{|i+j|}=\lim\limits _{\substack{i,j\tto \infty \\
(i,j)\Supp(a)}}\frac {|\sigma ^{-1}(i)-\sigma ^{-1}(j)|^\lambda }{|\sigma ^{-1}(i)+\sigma ^{-1}(j)|}=0
\]
by definition of $S_o$.

For $* = l$, the proof is quite similar.
\end{proof}

\section{Main Reduction}

The following theorem shows that the isomorphism problem for Lie superalgebras $\widetilde {\fgl }_{p,g}$, $\widetilde {\fgl }_{p,l}$, $\widetilde {\fgl }_{p,o}$, $\widetilde {\fgl }_{p,c}$ can be reduced to the equivalence problem for the parity functions on $\Zee$ with respect to permutations groups $S_g$, $S_l$, $S_o$ and $S_c$, respectively.

\ssbegin{Theorem} Let $\tau \in S_{\Zee}$ be given by the formula $\tau (i)=-i-1$. The Lie superalgebras $\widetilde {\fgl }_{p_1,g}$ and $\widetilde {\fgl }_{p_2,g}$ \textup{(}$\widetilde {\fgl }_{p_1,l}$ and $\widetilde {\fgl }_{p_2,l}$, $\widetilde {\fgl }_{p_1,o}$ and $\widetilde {\fgl }_{p_2,o}$, $\widetilde {\fgl }_{p_1,c}$ and $\widetilde {\fgl }_{p_2,c}$\textup{)} are isomorphic if and only if there exists $\sigma \in S_g$ (resp. $S_l$, $S_o$, $S_c$) such that $p_2=\sigma (p_1)$ \textup{(}respectively $p_2 =\tau \sigma (p_1)$\textup{)}.
\end{Theorem}

First, let us prove the following lemma.

\sssbegin{Lemma}\label{l2.1} If $\sigma \in S_g$ and for any $i,j\in \Zee$ we have
\[
|\sigma (i)-\sigma (j)|<c|i-j|\quad \text{and}\quad |\sigma ^{-1}(i)-\sigma ^{-1}(j)|<c|i-j| ,
\]
then there exists $C$ such that $|\sigma (i)-i|<C$ for any $i\in \Zee$.
\end{Lemma}

\begin{proof} We will find such a~constant $C$ for $i \geqslant 0$ and apply the same arguments to $i < 0$.

Let $C_0 = 1+\max\nolimits _{i \leqslant 0}\sigma (i)$. The maximum exists because the definition of $S_g$ implies that $\sigma (i)<0$ for almost all $i < 0$.

Consider the permutation $\sigma '(i)=\sigma (i)-C_0$. Since $\sigma '(i)<0$ for $i \leqslant 0$, we see that for any $k\in \Nee$, there exist $i$, $j$ such that $j > k > i$ and such that $0 <\sigma '(j)\leqslant k$, $\sigma '(i)\leqslant k$. Then, if $\sigma '(k)>k$, there exist $m$, $n$ such that
\[
m<k<n,\quad \sigma '(m)<k,\quad \sigma '(n)\leqslant k,\quad \sigma '(m+1)\geqslant k,\quad \sigma '(n+1)\geqslant k.
\]

Since
\[
\sigma '(a)-\sigma '(b)=\sigma (a)-\sigma (b)\quad \text{and}\quad |\sigma (i+1)-\sigma (i)|<c,
\]
we get
\[
|\sigma '(m+1)-\sigma '(m)|<c,\quad |\sigma '(n-1)-\sigma '(n)|<c.
\]
Thus,
\[
|\sigma '(m)-k|<c,\quad |\sigma '(n)-k|<c,
\]
and therefore
\[
|\sigma '(m)-\sigma '(n)|<2c \quad \text{and }|m-n|<2c^2.
\]
Since $m > k > n$, then $|m - k|<2c^2$ and $|\sigma '(m)-\sigma '(k)|<2c^3$.

Finally,
\begin{gather}\label{3}
|\sigma (k)-k|<|\sigma '(k)-k|+C_0 \leqslant{}\notag\\
 |\sigma '(m)-k|+|\sigma '(m)-\sigma '(k)|<C_1=2c^3+c+C_0. \label{3.1}
\end{gather}
If $\sigma '(k)<k<\sigma (k)$, the inequality \eqref{3.1} still holds because $\sigma (i)-\sigma '(i)=C_0$.

If $0 <\sigma (k)<k$, then we can obtain a~similar inequality for $\sigma ^{-1}$:
\[
|k-\sigma (k)|=|\sigma ^{-1}(\sigma (k))-\sigma (k)|<C_2.
\]

Since the estimate is done for all $i\in \Nee$ but a~finite set, we have:
\[
\card B < \infty ,\quad \text{where}\quad B = \{i \mid i\in \Nee,\sigma (i) \notin \Nee\}.
\]

Finally, set
\[
C := \max\{C_1 , C_2, \max\limits _{i\in B}|\sigma (i) - i|\}.\qed
\]
\noqed \end{proof}

\sssbegin{Lemma}\label{l2.2} Let $\phi \colon \fgl _{p,*} \tto \fgl _{p',*}$ be an isomorphism. Then, there exist $\gamma \in \Aut \fgl _{p',*}$ and $\sigma \in S_{\Zee}$ such that $\gamma \phi =\phi _\sigma $.
\end{Lemma}

\begin{proof} Let $(e_{ij})$ be an elementary matrix; $H$ and $H'$ the diagonal subalgebras in $\fgl _{p,*}$ and $\fgl _{p',*}$, respectively; 
\[
\text{$L_r := \{(a_{ij})_{i,j\in \Zee} \in \Mat_f$ such that $a_{ij}=0$ if $|i| > r$ or $|j| > r\}$.}
\]
 (Clearly, $L_r$ is a~subsuperalgebra.)

For any $k\in \Nee$, there exists $m\in \Nee$ such that $\phi (L_k)\subseteq L_m$, because $\phi (\fgl _{p,f})\subseteq \fgl _{p',f}$. It is possible to show that there exists $X_k\in L_m$ such that $1+X_k$ is invertible and 
\[
\text{$\Ad_{1 + X_k}\phi (e_{ii}) =e_{\sigma (i),\sigma (i)}$ for some function $\sigma \colon [-k,k] \tto \Zee$.}
\]

When $X_k$ is constructed, we can choose $X_{k+1}$ so that $\Ad_{1+X_{k+1}} (e_{\sigma (i),\sigma (i)})=e_{\sigma (i),\sigma (i)}$ for $|i| < k$. Then, we can set $\gamma :=\prod\nolimits _{k=1}^\infty \Ad_{1+X_{k}}$ and take the corresponding $\sigma \in S_{\Zee}$. It is easy to prove that $\gamma $ is extendable to an automorphism of $\fgl _{p',*}$ and satisfies Lemma's conditions.
\end{proof}

\begin{proof}[Proof of Theorem] By Lemma~\ref{l2.2} we can only consider the isomorphism $\phi _\sigma $ for some ${\sigma \in S_{\Zee}}$. Moreover, $\sigma \in S_g$ because $\phi _\sigma (\fgl _{p,c})\subseteq \fgl _{p',g}$. 

We have to prove that if $\phi _\sigma (\fgl _{p,*})\subseteq \fgl _{p',*}$, then $\sigma \in S_*$. For $* = o$, this follows from Lemma~\ref{l2.1}. For the other classes, it is not difficult to suitably adjust the formulation of Lemma~\ref{l2.1}.
\end{proof}

\section{Description of equivalence classes of parity functions}

For an arbitrary parity function $p$ and $-\infty \leqslant a<b \leqslant +\infty $, set:
\begin{gather*}
\Odd (p;a,b) := \card \{i\in \Zee\mid a~\leqslant i \leqslant b,p(i)=1\},\\
\Even (p; a, b):= \card \{i \in \Zee\mid a~\leqslant i \leqslant b,p(i) = 0\},\\
d(p;a,b)=d(p,b,a):=\frac {\Odd(p;a,b)}{(b-a)}.
\end{gather*}

We will call a~parity function $p$ \textit{finite} if
\[
\Odd(p; - \infty , +\infty ) < \infty \quad \text{or}\quad \Even(p; -\infty , +\infty ) < \infty .
\]
It is clear that both $\Odd(p; -\infty , +\infty )$ and $\Even(p; -\infty ,{+}\infty )$ are invariant under any of the permutation groups defined above. On the other hand, two finite parity functions $p$, $p'$ are $S_c$-equivalent if and only if
\[
\Odd(p; -\infty , +\infty ) = \Odd(p'; -\infty , +\infty )
\]
and
\[
\Even(p; -\infty , +\infty ) = \Even(p'; -\infty , +\infty ).
\]

We will denote the superalgebras corresponding to a~finite parity function $p$ by:
\[
\fgl _{p,*}(m|\infty ),\quad \text{where}\quad m = \card(\Even(p; -\infty , \infty ))
\]
or by
\[
\fgl _{p,*}(\infty |n),\quad \text{where}\quad n = \card(\Odd(p;-\infty ,\infty )) .
\]
\ssbegin{Proposition}\label{p3.1} For any finite parity function $p$, the cardinalities 
 $\Odd(p; -\infty, +\infty )$ and $\Even(p; -\infty ,{+}\infty )$ constitute the complete system of invariants with respect to $S_c$, $S_o$, $S_l$, and~ $S_g$.
\end{Proposition}

\begin{proof} It is obvious that both the cardinalities are $S_{\Zee}$-invariant. Therefore, we have to show that there exists a~belonging to $S_c$ isomorphism between two such parity functions. So take a~one-to-one map
\[
\sigma \colon \begin{cases}\Zee_{p'}^{+} \tto \Zee_{p'}^{-}=\Zee_p^{+}&\\ 
\Zee_p^{-} \tto \Zee_p^{+}=\Zee_{p'}^{-}&\\
\end{cases}
\]
arbitrary on two finite sets, where $p \ne p'$ and such that $\sigma (x) = x$ if $p(x) = p'(x)$.
\end{proof}

Non-finite parity functions are subdivided into five $S_g$-invariant classes according to how many infinite values (0 to 4) there are among
\[
\Odd(p;-\infty ,0),\ \Even(p;-\infty ,0),\ \Odd(p; 0,{+}\infty ) \text{ and } \Even(p; 0,{+}\infty ).
\]
Denote by $\text{Inf}$ the set of parity functions for which all the four values are infinite.

\subsection{Continuous invariants of certain parity functions} Let $\{a_i\}_{i\in\Zee}$, $\{b_i\}_{i\in\Zee}$ be two arbitrary series such that
\be\label{4}
\lim\limits_{i\to\infty} a_i=\lim\limits_{i\to\infty} b_i=\pm \infty ,\quad \lim\limits_{i\to\infty} (a_i-b_i)=\infty .
\ee

We will say that $\alpha $ is a~\textit{left \textup{(}right\textup{)} density point} for $p$ if there is a~$-$ (respectively, $+$) sign in eq.~\eqref{4} and $\alpha $ is a~limit point of the $d_i = d(p; a_i,b_i)$. We will call the set of all left (right) density points for all such series the \textit{left \textup{(}right\textup{)} density spectrum} and denote it by $D_l(p)$ and $D_r(p)$, respectively.

\sssbegin{Proposition} Both $D_l$ and $D_r$ are subsegments of $[\hspace*{.9pt}0, 1]$.
\end{Proposition}

\begin{proof} This follows from the fact that these two sets are closed and convex.

Let $d$ be a~limit point of $D_r$, i.e., $\lim_{i\to\infty} d_i = d$ for some $d_i\in D_r$. For each $d_i$, select
\[
\{(a_{ij},b_{ij}) \mid \lim\limits _{j\tto \infty }d(p,a_{ij},b_{ij})=d_i\}
\]
as in definition of $D_r$. Then, $d$ is a~density point for the set $\{a_{ii},b_{ii}\}_{i\in\Zee}$.

Let $d, d' \in D_r$ and $\{(a_i,b_i)\}_{i\in\Zee}$, $\{(a_i',b_i')\}_{i\in\Zee}$ be some series converging to them, respectively. Then, for any $\alpha$ such that $0 < a~< 1$, we can select $(a'', b'')$ such that $a_i< a''< a'$ and $b_i<b''<b'$.
\end{proof}

\sssbegin{Theorem}\label{Th} \textup{1)} The sets $D_l$ and $D_r$ are $S_c$- and $S_o$-invariant; i.e., if $D_l=[\alpha ^l,\beta ^l]$, $D_r=[\alpha ^r,\beta ^r]$ and parity functions $p$ and $p'$ are $S_o$-equivalent, then $\alpha _p^l=\alpha _{p'}^l$, $\alpha _p^r=\alpha _{p'}^r$, $\beta _p^l=\beta _{p'}^l$, $\beta _p^r=\beta _{p'}^r$.

\textup{2)} If $0<\alpha ^l=\beta ^l<1$ and $0 <\alpha ^r=\beta ^r<1$ for both $p$ and $p'$, then these two functions are $S_m$-equivalent if and only if they have the same density spectrum.

\textup{3)} If $0<\alpha ^l \leqslant \beta ^l<1$ and $0<\alpha ^r \leqslant \beta ^r<1$ for both $p$ and $p'$, then these two functions are $S_m$-equivalent.

\textup{4)} Any two non-finite parity functions that belong to the same invariance class are $S_g$-equivalent.
\end{Theorem}

\begin{proof} First, let us examine the case of $S_g$. Let us fix some invariance class of non-finite parity functions, say $\Inf$. For any parity function $p\in \Inf$, we can define $\sigma _p\colon \Zee\tto \Zee$ by the following formulas:
\[
\begin{array}{l}
\sigma _p(x) 
\footnotesize
:=\begin{cases}2(1-p(x))\Even(p;1,x)+p(x)(2 \Odd(p; 0, x) - 1)&\text{for }x \geqslant 0,\\
-2(1-p(x) )(\Even(p; x, 0) - 1) - p(x)
(2\Odd(p; x, -1) - 1)&\text{for }x < 0.\qed
\end{cases}
\end{array}
\]
\noqed \end{proof}

\sssbegin{Lemma} For some $\sigma _p\in S_g$, we have $\sigma _p(p)=p_{st}$
\end{Lemma}

\begin{proof} Observe that 
\[
\sigma _p(-\infty ,0) = \begin{cases}(-\infty ,0)&\text{if $p(0) = 0$,}\\
(-\infty ,0)&\text{if $p(0) = 1$},\\ 
\end{cases}
\]
and $\sigma _p(0,\infty ) = (0,\infty )$.

The function $p_{st}$ is therefore $S_g$-equivalent to any other parity function from Inf. So any two parity functions $p,p' \in \Inf$ are $S_g$-equivalent: $p' =\sigma _{p'}^{-1}\sigma _p(p)$.

It is easy to see that four parity functions, which are equal to $p_{st}$ either on $(-\infty ,0)$ or on $(0, +\infty )$ and are constant elsewhere, are representatives of the four other classes. The required isomorphisms are obtained in a~similar way.
\end{proof}

\subsubsection{}
Now, let us consider the smallest of our permutation groups, $S_c$. The main reason to investigate it is that for $p \equiv 0$, the Lie algebra $\widetilde {\fgl }_{p,c}$ has various nice properties, studied in~\cite{V}, and our $\widetilde {\fgl }_{p,c}$ for $p$ not identically equal to zero is its straightforward generalization.

For $\sigma \in S_c$, we have $|\sigma (a)-a|\leqslant c_\sigma $, and therefore we obviously have
\[
|d(\sigma (p);a,b)-d(p;a,b)|\leqslant \frac {2c_\sigma }{|a-b|}\text{ for any }p.
\]
Thus, for any two series $a_i$, $b_i$ such that $|a_i-b_i| \tto \infty $ as $i\tto \infty $, all density limit points are $\sigma $-invariant.

There exist continuous $S_c$-invariants other than just upper/lower left/right density limits, which describe the behavior of the density (for example, logarithmic density) in more detail, as well as involve other $S_c$-invariant characteristics (for example, $\limsup \ln \Odd(p; - x,x)$). The proof holds not only for $S_c$, but for $S_a$ as well. Still, I can only guarantee that two parity functions are non-$S_o$-equivalent if some of these invariants differ.

One can hope to describe parity functions from \text{Inf} in terms of density spectrum only for non-finite functions with the property that for some $c_p > 0$ there is no interval $(a, b)$ of more than $c_p$ units long on which $p$ is constant. In other words, the density spectrum should be separated from 0 and 1. Let us call such parity functions \textit{tight}.

A larger group, $S_n$, enables us to progress further. Namely, we have

\sssbegin{Proposition} If two parity functions have coinciding one-point left and right density spectra and are tight, then they are $S_n$-equivalent.
\end{Proposition}

\begin{proof}[Proof\nopoint] uses the construction of $\sigma _p$ described above. Although neither $\sigma _p$ nor $\sigma _{p'}$ belong to $S_n$, if $D^l_p =D_{p'}^l$ and $D_p^r=D_{p'}^r$, then $\sigma _{p'}^{-1}\sigma _p\in S_n$.
\end{proof}

\section{Miscellanea}

For complete list of simple (twisted) loop superalgebras see~\cite{S3}. The list of all their central extensions (not all of which are affine Kac-Moody superalgebras a.k.a. double extensions with Cartan matrix, see \cite{BLS}) is given in~\cite{FLS}. For the definition of Lie superalgebras with Cartan matrix, in particular, of affine Kac-Moody superalgebras, see \cite{CCLL}.

\subsection{Embeddings of affine Kac-Moody superalgebras into $\widetilde {\fgl }_{p,c}$} Recall the definition of the double extension of the loop (super)algebra $\fg^{\ell(1)}$ or of its twisted version $\fg^{\ell(k)}$, see \cite{BLS}. Let $p$ be a~parity function such that $p(i) = p(i + k)$ and set
\[
A^k:=\{(a_{ij})_{i,j\in \Zee} \in \widetilde {\fgl }_{p,c} \mid a_{ij}=a_{i+k,j+k}\}.
\]
Let $p$ be a~parity function such that $p(i) = p(i + k) + 1$, and set
\[
B^k := \{(a_{ij})_{i,j\in \Zee} \in \widetilde {\fgl }_{p,c} \mid a_{ij}=a_{i+k,j+k}\}.
\]

\sssbegin{Proposition} \textup{1)} $A^k \cong \fgl (m|n)^{\ell(1)}$, where $m=\Even(p; 1,k)$, $n =\Odd(p; 1,k)$.

\textup{2)} $B^k \cong \fq(k)^{\ell(2)}$.
\end{Proposition}

\subsection{Analogues of classical subalgebras for $\widetilde {\fgl }_{p,*}$}
Provided the parity function meets the requirements in curly brackets below, set
\begin{gather*}
B_{p,*}:=\{(a_{ij})_{i,j\in \Zee} \in \widetilde {\fgl }_{p,*} \mid a_{ij}=-(-1)^{p(i)}a_{-j,{-}i}\text{ for }p(i)=p(-i),p(0)=0\}.\\
D_{p,*}:=\{(a_{ij})_{i,j\in \Zee}\in \widetilde {\fgl }_{p,*} \mid a_{ij}=-(-1)^{p(i)}a_{1-j,1-i}\text{ for }p(i)=p(1-i)\}.\\
\fq_{p;*}:=\{a\in \widetilde {\fgl }_{p,*} \mid [a,P]=0 \text{ for an odd matrix $P$ such that }P^2 = \lambda \pmb1\}.
\end{gather*}

The Dynkin diagrams corresponding to $B_{p_{st},*}$ are of the form:
\[
\raisebox{-.25em}[0pt][0pt]{\text{\huge $\bullet $}}-\otimes -\cdots-\otimes -\cdots
\]
The Dynkin diagrams corresponding to $D_{p_{st},*}$ are of the form:
\[
\begin{array}{@{}r@{}c@{}l@{}}
\otimes \\[-.3em]
&\diagdown \\[-.16em]
\raisebox{0pt}[0pt][0pt]{\rule[-1em]{0.05em}{2.5em}\,\rule[-1em]{.05em}{2.5em}}\hspace*{.2em}\mbox{}&&\otimes -\cdots-\otimes -\cdots\\[-.16em]
&\diagup\\[-.3em]
\otimes 
\end{array}
\]
The Dynkin diagram corresponding to $\fq_{p_{st},*}$ is of the form (for a definition of the base described by a~square circle, see~\cite{LSe2}:
\[
-\ldots -\lozenge-\ldots 
\]
There is also the periplectic superalgebra without a Dynkin diagram:
\[
\mathfrak {pe}_{p,*}:=\{(a_{ij})_{i,j\in \Zee} \in \widetilde {\fgl }_{p,*} \mid a_{ij}=-(-1)^{p(i)}a_{1-j,1-i}\text{ for }p(i)= p(1-i)+1\}.
\]

\subsection*{Acknowledgements} I am thankful to D. Leites, who raised the problem, and to V. Serganova, who defined $\fgl_{p,g}$, for help.


\begin{landscape}
\begin{table}
\caption{Coxeter diagrams of universal super Weyl groups}

Notation: $\otimes $ stands for an odd reflection.

\medskip 

\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$\fg$&$\fsl(m | n)$&$\fosp(3 | 2)$&$\fosp(3 | 2n)$; $n> 1$\\
\hline
$UW_{\fg}$&$\underbrace {\bigcirc -\cdots-\bigcirc }_{m-1}\overset \infty {-}\otimes \overset \infty {-}\underbrace {\bigcirc -\cdots-\bigcirc }_{n-1}$&$\bigcirc \overset \infty {-}\otimes \overset \infty {-}\bigcirc $&$\bigcirc \overset \infty {-}\otimes \overset \infty {-}\underbrace {\bigcirc -\cdots-\bigcirc \overset 4{-}\bigcirc }_n$\\
\hline
\end{tabular}

\vspace{5pt}

\begin{tabular}{|c|c|c|c|}
\hline
$\fg$&$\fosp(2m+1 | 2)$, $m>1$&$\fd(\alpha )$&$\fosp(2 | 2n)$, $n>1$\\
\hline
$UW_{\fg}$&$\underbrace {\bigcirc \overset 4{-}\bigcirc -\cdots-\bigcirc }_m \overset \infty {-}\otimes \overset \infty {-}\bigcirc $&$\underset {\displaystyle \overset {\displaystyle \hphantom {\scriptstyle \infty }|{\scriptstyle \infty }}{\displaystyle \bigcirc }}{\bigcirc \overset \infty {-}\otimes \overset \infty {-}\bigcirc }$&$\otimes -\otimes \overset \infty {-}\underbrace {\bigcirc -\cdots-\bigcirc \overset 4{-}\bigcirc }_n$\\
\hline
\end{tabular}


\vspace{5pt}

\begin{tabular}{|c|c|c|c|}
\hline
$\fg$&$\fosp(2m+1 | 2n)$; $m,n>1$&$\fosp(4 | 2n)$, $n>1$&$\fosp(2m|2)$, $m>2$\\
\hline
$UW_{\fg}$&$\underbrace {\bigcirc \overset 4{-}\bigcirc -\cdots-\bigcirc }_m \overset \infty {-}\otimes \overset \infty {-}\underbrace {\bigcirc -\cdots-\bigcirc \overset 4{-}\bigcirc }_n$&
$\begin{array}{@{}r@{}l@{}l@{}}
\bigcirc \\[-.16em]
&\hspace*{-.145em}\diagdown \hspace*{-.3em}\raisebox{.3em}[0pt][0pt]{$\scriptstyle \infty $}\\[-.7em]
&&\hspace*{-.4em}\otimes\overset \infty {-}\underbrace {\bigcirc -\cdots-\bigcirc \overset 4{-}\bigcirc }_n \\[-1.3em]
&\hspace*{-.145em}\diagup \hspace*{-.3em}\raisebox{-.3em}[0pt][0pt]{$\scriptstyle \infty $}\\[-.16em]
\bigcirc 
\end{array}
$&$\underbrace {\bigcirc -\cdots-\bigcirc }_{m-2}\overset \infty {-}\otimes \overset \infty {-}\bigcirc $\\
\hline
\end{tabular}

\vspace{5pt}

\begin{tabular}{|c|c|c|c|}
\hline
$\fg$&$\fosp(2m|2n)$; $m>2$, $n>1$&$\fag(2)$&$\fab(3)$\\
\hline
$UW_{\fg}$&$\begin{array}{@{}r@{}c@{}l@{}}
\bigcirc \\[-.16em]
&\hspace*{-.145em}\diagdown \\[-1em]
&&\underbrace {\bigcirc -\cdots-\bigcirc }_{m-2}\overset \infty {-}\otimes \overset \infty {-}\underbrace {\bigcirc -\cdots-\bigcirc \overset 4{-}\bigcirc }_n\\[-1.65em]
&\hspace*{-.145em}\diagup\\[-.16em]
\bigcirc 
\end{array}
$&$\bigcirc \overset \infty {-}\otimes \overset \infty {-}\bigcirc \overset 6{-}\bigcirc $&$\bigcirc \overset \infty {-}\otimes \overset \infty {-}\bigcirc \overset 4{-}\bigcirc -\bigcirc $\\
\hline
\end{tabular}
\end{center}
\end{table}
\end{landscape}



