# Definition of orthosymplectic supergroups

I found two versions of definitions of orthosymplectic supergroups. It seems that they are not equivalent. I don't know which version of the definition is standard.

The first version of the definition is in the paper. The orthosymplectic supergroup $OSP(2p|n)$ is defined as follows.

Let $A = A_0 \oplus A_1$ be a supercommutative superalgebra, where elements in $A_0$ are even and elements in $A_1$ are odd, and \begin{align} A_i A_j \subset A_{i+j \ (\text{mod} \ 2)}, \quad i, j \in \{0, 1\}. \end{align}

The general linear Lie supergroup $GL(m|n)$ is defined by \begin{align} & GL(m|n) = \left\{ \left( \begin{array}{c|c} X_{11} & Y_{12} \\ \hline Y_{21} & X_{22} \end{array} \right) \right\}, \\ & X_{11}=(x_{ij})_{i,j=1,\ldots,m}, X_{22}=(x_{ij})_{i,j=m+1,\ldots,m+n}, \\ & Y_{12}=(y_{ij})_{\substack{ i=1,\ldots,m \\ j = m+1, \ldots, m+n}}, Y_{21}=(y_{ij})_{\substack{i=m+1,\ldots,m+n \\ j = 1, \ldots, m}}, \end{align} where $x_{ij} \in A_0$ and $y_{ij} \in A_1$ and $\det(X_{11})\neq 0$, $\det(X_{22}) \neq 0$.

We have \begin{align} x_{ij} x_{kl} = x_{kl} x_{ij} \\ y_{ij} y_{kl} = - y_{kl} y_{ij} \\ y_{ij} x_{kl} = - x_{kl} y_{ij}. \end{align}

The Lie supergroup $OSP(m|n)$ is defined by \begin{align} & OSP(m|n) = \{ M \in GL(m|n): m=2p, M^{\text{st}}HM = H \}, \end{align} where \begin{align} & H = \left( \begin{array}{c|c} Q & 0 \\ \hline 0 & I_n \end{array} \right), \\ & Q = \left( \begin{array}{c|c} 0 & I_p \\ \hline -I_p & 0 \end{array} \right), \end{align} $I_n$ is the identity matrix of order $n$, and for \begin{align} M = \left( \begin{array}{c|c} A & B \\ \hline C & D \end{array} \right), \end{align} \begin{align} M^{\text{st}} = \left( \begin{array}{c|c} A^T & -C^T \\ \hline B^T & D^T \end{array} \right). \end{align}

The second version of the definition is in the paper. The definition is in Section 3.1. It is similar as the first definition above. But the matrix $H$ is different. It is defined as follows (formulas (3.2), (3.3), it is denoted by $\mathfrak{J}_{2n|2m+1}, \mathfrak{J}_{2n|2m}$). \begin{align} \mathfrak{J}_{2n|2m+1} = \left( \begin{array}{c|c|c|c|c} 0 & I_n & 0 & 0 & 0 \\ \hline -I_n & 0 & 0 & 0 & 0 \\ \hline 0 & 0 & 0 & I_m & 0 \\ \hline 0 & 0 & I_m & 0 & 0 \\ \hline 0 & 0 & 0 & 0 & 1 \end{array} \right) \end{align} \begin{align} \mathfrak{J}_{2n|2m} = \left( \begin{array}{c|c|c|c} 0 & I_n & 0 & 0 \\ \hline -I_n & 0 & 0 & 0 \\ \hline 0 & 0 & 0 & I_m \\ \hline 0 & 0 & I_m & 0 \end{array} \right) \end{align}

Which definition is standard? Thank you very much.

They are both standard. It depends on your choice of non-degenerate supersymmetric bilinear form, which doesn't matter. It seems that your definition is for $SPO(2p|n)$. SPO means symplectic-orthogonal, while OSP means othogional-symplectic. Be careful with non-degenerate bilinear form: $H=diag(Q,I_n)$ in the case of $SPO(2p|n)$, and $H=diag(I_n,Q)$ in the case of $OSP(n|2p)$.
• thank you very much. The lower part of the matrix $H$ in both cases are different: one is diagonal matrix and the other is \begin{align} \left( \begin{array}{c|c} 0 & I_m \\ \hline I_m & 0 \end{array} \right), \end{align} or \begin{align} \left( \begin{array}{c|c|c} 0 & I_m & 0 \\ \hline I_m & 0 & 0 \\ \hline 0 & 0 & 1 \end{array} \right), \end{align} Are the coordinate ring of $SPO(2p|n)$ of the definitions in both cases the same? Mar 17, 2017 at 6:49