Xu, Xiaoping Quadratic conformal superalgebras. (English) Zbl 1001.17024 J. Algebra 231, No. 1, 1-38 (2000). A conformal \(\mathbb C\)-superalgebra is a \(\mathbb Z_2\)-graded \(\mathbb C[\partial]\)-module \(R=R_0\oplus R_2\) provided with a \(\mathbb Z_2\)-linear map \(Y^+(-,z)\) from \(R\) to the space of all linear mappings from \(R\) to \(R[z^{-1}]z^{-1}\) such that \[ Y^+(\partial u,z) = \frac{dY^+(-,z)}{dz},\qquad u\in R; \]\[ Y^+(u,z)v = (-1)^{ij}\text{Res}_x\frac{e^{x\partial}Y^+(v,-x)u}{z-x}, \]\[ Y^+(u,z_1)Y^+(v,z_2)-(-1)^{ij}Y^+(v,z_2)Y^+(u,z_1) = \text{Res}_x\frac{Y^+(Y^+(u,z_1-x)v,x)}{z_2-x} \] for all \(u\in R_i, v\in R_j\). A Novikov superalgebra is a \(\mathbb Z_2\)-graded nonassociative \(\mathbb C\)-algebra \(A=A_0\oplus A_1\) with a multiplication \(x\circ y\) such that \[ (u\circ v)\circ w =(-1)^{jl}( u\circ w)\circ v,\quad (u,v,w)=(-1)^{ij}(v,u,w) \] for all \(u\in A_i, v\in A_j, w\in A_l\), where \((u,v,w)=(u\circ v)\circ w -u\circ (v\circ w)\). A quadratic conformal superalgebra is a conformal superalgebra \(R\) which is a free \(\mathbb C[\partial]\)-module over a \(\mathbb Z_2\)-graded subspace \(V\) such that for all \(u,v\in V\) there exist elements \(w_i\in V_i\) for which \[ Y^+(u,z)v= (w_1+\partial w_2)z^{-1}+w_3z^{-2}. \] A super Gelfand-Dorfman bialgebra is a Lie superalgebra \(L=L_0\oplus L_1\) with multiplication \([x,y]\) having an additional structure of Novikov superalgebra \((L,\circ)\) such that \[ [w\circ u,v]-(-1)^{ij}[w\circ v,u]+[w,u]- (-1)^{ij}[w,v]\circ u -w\circ [u,v]=0 \] for all \(u\in L_i\), \(v\in L_j\), \(w\in L\). The main result of the paper states that a quadratic conformal superalgebra is equivalent to a Gelfand-Dorfman bialgebra. The author gives a way to construct Gelfand-Dorfman bialgebras and to classify these bialgebras based on simple Novikov algebras. 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