The author introduces Poisson-Malcev manifolds and Malcev bialgebras which are characterized in terms of a diagonal mapping. In more detail, an associative commutative algebra \(A\) endowed with a bilinear map \(\{, \}: A\times A\to A\) satisfying \[ \{f_1f_2,f_3\}= f_1\{f_2,f_3\}+ \{f_1,f_3\} f_2,\;\{f_1,f_2\}= -\{f_2,f_1\}, \]
\[ J\bigl(f_1,f_2, \{f_1,f_3\}\bigr)= \bigl\{J (f_1,f_2,f_3), f_1\bigr\} \] is called a Poisson-Malcev algebra; here \[ J(f_1,f_2, f_3)=\{\{f_1,f_2\},f_3\}+ \{\{f_2,f_3\}, f_1\}+\{\{f_3,f_1\},f_2\} \] is the Jacobi expression. A manifold \(M\) is called a Poisson-Malcev space if the algebra \(A= C^\infty(M)\) is endowed with a Poisson-Malcev bracket. Let \(A\) be a Malcev algebra with comultiplication \(\Delta:A\to A\otimes A\) \((\Delta(a)=\sum a(1)\otimes a(2))\). Then a multiplication on \(A^*\) is defined by \(\langle fg,a \rangle=\sum \langle f,a(1)\rangle\langle g,a(2)\rangle\) and if the algebra \(A\oplus A^*\) is a Malcev algebra (we cannot state the multiplication on \(A \oplus A^*\) here), then \(A\) is called a Malcev bialgebra. The relevant conditions are expressed in terms of \(\Delta\).