  Braided m-Lie algebras induced by multiplication are introduced, which
generalize Lie algebras, Lie color algebras and quantum Lie algebras. The
necessary and sufficient conditions for the braided m-Lie algebras to be strict
Jacobi braided Lie algebras are given. Two classes of braided m-Lie algebras
are given, which are generalized matrix braided m-Lie algebras and braided
m-Lie subalgebras of $End_F M$, where $M$ is a Yetter-Drinfeld module over $B$
with dim $B< \infty $ . In particular, generalized classical braided m-Lie
algebras $sl_{q, f}(GM_G(A), F)$ and $osp_{q, t} (GM_G(A), M, F)$ of
generalized matrix algebra $GM_G(A)$ are constructed and their connection with
special generalized matrix Lie superalgebra $sl_{s, f}(GM_{{\bf Z}_2}(A^s), F)$
and orthosymplectic generalized matrix Lie super algebra $osp_{s, t} (GM_{{\bf
Z}_2}(A^s), M^s, F)$ are established. The relationship between representations
of braided m-Lie algebras and their associated algebras are established.
