  Vertex operator approach is a powerful method to study exactly solvable
models. We review recent progress of vertex operator approach to semi-infinite
spin chain. (1) The first progress is a generalization of boundary condition.
We study $U_q(\widehat{sl}(2))$ spin chain with a triangular boundary, which
gives a generalization of diagonal boundary [Baseilhac and Belliard 2013,
Baseilhac and Kojima 2014]. We give a bosonization of the boundary vacuum
state. As an application, we derive a summation formulae of boundary
magnetization. (2) The second progress is a generalization of hidden symmetry.
We study supersymmetry $U_q(\widehat{sl}(M|N))$ spin chain with a diagonal
boundary [Kojima 2013]. By now we have studied spin chain with a boundary,
associated with symmetry $U_q(\widehat{sl}(N))$, $U_q(A_2^{(2)})$ and
$U_{q,p}(\widehat{sl}(N))$ [Furutsu-Kojima 2000, Yang-Zhang 2001, Kojima 2011,
Miwa-Weston 1997, Kojima 2011], where bosonizations of vertex operators are
realized by "monomial" . However the vertex operator for
$U_q(\widehat{sl}(M|N))$ is realized by "sum", a bosonization of boundary
vacuum state is realized by "monomial".
