This paper deals with multidimensional dynamic risk measures induced by
conditional $g$-expectations. A notion of multidimensional $g$-expectation is
proposed to provide a multidimensional version of nonlinear expectations. By a
technical result on explicit expressions for the comparison theorem, uniqueness
theorem and viability on a rectangle of solutions to multidimensional backward
stochastic differential equations, some necessary and sufficient conditions are
given for the constancy, monotonicity, positivity, homogeneity and
translatability properties of multidimensional conditional $g$-expectations and
multidimensional dynamic risk measures; we prove that a multidimensional
dynamic $g$-risk measure is nonincreasingly convex if and only if the generator
$g$ satisfies a quasi-monotone increasingly convex condition. A general dual
representation is given for the multidimensional dynamic convex $g$-risk
measure in which the penalty term is expressed more precisely. Similarly to the
one dimensional case, a sufficient condition for a multidimensional dynamic
risk measure to be a $g$-expectation is also explored. As to applications, we
show how this multidimensional approach can be applied to measure the
insolvency risk of a firm with interacted subsidiaries.