What is the relation between the Lagrange multipliers and the solution of the dual in a linear optimisation problem?

Consider the following linear minimisation problem
$$
(1) quad min_{xgeq 0} a^top x,\
quad quad quad text{s.t. }B^top x=c
$$

where

  • $x$ is the $ptimes 1$ vector of unknowns.

  • $a$ is a $ptimes 1$ vector of real numbers.

  • $B$ is a $ptimes k$ matrix of real scalars.

  • $c$ is a $ktimes 1$ vector of real scalars.

  • $p>k$.

Consider the Lagrangian of (1)

$$
(2) quad L(x,mu,nu)= a^top x+mu^{top}left(B^top x-cright)+nu^{top}x,
$$

Consider also the dual of (1)
$$
(3) quad max_{y} c^top y,\
quad quad quad text{s.t. } By leq a
$$

Question: what is the relation between the dual of (1) and the Lagrange multipliers of (1)?

Let $mathcal{X}^*$ be the set of argmin of (1).

Let $mathcal{Y}^*$ be the set of argmax of (3).

Let $mathcal{M}^*$ be the set of Lagrange multipliers $mu$ corresponding to each element of $mathcal{X}^*$.

Let $mathcal{V}^*$ be the set of Lagrange multipliers $nu$ corresponding to each element of $mathcal{X}^*$.

Is $mathcal{Y}^*=mathcal{M}^*$? What is the relation between $mathcal{Y}^*$ and $mathcal{V}^*$?