Let $\mathfrak g:=(V,d,[\cdot,\cdot])$ be a differential graded Lie algebra (DGLA) where $d$ is the zero differential.

Intrinsic formality: The DGLA $\mathfrak g$ will be said intrinsically formal if any $L_\infty$-algebra $l$ admitting $\mathfrak g$ in cohomology (i.e. $H(l)=\mathfrak g$) is formal (i.e. there exists a $L_\infty$-quasi isomorphism $\mathfrak g\to l$).

Rigidity: The DGLA $\mathfrak g$ will be said rigid if any $L_\infty$-algebra $l'$ such that $l'_1=0$, $l'_2=[\cdot,\cdot]$ is isomorphic to $\mathfrak g$. In other words, $\mathfrak g$ does not admit non-trivial deformations as a $L_\infty$-algebra.

Question 1: What is the relation (equivalence, implication) between these two notions?

Let me denote $NR(\mathfrak g)$ the DGLA controlling the deformations of $\mathfrak g$ as a $L_\infty$-algebra and let $H^1(NR(\mathfrak g))$ be the associated first cohomology group.

Question 2: Does any of the previous notions entails the vanishing of $H^1(NR(\mathfrak g))$, or conversely ?

Let $\mathfrak g:=(V,d,[\cdot,\cdot])$ be a differential graded Lie algebra (DGLA) where $d$ is the zero differential.

Intrinsic formality: The DGLA $\mathfrak g$ will be said intrinsically formal if any $L_\infty$-algebra $l$ admitting $\mathfrak g$ in cohomology (i.e. $H(l)=\mathfrak g$) is formal (i.e. there exists a $L_\infty$-quasi isomorphism $\mathfrak g\to l$).

Rigidity: The DGLA $\mathfrak g$ will be said rigid if any $L_\infty$-algebra $(V,l')$ such that $l'_1=0$, $l'_2=[\cdot,\cdot]$ is isomorphic to $\mathfrak g$. In other words, $\mathfrak g$ does not admit non-trivial deformations as a $L_\infty$-algebra.

Question 1: What is the relation (equivalence, implication) between these two notions?

Let me denote $NR(\mathfrak g)$ the DGLA controlling the deformations of $\mathfrak g$ as a $L_\infty$-algebra and let $H^1(NR(\mathfrak g))$ be the associated first cohomology group.

Question 2: Does any of the previous notions entails the vanishing of $H^1(NR(\mathfrak g))$, or conversely ?