Homotopy Lie algebra

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, in particular <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a> and <a href="/facts/Topology/2EqW47cU">topology</a>, a homotopy Lie algebra (or 
  
    
      
        
          L
          
            ∞
          
        
      
    
    {\displaystyle L_{\infty }}
  
-algebra) is a generalisation of the concept of a <a href="/facts/Differential_graded_Lie_algebra/KfMGlkQr">differential graded Lie algebra</a>. To be a little more specific, the <a href="/facts/Jacobi_identity/C9HoyQ3z">Jacobi identity</a> only holds up to homotopy. Therefore, a differential graded Lie algebra can be seen as a homotopy Lie algebra where the Jacobi identity holds on the nose. These homotopy algebras are useful in classifying deformation problems over characteristic 0 in <a href="/facts/Deformation_(mathematics)/tVLS3kkG">deformation theory</a> because deformation functors are classified by quasi-isomorphism classes of 
  
    
      
        
          L
          
            ∞
          
        
      
    
    {\displaystyle L_{\infty }}
  
-algebras. This was later extended to all characteristics by Jonathan Pridham.
</p><p>Homotopy Lie algebras have applications within mathematics and <a href="/facts/Mathematical_physics/sLKg8T1G">mathematical physics</a>; they are linked, for instance, to the <a href="/facts/Batalin%25E2%2580%2593Vilkovisky_formalism/E6b8pdmw">Batalin–Vilkovisky formalism</a> much like differential graded Lie algebras are.
</p>

Homotopy Lie algebra open-in-new

Homotopy Lie algebra