Since a Lie algebra is also a vector space, we can see that a Lie algebra has variety structure.

Is the bracket a morphism of varieties, and if it isn't always the case, is there a name for a structure that does have that property?


  • If a variety is also a group, where the multiplication and inverse are morphisms of varieties, we have an algebraic group.
  • If a variety is also a Lie algebra, where the Lie bracket is a morphism of varieties. What do we call this?
  • 4
    $\begingroup$ Yes, the Lie bracket is bilinear, hence is a morphism $V \times V \rightarrow V$. $\endgroup$ – Nefertiti Aug 4 '17 at 12:40

As Nefertiti said, in an arbitrary finite-dimensional algebra, the bilinear map is a morphism of varieties. Indeed in a basis, the bracket takes the form $$(x_1,\dots,x_n,y_1,\dots,y_n)\mapsto (z_1,\dots,z_n)$$ $$z_k=\sum_{(i,j)}a_{i,j,k}x_iy_j,$$ where $a_{i,j,k}$ are the structure constants. So each coordinate $z_k$ is indeed polynomial in the variables $(x_1,\dots,y_n)$.


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