nLab tensor product of Lie algebras

Definition

We first recall the definition of an action of a Lie algebra.

Definition

Let $\mathfrak{m},\mathfrak{n}$ be a pair of Lie algebras over some field $k$. An action of $\mathfrak{m}$ on $\mathfrak{n}$ is a $k$-bilinear map of $k$-vector spaces $\rho:\mathfrak{m}\otimes \mathfrak{n}\to \mathfrak{n}$ satisfying

• $\rho([m,m'])(n) = \rho(m)(\rho(m')(n))-\rho(m')(\rho(m)(n))$

• $\rho(m)([n,n']) = [\rho(m)(n),n']+[n,\rho(m)(n')]$

for $m\in \mathfrak{m}$ and $n,n'\in \mathfrak{n}$.

Definition

Let $\mathfrak{m},\mathfrak{n}$ be a pair of Lie algebras, endowed with an action $\rho$ of $\mathfrak{m}$ on $\mathfrak{n}$, and an action $\sigma$ of $\mathfrak{n}$ on $\mathfrak{m}$. Their tensor product $\mathfrak{m}\otimes \mathfrak{n}$ is generated by elements $m\otimes n$ such that

• $\lambda(m\otimes n) = (\lambda m)\otimes n = m\otimes (\lambda n),$

• $(m+m')\otimes n = m\otimes n + m'\otimes n,$

• $(m\otimes (n+n') = m\otimes n + m\otimes n',$

• $[m,m']\otimes n = m\otimes \rho(m')(n) - m'\otimes \rho(m)(n)$

• $m\otimes [n,n'] = \sigma(n')(m)\otimes n - \sigma(n)(m)\otimes n',$

• $[m\otimes n, m'\otimes n'] = - (\sigma(n)(m))\otimes (\rho(m')(n')),$

for $\lambda\in k$, $m,m'\in \mathfrak{m}$, and $n,n'\in \mathfrak{n}$.

This definition, due to Ellis (1989), is to be thought of as the infinitesimal version of the tensor product of groups. It was further extended in Garcia-Martinez et al. (2015) to super Lie algebras.

References

This definition first appears in

• Graham Ellis. A non-abelian tensor product of Lie algebras. Glasgow Mathematical Journal, 33(1), pp.101-120. (doi)

The corresponding definition for super Lie algebras is in

• Xabier García-Martínez, Emzar Khmaladze, Manuel Ladra. Non-abelian tensor product and homology of Lie superalgebras. Journal of Algebra Volume 440, 15 October 2015, Pages 464-488. (doi)