nLab tensor product of Lie algebras




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


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

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

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

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


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

  • λ(mn)=(λm)n=m(λn),\lambda(m\otimes n) = (\lambda m)\otimes n = m\otimes (\lambda n),

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

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

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

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

  • [mn,mn]=(σ(n)(m))(ρ(m)(n)),[m\otimes n, m'\otimes n'] = - (\sigma(n)(m))\otimes (\rho(m')(n')),

for λk\lambda\in k, m,m𝔪m,m'\in \mathfrak{m}, and n,n𝔫n,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.


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)

Created on January 25, 2024 at 19:18:53. See the history of this page for a list of all contributions to it.