Tensor product of Lie rings

Definition

Suppose $M$ and $N$ are Lie rings and $\alpha :M\to \mathrm{D}\mathrm{e}\mathrm{r}(N)$ and $\beta :N\to \mathrm{D}\mathrm{e}\mathrm{r}(M)$ is a compatible pair of actions of Lie rings. We define the tensor product $M\otimes N$ for this pair of actions as follows. It is the quotient of the free Lie ring on formal symbols of the form $m\otimes n$ ($m\in M,n\in N$) by the following relations:

• Additive in $M$: $(m_1+m_2)\otimes n=(m_1\otimes n)+(m_2\otimes n)\mathrm{\forall }{m}_1,m_2\in M,n\in N$. Note that if we are dealing with Lie algebras instead of Lie rings, we will replace additivity by linearity in $M$ with respect to the ground ring.
• Additive in $N$: $m\otimes (n_1+n_2)=(m\otimes n_1)+(m\otimes n_2)\mathrm{\forall }m\in M,n_1,n_2\in N$. Note that if we are dealing with Lie algebras instead of Lie rings, we will replace additivity by linearity in $N$ with respect to the ground ring.
• Expanding a tensor product involving one Lie bracket:
  • $[m_1,m_2]\otimes n=m_1\otimes \alpha (m_2)(n)-m_2\otimes \alpha (m_1)(n)\mathrm{\forall }{m}_1,m_2\in M,n\in N$
  • $m\otimes [n_1,n_2]=\beta (n_2)m\otimes n_1-\beta (n_1)(m)\otimes n_2\mathrm{\forall }m\in M,n_1,n_2\in N$

If both the actions are rewritten using $\cdot$, this simplifies to:

• • $[m_1,m_2]\otimes n=m_1\otimes (m_2\cdot n)-m_2\otimes (m_1\cdot n)\mathrm{\forall }{m}_1,m_2\in M,n\in N$
  • $m\otimes [n_1,n_2]=(n_2\cdot m)\otimes n_1-(n_1\cdot m)\otimes n_2\mathrm{\forall }m\in M,n_1,n_2\in N$
• Expanding a Lie bracket of two pure tensors:

$[(m_1\otimes n_1),(m_2\otimes n_2)]=-(\beta (n_1)(m_1))\otimes (\alpha (m_2)(n_2))$

If both the actions are rewritten using $\cdot$, this becomes:

$[(m_1\otimes n_1),(m_2\otimes n_2)]=-(n_1\cdot m_1)\otimes (m_2\cdot n_2)$

Facts

Maps and constructions

For the statements in these facts, we will use the same notation as in the definition above: $M$ and $N$ are Lie rings with a compatible pair of actions of Lie rings $\alpha :M\to \mathrm{D}\mathrm{e}\mathrm{r}(N)$ and $\beta :N\to \mathrm{D}\mathrm{e}\mathrm{r}(M)$.

Name	The kind of map or construction	Explicit description using named actions	Explicit description using $\cdot$
Tensor product of Lie rings is commutative up to natural isomorphism	A natural isomorphism $M\otimes N\to N\otimes M$. If the isomorphism is applied twice, it gives the identity mapping.	$m\otimes n\mapsto -(n\otimes m)$	$m\otimes n\mapsto -(n\otimes m)$
Tensor product of Lie rings maps to both Lie rings	Homomorphisms: ${\lambda }_M:M\otimes N\to M$ ${\lambda }_N:M\otimes N\to N$	${\lambda }_M(m\otimes n)=-(\beta (n)(m))$ ${\lambda }_N(m\otimes n)=\alpha (m)(n)$	${\lambda }_M(m\otimes n)=-(n\cdot m)$ ${\lambda }_N(m\otimes n)=m\cdot n$
Lie ring acts naturally on its tensor product with any Lie ring	Homomorphisms: $M\to \mathrm{D}\mathrm{e}\mathrm{r}(M\otimes N)$ $N\to \mathrm{D}\mathrm{e}\mathrm{r}(M\otimes N)$	$m_1$ sends $m_2\otimes n$ to $[m_1,m_2]\otimes n+m_2\otimes \alpha (m_1)(n)$ $n_1$ sends $m\otimes n_2$ to $\beta (n_1)(m)\otimes n_2+m\otimes [n_1,n_2]$	$m_1\cdot (m_2\otimes n)=[m_1,m_2]\otimes n+m_2\otimes (m_1\cdot n)$ $n_1\cdot (m\otimes n_2)=(n_1\cdot m)\otimes n_2+m\otimes [n_1,n_2]$

References

Journal references

Original use

• A non-abelian tensor product of Lie algebras by Graham J. Ellis, Glasgow Journal of Math, Volume 33, Page 101 - 120(Year 1991): ^{Official copy (PDF)More info}