Tensor product of Lie rings

Definition

Suppose $M$ and $N$ are Lie rings and $\alpha :M\to \operatorname {Der} (N)$ and $\beta :N\to \operatorname {Der} (M)$ 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 $M$: $(m_{1}+m_{2})\otimes n=(m_{1}\otimes n)+(m_{2}\otimes n)\ \forall \ m_{1},m_{2}\in M,n\in N$. 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 $N$: $m\otimes (n_{1}+n_{2})=(m\otimes n_{1})+(m\otimes n_{2})\ \forall \ m\in M,n_{1},n_{2}\in N$. 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:
  • $[m_{1},m_{2}]\otimes n=m_{1}\otimes \alpha (m_{2})(n)-m_{2}\otimes \alpha (m_{1})(n)\ \forall \ m_{1},m_{2}\in M,n\in N$
  • $m\otimes [n_{1},n_{2}]=\beta (n_{2})m\otimes n_{1}-\beta (n_{1})(m)\otimes n_{2}\ \forall m\in M,n_{1},n_{2}\in N$

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

• • $[m_{1},m_{2}]\otimes n=m_{1}\otimes (m_{2}\cdot n)-m_{2}\otimes (m_{1}\cdot n)\ \forall \ m_{1},m_{2}\in M,n\in N$
  • $m\otimes [n_{1},n_{2}]=(n_{2}\cdot m)\otimes n_{1}-(n_{1}\cdot m)\otimes n_{2}\ \forall m\in M,n_{1},n_{2}\in N$
• 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 $\alpha :M\to \operatorname {Der} (N)$ and $\beta :N\to \operatorname {Der} (M)$.

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: $M\to \operatorname {Der} (M\otimes N)$ $N\to \operatorname {Der} (M\otimes N)$	$m_{1}$ 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}