# Schur multiplier of a Lie ring

From Groupprops

## Definition

The notion of Schur multiplier of a Lie ring is defined analogously to the notion of Schur multiplier of a group. In fact, each of the equivalent formulations of the definition of Schur multiplier of a group has an analogous definition in the Lie ring context.

No. | Shorthand | Full definition |
---|---|---|

1 | second homology group | It is the second homology group for trivial Lie ring action . |

2 | second cohomology group | (?) It is the second cohomology group for trivial Lie ring action of on , i.e., the group . Note that we could replace by any divisible abelian group viewed as an abelian Lie ring. |

3 | Baer invariant | It is the Baer invariant of corresponding to the subvariety of abelian Lie rings in the variety of Lie rings. |

4 | Kernel of Lie bracket | It is the kernel of the Lie bracket homomorphism from exterior square to derived subgroup, i.e., it is the kernel of the map given by . Here is the exterior square of and is the derived subring of . |

5 | Hopf's formula | It is given by Hopf's formula for Schur multiplier: If is isomorphic to the quotient of a free Lie ring by an ideal , then . |