Nilpotent multiplier of abelian group is graded component of free Lie ring
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Statement
Suppose is an abelian group and is a positive integer. Denote by the free Lie ring on . Note that naturally has the structure of a graded Lie ring. The claim is that the -nilpotent multiplier (defined as the Baer invariant of with respect to the subvariety of the variety of groups given by groups of nilpotency class at most ) is isomorphic, as an abelian group, to the graded component of :
Note that the first graded component is , so we get that:
Interpretation in Schur functor terms
The graded components of the free Lie ring of can be viewed as functors of . Each of these functors can explicitly described in Schur functor terms: see induction formula for Lie operad.
Related facts
- Formula for dimension of graded component of free Lie algebra, along with this fact, allows us to explicitly compute the nilpotent multipliers for a free abelian group.
- Nilpotent multiplier of perfect group equals Schur multiplier
Particular cases
Value of | Term for group of class at most | Term for -nilpotent multiplier | Statement | How it follows |
---|---|---|---|---|
1 | abelian group | Schur multiplier | Schur multiplier of abelian group is its exterior square | We have is the degree two graded component of . This can be described as the quotient of by all the degree two relations. The degree two relations are all generated by alternation, so the quotient is the exterior square . |
2 | group of nilpotency class two | 2-nilpotent multiplier | Explicit description of 2-nilpotent multiplier of abelian group in terms of relations of the tensor cube | is the degree three graded component of . This is a quotient of by a bunch of relations: alternation in the inner two variables and Jacobi identity in all three variables. The quotient by these relations is the 2-nilpotent multiplier. |
References
Journal references
- On five well-known commutator identities by Graham Ellis, Journal of the Australian Mathematical Society, Volume 54, Page 1 - 19(Year 1993): ^{}^{More info}
- On the nilpotent multipliers of a group by John Burns and Graham Ellis, Math. Zeitschr., Volume 226, Page 405 - 428(Year 1997): ^{Official page (PDF downloadable)}^{More info}, Proposition 2.1o (not proved here, but referenced with context)