Fixed points of automorphisms of free Lie algebras

Bryant, R. M.; Stöhr, Ralph

doi:10.1007/bf01197591

Cited by 12 publications

(24 citation statements)

References 2 publications

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“…This is, in fact, not hard to show, and will be established in Section 2 (Corollary 2.2). For p = 2, Theorem 1 was proved in our earlier paper [4]. An easy consequence of the theorem is the following result (derived in Section 6).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 69%

On the module structure of free Lie algebras

Bryant

Stöhr

1999

Trans. Amer. Math. Soc.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 69%

On the module structure of free Lie algebras

Bryant

Stöhr

1999

Trans. Amer. Math. Soc.

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“…Also, a n − b n + 2c n = λ n s , while a n + b n + c n is the dimension of the fixed point space of t in L n . By Theorem 1 of Bryant and Stöhr [5], the latter is 1 2 ψ n 2 . Thus we have three simultaneous linear equations which determine the three multiplicities: we may express a n from the first and last, b n from the first and second, and then c n from the last.…”

Section: The Case Of Gl 2mentioning

confidence: 94%

“…Its integral representation theory is well understood: see Lee [20] or p. 752 in Curtis and Reiner [7]. There are 10 isomorphism types of indecomposables, and one can readily check that only 4 of those satisfy the two conditions that we know must hold for every direct summand of an L n with n = 2: as a module for the subgroup t , it can have no trivial direct summand (because of the corollary in Bryant and Stöhr [5]), and upon reduction modulo 3 it must not acquire a one-dimensional direct summand (because of the present Theorem 5.2). It follows that an unrefinable direct decomposition of an L n (with n = 2) can involve only those four indecomposables.…”

Section: The Conclusion For Gl 2mentioning

confidence: 95%

“…Problem 11.47 in the Kourovka notebook [17] was posed with this in mind, and its recent solution in [5] finally provided the missing step. In turn, the extension of that work in [6] has provided the basis for our next theorem, which will play a similar role in later sections.…”

Section: The Case Of Gl 2mentioning

confidence: 99%

“…The first step in this direction is implicit in Bryant and Stöhr [5]: for the case r = 2, the results of that paper readily yield the structure of these homogeneous components as modules for an indecomposable subgroup of order 2 in GL 2 (all such subgroups are conjugate). In Corollary 7.3 and Theorem 7.4, we extend this result to the maximal finite subgroups of GL 2 (which fall into two conjugacy classes).…”

Section: Introductionmentioning

confidence: 97%

See 2 more Smart Citations

Lie Powers of the Natural Module for GL(2)

Kovács

Stöhr

2000

Journal of Algebra

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Invariant Bases for Free Lie Algebras

Guilfoyle

Stöhr

1998

Journal of Algebra

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1Let L be a free Lie algebra of finite rank over a field K and let X be a free generating set of L. Then L is a graded algebra,where L n is the K-span of all left-normed Lie productsLet g L → L be a graded algebra automorphism of order 2. If 2 is invertible in K then L has a homogeneous basis consisting of eigenvectors of g with eigenvalues ±1. For example, every Hall basis constructed from a free generating set consisting of eigenvectors has this property. Here we are concerned with the case where K is of characteristic 2. Then L has a homogeneous basis which is invariant under the action of g. In this paper we give an explicit construction of such a basis in terms of a g-invariant free generating set X. In fact, we first construct a g-invariant basis * for the free restricted Lie algebra on X (Theorem 1), and then we get as a subset of that basis (Theorem 2).The motivation for this work came from a problem posed by Bryant [2]. If the field K is replaced by the ring of integers , that is L is the free Lie ring on X, and g is a graded algebra automorphism of order 2, then L has a -basis such that ∪ − is a g-invariant set. Bryant asked if one could actually find a basis with this high degree of symmetry (Problem D in [2]). Theorem 2 solves this problem modulo 2.Our construction yields a new proof of results from [3,7] about the structure of L (with K a field of characteristic 2) as a module for the cyclic group of order 2 acting on L via a graded algebra automorphism. We mention that in the case where K = GF 2 and X = 2, the results of [3] have been exploited in [6] to determine the multiplicities of the indecomposable GL 2 2 -modules (induced from the natural action of GL 2 2 337

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Fixed points of automorphisms of free Lie algebras

Cited by 12 publications

References 2 publications

On the module structure of free Lie algebras

On the module structure of free Lie algebras

Lie Powers of the Natural Module for GL(2)

Invariant Bases for Free Lie Algebras

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