Lie identities in symmetric elements in group rings

Cristo, Osnel Broche; Mar√≠n, Manuel Ruiz

doi:10.1201/9781420010961.ch5

Cited by 2 publications

(2 citation statements)

References 23 publications

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“…Notice too that the set (RG) − is closed under the Lie bracket [α, β] = αβ − βα, and this Lie bracket is trivial if and only if for α, β ∈ (RG) − , αβ − βα = 0, that is, if and only if the skew-symmetric elements commute. This problem has also received much attention [3][4][5][6]16].…”

Section: Introductionmentioning

confidence: 99%

Involutions and Anticommutativity in Group Rings

Goodaire¹,

Milies²

2013

Can. math. bull.

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Abstract. Let g → g * denote an involution on a group G. For any (commutative, associative) ring R (with 1), * extends linearly to an involution of the group ring RG. An element α ∈ RG is symmetric if α * = α and skew-symmetric if α * = −α. The skew-symmetric elements are closed under the Lie bracket, [α, β] = αβ − βα. In this paper, we investigate when this set is also closed under the ring product in RG. The symmetric elements are closed under the Jordan product, α • β = αβ + βα. Here, we determine when this product is trivial. These two problems are analogues of problems about the skew-symmetric and symmetric elements in group rings that have received a lot of attention.

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Section: Introductionmentioning

confidence: 99%

Involutions and Anticommutativity in Group Rings

Goodaire¹,

Milies²

2013

Can. math. bull.

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“…This is a contribution to the volume of recent papers that consider involutions of group rings and, specifically, the sets of elements that are symmetric [Cri,CM06,Lee03,Lee99,GSV98] or skew-symmetric [CM,JM05,GM03] relative to an involution. The twist here is that we focus attention on RA loops and their loop rings.…”

Section: Introductionmentioning

confidence: 99%

Involutions of RA Loops

Goodaire¹,

Milies²

2009

Can. math. bull.

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Abstract. Let L be an RA loop, that is, a loop whose loop ring over any coefficient ring R is an alternative, but not associative, ring. Let ℓ → ℓ θ denote an involution on L and extend it linearly to the loop ring RL. An element α ∈ RL is symmetric if α θ = α and skew-symmetric if α θ = −α. In this paper, we show that there exists an involution making the symmetric elements of RL commute if and only if the characteristic of R is 2 or θ is the canonical involution on L, and an involution making the skew-symmetric elements of RL commute if and only if the characteristic of R is 2 or 4.

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Lie identities in symmetric elements in group rings

Cited by 2 publications

References 23 publications

Involutions and Anticommutativity in Group Rings

Involutions and Anticommutativity in Group Rings

Involutions of RA Loops

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