Special identities for the pre-Jordan product in the free dendriform algebra
- Bremner, M.R. 1
- Madariaga, S. 1
-
1
University of Saskatchewan
info
ISSN: 0024-3795
Year of publication: 2013
Volume: 439
Issue: 2
Pages: 435-454
Type: Article
More publications in: Linear Algebra and Its Applications
Metrics
JCR (Journal Impact Factor)
- Year 2013
- Journal Impact Factor: 0.983
- Journal Impact Factor without self cites: 0.632
- Article influence score: 0.682
- Best Quartile: Q2
- Area: MATHEMATICS, APPLIED Quartile: Q2 Rank in area: 84/251 (Ranking edition: SCIE)
SCImago Journal Rank
- Year 2013
- SJR Journal Impact: 0.985
- Best Quartile: Q1
- Area: Algebra and Number Theory Quartile: Q1 Rank in area: 20/93
- Area: Discrete Mathematics and Combinatorics Quartile: Q1 Rank in area: 11/66
- Area: Geometry and Topology Quartile: Q2 Rank in area: 22/85
- Area: Numerical Analysis Quartile: Q2 Rank in area: 15/65
Scopus CiteScore
- Year 2013
- CiteScore of the Journal : 1.8
- Area: Algebra and Number Theory Percentile: 90
- Area: Discrete Mathematics and Combinatorics Percentile: 85
- Area: Geometry and Topology Percentile: 76
- Area: Numerical Analysis Percentile: 54
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Abstract
Pre-Jordan algebras were introduced recently in analogy with pre-Lie algebras. A pre-Jordan algebra is a vector space A with a bilinear multiplication x · y such that the product x {ring operator} y = x · y + y · x endows A with the structure of a Jordan algebra, and the left multiplications L· (x) : y {mapping} x · y define a representation of this Jordan algebra on A. Equivalently, x · y satisfies these multilinear identities:((x {ring operator} y) · (z · u) + (y {ring operator} z) · (x · u) + (z {ring operator} x) · (y · u); ≡ z · [(x {ring operator} y) · u] + x · [(y {ring operator} z) · u] + y · [(z {ring operator} x) · u],; x · [y · (z · u)] + z · [y · (x · u)] + [(x {ring operator} z) {ring operator} y] · u; ≡ z · [(x {ring operator} y) · u] + x · [(y {ring operator} z) · u] + y · [(z {ring operator} x) · u] .). The pre-Jordan product x · y = x ≻ y + y ≺ x in any dendriform algebra also satisfies these identities. We use computational linear algebra based on the representation theory of the symmetric group to show that every identity of degree ≤ 7 for this product is implied by the identities of degree 4, but that there exist new identities of degree 8 which do not follow from those of lower degree. There is an isomorphism of S8-modules between these new identities and the special identities for the Jordan diproduct in an associative dialgebra. © 2013 Elsevier Inc. All rights reserved.