# Special identities for the pre-Jordan product in the free dendriform algebra

1. Bremner, M.R. 1
info

GRID grid.25152.31 Journal:
Linear Algebra and Its Applications

ISSN: 0024-3795

Year of publication: 2013

Volume: 439

Issue: 2

Pages: 435-454

Type: Article

Export: RIS
SCOPUS: 2-s2.0-84879889505 arXiv: 1212.5631v1 lock_openOpen access editor

## Metrics

### Cited by

• Scopus Cited by: 5 (14-07-2021)

### Journal Citation Reports

• Year 2013
• Journal Impact Factor: 0.983
• 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: 19/91
• Area: Discrete Mathematics and Combinatorics Quartile: Q1 Rank in area: 10/62
• Area: Geometry and Topology Quartile: Q2 Rank in area: 22/86
• Area: Numerical Analysis Quartile: Q2 Rank in area: 15/66

### 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

## 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.
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