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- W2963746732 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 ∘ y = x · y + y · x endows A with the structure of a Jordan algebra, and the left multiplications L · ( x ) : y ↦ x · y define a representation of this Jordan algebra on A . Equivalently, x · y satisfies these multilinear identities: ( x ∘ y ) · ( z · u ) + ( y ∘ z ) · ( x · u ) + ( z ∘ x ) · ( y · u ) ≡ z · [ ( x ∘ y ) · u ] + x · [ ( y ∘ z ) · u ] + y · [ ( z ∘ x ) · u ] , x · [ y · ( z · u ) ] + z · [ y · ( x · u ) ] + [ ( x ∘ z ) ∘ y ] · u ≡ z · [ ( x ∘ y ) · u ] + x · [ ( y ∘ z ) · u ] + y · [ ( z ∘ 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 S 8 -modules between these new identities and the special identities for the Jordan diproduct in an associative dialgebra." @default.
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- W2963746732 date "2013-07-01" @default.
- W2963746732 modified "2023-09-25" @default.
- W2963746732 title "Special identities for the pre-Jordan product in the free dendriform algebra" @default.
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- W2963746732 doi "https://doi.org/10.1016/j.laa.2013.03.010" @default.
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