[Paper Review] Operations and Identities in Tensor Algebra

Yu. B. Chernyakov, V. Dolotin|ArXiv.org|Jan 13, 2005

Algebraic structures and combinatorial models2 references3 citations

TL;DR

This paper introduces non-decomposable $N$-ary operations in mixed tensor algebras, focusing on a 3-ary operation in $A^1_2 \oplus A^2_1$ that generalizes the Jacobi identity. It derives higher identities for deformed matrix commutators and proves that a specific 3-ary operation satisfies a generalized Jacobi-like identity, establishing a new class of non-associative algebras closed under such operations.

ABSTRACT

We define the class of non-decomposable $N$-ary operations in the mixed tensor algebra $\bigoplus\limits_{i,j=0}^\infty A_i^j$. There are higher Jacobi-like identities for (binary) deformed matrix commutator and a 3-ary operation which is non-decomposable into binary ones.

Motivation & Objective

• To identify and characterize non-decomposable $N$-ary operations in the mixed tensor algebra $\bigoplus_{i,j=0}^\infty A^j_i$.
• To determine subspaces invariant under these operations and to derive the identities they satisfy.
• To generalize the classical Jacobi identity to a 3-ary operation in $A^1_2 \oplus A^2_1$, extending the structure of Lie algebras.
• To explore the algebraic structure of $n$-ary operations and their closure properties under tensor contractions and products.
• To establish a diagrammatic language for tracking tensor contractions and operations in the tensor algebra.

Proposed method

• Uses a diagrammatic notation with arrows to represent (1,1)-tensors and contractions between upper and lower indices.
• Defines elementary operations via tensor products and contractions, classifying all possible compositions of two (1,1)-tensors.
• Constructs a generic binary operation $A \circ B = \alpha AB + \beta BA + \gamma A \operatorname{Tr} B + \delta B \operatorname{Tr} A$ with complex coefficients.
• Derives identities for the 3-ary operation by computing cyclic sums of triple compositions and simplifying using trace identities.
• Applies group-theoretic reasoning (permutations of five symbols) to classify terms and assign weights to words in the identity expansion.
• Uses symmetry and weight analysis to derive a system of equations (e.g., $2(\beta\gamma + \gamma\alpha + \alpha\beta) = 0$) that ensure the total identity vanishes.

Experimental results

Research questions

• RQ1 What are the conditions under which a 3-ary operation in $A^1_2 \oplus A^2_1$ is non-decomposable into binary operations?
• RQ2 What generalized Jacobi-type identity does this 3-ary operation satisfy?
• RQ3 How do trace terms and cyclic compositions interact in higher-order identities?
• RQ4 What subspaces of the mixed tensor algebra are invariant under such $N$-ary operations?
• RQ5 Can the full set of identities for $n$-ary operations be systematically derived using diagrammatic and group-theoretic methods?

Key findings

• A 3-ary operation in $A^1_2 \oplus A^2_1$ is constructed that is non-decomposable into binary operations and satisfies a generalized Jacobi identity.
• The identity $ (A\circ B)\circ C + (C\circ A)\circ B + (B\circ C)\circ A = \operatorname{Tr}A(BC - CB) + \operatorname{Tr}B(AC - CA) + \operatorname{Tr}C(BA - AB) $ generalizes the classical Jacobi identity.
• The cyclic sum of triple compositions vanishes when the coefficients satisfy $ 2(\beta\gamma + \gamma\alpha + \alpha\beta) = 0 $, $ \gamma\gamma + \gamma\alpha + \gamma\beta + \beta\beta + \beta\alpha + \beta\gamma = 0 $, etc.
• The full identity for the 3-ary operation is verified by decomposing all 120 permutations of five symbols into 10 classes of 12 words, each class corresponding to a pair of tensor types.
• The structure of the identity is shown to be consistent across all 10 classes, with weights assigned via coefficient systems ensuring cancellation.
• The paper demonstrates that the 3-ary operation generates a non-associative algebra closed under the derived identities, extending the framework of Lie and Jordan algebras.