Yu. Chernyakov, V. Dolotin
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.
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Tensor polynomial identities generalize the concept of polynomial identities on $d \times d$ matrices to identities on tensor product spaces. Here we completely characterize a certain class of tensor polynomial identities in terms of their associated Young tableaux. Furthermore, we provide a method to evaluate arbitrary alternating tensor polynomials in $d^2$ variables.
We consider the algebra $E\otimes E$ over an infinite field equipped with a $\mathbb{Z}_2$-grading where the canonical basis is homogeneous and prove that in various cases the graded identites are just the ordinary ones. If the grading is a non-canonical grading obtained as a quotient grading of the natural $\mathbb{Z}_2\times\mathbb{Z}_2$-grading we exhibit a basis for the graded identities.
Derived operations satisfy standard identities
研究导出操作的左乘或右乘操作满足特定阶的标准恒等式，证明模形式的Rankin-Cohen括号和Kontsevich的Poisson结构变形公式高阶括号均满足此类恒等式。
This is a survey of some recent developments in the theory of associative and nonassociative dialgebras, with an emphasis on polynomial identities and multilinear operations. We discuss associative, Lie, Jordan, and alternative algebras, and the corresponding dialgebras; the KP algorithm for converting identities for algebras into identities for dialgebras; the BSO algorithm for converting operations in algebras into operations in dialgebras; Lie and Jordan triple systems, and the corresponding disystems; and a noncommutative version of Lie triple systems based on the trilinear operation abc-b…
Special Identities for Comtrans Algebras
该研究聚焦Comtrans代数的恒等式问题，确定了comtrans operad的Gröbner基并提出维数猜想，分析了不同次数下特殊换位子与平移子的恒等式，还构造了2×2矩阵特殊comtrans代数的泛结合包络。
Gelfand--Dorfman bialgebras (GD-algebras) are nonassociative systems with two bilinear operations satisfying a series of identities that express Hamiltonian property of an operator in the formal calculus of variations. The paper is devoted to the study of GD-algebras related with differential Poisson algebras. As a byproduct, we obtain a general description of identities that hold for operations $a\succ b = d(a)b$ and $a\prec b = ad(b)$ on a (non-associative) differential algebra with a derivation~$d$.
We exhibit a finite basis M for a certain variety $\mathbf{V}$ of medial groupoids. The set M consists of the medial law (xy)(zt)=(xz)(yt) and five other identities involving four variables. The variety $\mathbf{V}$ is generated by the four groupoids $\pm x\pm y$ on the integers. Since $\mathbf{V}$ is a very natural variety, proving it to be finitely based should be of interest. In an earlier paper, we made a conjecture which implies that $\mathbf{V}$ is finitely based. In this paper, we show that $\mathbf{V}$ is finitely based by proving that M is a basis. Based on our proof, we think that ou…
The operation of binary intermolecular recombination, originating in the theory of DNA computing, permits a natural generalization to n-ary operations which perform simultaneous recombination of n molecules. In the case n = 3, we use computer algebra to determine the polynomial identities of degree <= 9 satisfied by this trilinear nonassociative operation. Our approach requires computing a basis for the nullspace of a large integer matrix, and for this we compare two methods: (i) the row canonical form, and (ii) the Hermite normal form with lattice basis reduction. In the conclusion, we formul…
Jacobson identities for post-Lie algebras in positive characteristic
本文研究了正特征下后李代数的雅可比恒等式，通过构造子邻近p-结构，探讨了受限李代数的条件，并引入受限后李代数的概念。
Polynomial identities for tangent algebras of monoassociative loops
本文通过引入n次Sabinin代数并利用交换子、结合子与四元子的多项式恒等式刻画4次情形，构造了满足单结合性a^2 a = a a^2的解析环切空间上的自然代数结构，并给出一个不能由低次恒等式推出的5次恒等式。
Operations of arbitrary arity expressible via addition modulo 2^n and bitwise addition modulo 2 admit a simple description. The identities connecting these two additions have finite basis. Moreover, the universal algebra with these two operations is rationally equivalent to a nilpotent ring and, therefore, generates a Specht variety.
We use computer algebra to demonstrate the existence of a multilinear polynomial identity of degree 8 satisfied by the bilinear operation in every Lie-Yamaguti algebra. This identity is a consequence of the defining identities for Lie-Yamaguti algebras, but is not a consequence of anticommutativity. We give an explicit form of this identity as an alternating sum over all permutations of the variables in a polynomial with 8 terms. Our computations also show that such identities do not exist in degrees less than 8.
We take an algorithmic and computational approach to a basic problem in abstract algebra: determining the correct generalization to dialgebras of a given variety of nonassociative algebras. We give a simplified statement of the KP algorithm introduced by Kolesnikov and Pozhidaev for extending polynomial identities for algebras to corresponding identities for dialgebras. We apply the KP algorithm to the defining identities for Jordan triple systems to obtain a new variety of nonassociative triple systems, called Jordan triple disystems. We give a generalized statement of the BSO algorithm intro…
In our paper arXiv: math.RA/0110333 v1 Oct 2001 we showed that the number of algebras defined by a binary operation satisfying a formally irreducible identity between two n-iterates is O( e^{-n/16}S_{n}^{2} for n --> infinity, S_{n} being the nth-Catalan number. This was proved by using exclusively the series of tableaux A_{n}. By using also the series of tableaux B_{n}, we now sharpen this result to O{(n+2)/n|e^{-n/16}-2/n)|S_{n}^{2}}
We propound the thesis that there is a limitation to the number of possible structures which are axiomatically endowed with identities involving operations. In the case of algebras with a binary operation satisfying a formally reducible (to be explained) identity between two n-iterates of the operation, it is established that the frequency of such algebras goes to zero as n tends to infinity. This is proved by a suitable ordering and labeling of the expressions (words) of the corresponding free algebra and the formation of a series of tableaux whose entries are the labels. The tableaux reveal …
Let $G$ be an abelian group and $\mathbb{K}$ an algebraically closed field of characteristic zero. A. Valenti and M. Zaicev described the $G$-gradings on upper block-triangular matrix algebras provided that $G$ is finite. We prove that their result holds for any abelian group $G$: any grading is isomorphic to the tensor product $A\otimes B$ of an elementary grading $A$ on an upper block-triangular matrix algebra and a division grading $B$ on a matrix algebra. We then consider the question of whether graded identities $A\otimes B$, where $B$ is an algebra with a division grading, determine $A\o…
We use computer algebra to study polynomial identities for the trilinear operation [a,b,c] = abc - acb - bac + bca + cab - cba in the free associative algebra. It is known that [a,b,c] satisfies the alternating property in degree 3, no new identities in degree 5, a multilinear identity in degree 7 which alternates in 6 arguments, and no new identities in degree 9. We use the representation theory of the symmetric group to demonstrate the existence of new identities in degree 11. The only irreducible representations of dimension < 400 with new identities correspond to partitions 2^5 1 and 2^4 1…
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