Simple Non-associative Algebra Research Articles

Classification of D-bialgebra structures on power series algebras

In this paper, we use algebro-geometric methods in order to derive classification results for so-called D-bialgebra structures on the power series algebra A〚z〛 for certain central simple non-associative algebras A. These structures are closely related to a version of the classical Yang-Baxter equation (CYBE) over A.If A is a Lie algebra, we obtain new proofs for pivotal steps in the known classification of non-degenerate topological Lie bialgebra structures on A〚z〛 as well as of non-degenerate solutions of the usual CYBE.If A is associative, we achieve the classification of non-triangular topological balanced infinitesimal bialgebra structures on A〚z〛 as well as of all non-degenerate solutions of an associative version of the CYBE.

Local and 2-local derivations on octonion algebras

This paper is devoted to study of local and 2-local derivations on octonion algebras. We shall give a general form of local derivations on the real octonion algebra [Formula: see text]. This description implies that the space of all local derivations on [Formula: see text] when equipped with Lie bracket is isomorphic to the Lie algebra [Formula: see text] of all real skew-symmetric [Formula: see text]-matrices. We also consider [Formula: see text]-local derivations on an octonion algebra [Formula: see text] over an algebraically closed field [Formula: see text] of characteristic zero and prove that every [Formula: see text]-local derivation on [Formula: see text] is a derivation. Further, we apply these results to similar problems for the simple seven-dimensional Malcev algebra. As a corollary, we obtain that the real octonion algebra [Formula: see text] and Malcev algebra [Formula: see text] are simple non-associative algebras which admit pure local derivations, that is, local derivations which are not derivation.

A class of continuous non-associative algebras arising from algebraic groups including

Abstract We give a construction that takes a simple linear algebraic group G over a field and produces a commutative, unital, and simple non-associative algebra A over that field. Two attractions of this construction are that (1) when G has type $E_8$ , the algebra A is obtained by adjoining a unit to the 3875-dimensional representation; and (2) it is effective, in that the product operation on A can be implemented on a computer. A description of the algebra in the $E_8$ case has been requested for some time, and interest has been increased by the recent proof that $E_8$ is the full automorphism group of that algebra. The algebras obtained by our construction have an unusual Peirce spectrum.

Combinatorial Algebra and Its Antisymmetrized Algebra I

The simple non-associative algebra N(eAS, q, n, t)k and its simple subalgebras are defined in [1, 3, 5–7, 13]. In this work, we define the combinatorial algebra N(e𝔄p, n, t)k and its antisymmetrized algebra [Formula: see text] and their subalgebras. We prove that these algebras are simple. Some authors [2, 5–7, 10, 13, 14, 16, 17] found all the derivations of an associative algebra, a Lie algebra, and a non-associative algebra. We find all the derivations of the subalgebra N(e±x1x2 ⋯xn, 0, n)[1] of N(e𝔄p, n, t)k and the Lie subalgebra [Formula: see text] of [Formula: see text].

DERIVATIONS OF A COMBINATORIAL LIE ALGEBRA

We consider the simple antisymmetrized algebra <TEX>$N(e^{A_P},n,t)_1^-$</TEX>. The simple non-associative algebra and its simple subalgebras are defined in the papers [1], [3], [4], [5], [6], [8], [13]. Some authors found all the derivations of an associative algebra, a Lie algebra, and a non-associative algebra in their papers [2], [3], [5], [7], [9], [10], [13], [15], [16]. We find all the derivations of the Lie subalgebra <TEX>$N(e^{{\pm}x_1x_2x_3},0,3)_{[1]}{^-}$</TEX> of <TEX>$N(e^{A_p},n,t)_k{^-}$</TEX> in this paper.

NOTES ON AN ALGEBRA WITH SCALAR DERIVATIONS

In this paper, we consider the simple non-associative algebra WN(F(e x r ;0;1)(@;@2)). There are many papers on nding

Central Simple Nonassociative Algebras with Operators

The classical central simple theory of associative algebras generalizes, in this article, to a central simple theory of nonassociative algebras with operators and a related central irreducible theory of modules. These theories are motivated by, and apply to, problems of constructing and classifying simple Jordan Lie algebras, irreducible modules, and birings.

NEW ALGEBRAS USING ADDITIVE ABELIAN GROUPS I

The simple non-associative algebra <TEX>$N(e^{A_S},q,n,t)_k$</TEX> and its simple sub-algebras are defined in the papers [1], [3], [4], [5], [6], [12]. We define the non-associative algebra <TEX>$\overline{WN_{(g_n,\mathfrak{U}),m,s_B}}$</TEX> and its antisymmetrized algebra <TEX>$\overline{WN_{(g_n,\mathfrak{U}),m,s_B}}$</TEX>. We also prove that the algebras are simple in this work. There are various papers on finding all the derivations of an associative algebra, a Lie algebra, and a non-associative algebra (see [3], [5], [6], [9], [12], [14], [15]). We also find all the derivations <TEX>$Der_{anti}(WN(e^{{\pm}x^r},0,2)_B^-)$</TEX> of te antisymmetrized algebra <TEX>$WN(e^{{\pm}x^r}0,2)_B^-$</TEX> and every derivation of the algebra is outer in this paper.

Lie Admissible Non-Associative Algebras

A non-associative ring which contains a well-known associative ring or Lie ring is interesting. In this paper, a method to construct a Lie admissible non-associative ring is given; a class of simple non-associative algebras is obtained; all the derivations of the non-associative simple [Formula: see text] algebra defined in this paper are determined; and finally, a solid algebra is defined.

A Note on Derivations of Simple Algebras

Necessary and sufficient conditions for all the derivations of a finite dimensional simple nonassociative algebra, over a field of characteristic zero, to be inner are given in terms of the Lie multiplication algebra and the trace of the derivations.

Specht ideals of identities of certain simple nonassociative algebras

Specht ideals of identities of certain simple nonassociative algebras

Radicals related to the Brown-McCoy radical in some varieties of algebras

AbstractThe Brown-McCoy radical is the upper radical defined by the class of simple rings with identities. For associative or alternative rings the Brown-McCoy radical is hereditary, and its semi-simple class consists of all subdirect products of simple rings with identities. In this paper we present some classes of simple non-associative algebras whose upper radicals behave similarly. Classifications are then obtained of ‘most’ semi-simple radical classes of (γ, δ) and right alternative rings.

A note on a theorem of Jacobson

PROOF. In the first instance, let A have an identity or not. Let R. (L.) denote the right (left) multiplication in A, and D be a derivation of A. Then A has neither proper ideals nor proper D-ideals. In other words, the Lie algebra L (L') generated by Rz, L. (R., L. and the derivation D) is irreducible. Moreover, L'=L+ {aD } F (a vector space sum and not necessarily a direct sum), and L is an ideal of L'. Further, by a theorem of Jacobson [1, p. 47], we have L = C? [LL]; L' = C' E [L'L'] for centers C, C' of L, L'; [LL], [L'L'] are semisimple; [LL] is an ideal of [L'L']. Any transformation T in C commutes with the irreducible associative algebra generated by RX, L. and hence should be a multiple of the identity transformation I, by Schur's lemma. Now, if A contains an identity, C= FI, C'= FI; since the dimension of L' is at the most greater by unity than that of L, and since [L'L'] cannot have a one dimensional (abelian) ideal complementary to [LL], [LL] = [L'L']; i.e., L=L', or, DEL. Thus, every derivation of A is inner. REMARK 1. In case A is any simple nonassociative algebra, then CCC' and therefore [LL] = [L'L'] in this case also. If, in addition C = C' for every derivation D of A, then every derivation of A will be inner. Because F is algebraically closed, C=O or C=FI, C'=O or C'= FI. Since C = FI implies C'= FI = C and since every derivation is inner in this case as well as when C' = 0, the question raised at the outset boils down to the consideration of the only case C= 0, C' = FI. The plausibility of this case remains to be seen. Now, in the case of simple Lie algebra A over a field F of characteristic zero, L= {ad x}eA; L'=L+ { aD}JaEF. Since A is simple, the center of A = { x EA I ad x = 0 } = {0}. If ad y Ecenter C of L, then