Lie-admissible Algebras Research Articles

A subalgebra condition in Lie-admissible algebras

Let A A be a finite-dimensional, flexible, Lie-admissible algebra over a field Φ \Phi of characteristic ≠ 2 \ne 2 . Let S S be a subalgebra of A − {A^ - } and H H be a Cartan subalgebra of S S . It is shown that S S is a subalgebra of A A if and only if H H ⊆ S HH \subseteq S .

Convolution and correlation algebras

An algebraic characterization of convolution and correlation is outlined. The basic algebraic structures generated on a suitable vector space by the two operations are described. The convolution induces an associative Abelian algebra over the real field; the correlation induces a not-associative, not-commutative — but Lieadmissible algebra — with a left unity. The algebraic connection between the two algebras is found to coincide with the relation of isotopy, an extension of the concept of equivalence. The interest of these algebraic structures with respect to information processing is discussed.

Some Classes of Flexible Lie-Admissible Algebras

Some Classes of Flexible Lie-Admissible Algebras

A note on Lie-admissible nilalgebras

It is shown that a finite dimensional, flexible, powerassociative, Lie-admissible algebra W over a field of characteristic 0 is a nilalgebra and there exists a Cartan subalgebra of %which is nil in W. Let W be a flexible algebra, that is, a nonassociative algebra satisfying the flexible law (xy)x = x(yx). The algebra 9is defined as the same vector space as 9t but with a multiplication given by [x, y] = xy yx. Then 9 is said to be Lie-admissible 9is a Lie algebra. If W is finite dimensional, we consider a Cartan subalgebra $ of 9-. Since Dh Rh L7, is a derivation of 9t for every h in $ and $ is the Fitting null component of 9for D($5), it follows that $ is a subalgebra of 9. If 9is a simple Lie algebra, it is shown that 91 is a nilalgebra ([3] and [4]). In case Wis simple, the structure of 91 has been studied in [2] by using a Cartan subalgebra of 9which is nil in 9t. In this note we give a condition that 9 be a nilalgebra in terms of a Cartan subalgebra of 9-. THEOREM. Suppose that 91 is a finite dimensional, flexible, powerassociative, Lie-admissible algebra over a field of characteristic 0. Then 9 is a nilalgebra and there exists a Cartan subalgebra of 9which is nil in W. PROOF. The only if part is obvious. Let $ be a Cartan subalgebra of 9which is nil in 9. Let $1 be the nil radical of W (the maximal nil ideal of 9). Suppose that 9 is not a nilalgebra. Then the quotient algebra W = W/91 satisfies the assumptions in the theorem and is semisimple. Since any flexible power-associative algebra of characteristic 0 is strictly power-associative, it follows from [4] that 9 has an identity T. Since the characteristic is 0, it also follows from [1, p. 379] that the homomorphic image 5 of $ is a Cartan subalgebra of the Lie algebra W-. But then 1 is in 5, and since t is nil in 9, this is a contradiction. Therefore 9 is a nilalgebra. Received by the editors February 24, 1971. AMS 1970 subject classifications. Primary 17A20, 17B05.

Some classes of flexible Lie-admissible algebras

Let A \mathfrak {A} be a finite-dimensional, flexible, Lie-admissible algebra over a field of characteristic ≠ 2 \ne 2 . Suppose that A − {\mathfrak {A}^ - } has a split abelian Cartan subalgebra H \mathfrak {H} which is nil in A \mathfrak {A} . It is shown that if every nonzero root space of A − {\mathfrak {A}^ - } for H \mathfrak {H} is one-dimensional and the center of A − {\mathfrak {A}^ - } is 0, then A \mathfrak {A} is a Lie algebra isomorphic to A − {\mathfrak {A}^ - } . This generalizes the known result obtained by Laufer and Tomber for the case that A − {\mathfrak {A}^ - } is simple over an algebraically closed field of characteristic 0 and A \mathfrak {A} is power-associative. We also give a condition that a Levi-factor of A − {\mathfrak {A}^ - } be an ideal of A \mathfrak {A} when the solvable radical of A − {\mathfrak {A}^ - } is nilpotent. These results yield some interesting applications to the case that A − {\mathfrak {A}^ - } is classical or reductive.

On Reductive Lie Admissible Algebras

A Lie admissible algebra is a non-associative algebra A such that A− is a Lie algebra where A− denotes the anti-commutative algebra with vector space A and with commutation [X, Y] = XY – YX as multiplication; see [1; 2; 5]. Next let L−(X): A− → A−: Y → [X, Y] and H = {L−(X): X ∊ A−}; then, since A− is a Lie algebra, we see that H is contained in the derivation algebra of A− and consequently the direct sum g = A − ⊕ H can be naturally made into a Lie algebra with multiplication [PQ] given by: P = X + L−(U), Q = Y + L−(V) ∊ g, thenand note that for any P, [PP] = 0 so that [PQ] = −[QP] and the Jacobi identity for g follows from the fact that A− is Lie.

Nodal algebras of dimension 𝑝³

An algebra A is in K if and only if there exist p, n, and P such that A+ = Bn where multiplication in A is defined by 1 » df dg 2 ,-,/_i dXi OX/ in terms of the (dot) product in Bn, and the [xt, Xj]=XiX/—XjXi are arbitrary except for the condition that at least one of these commutators be nonsingular. Every simple nodal noncommutative Jordan algebra is in K, but not all the algebras in K are simple [3]. Properties of K have been studied in [2], [3], [4], and in several other papers, but the ideal structure of the nonsimple algebras of K is unknown. In this paper we give a complete list of the ideals of a Lie-admissible algebra in K with three generators. Theorem. // A EK is Lie-admissible and of dimension ps, then there exist generators x, y, z such that A = Fl-\-F[x, y, z] (vector space direct sum) with [y, x] = 1 + c-xp-l-yp~1, c E A, (1) [z, x] = y'T~1-m(z), m(z) E P[l, z], and [z, y] = x*-l-n(z), n(z) E F[l, z].

Dissipativity and Lie-admissible algebras

The validity of Hamiltonian mechanics and Lie algebra invariance for free fields and their difficulties for interacting fields are recalled. The role of dissipativity is discussed as an approach allowing the introduction for interacting regions of larger analytical dynamics and algebraic formulations related by the enlarged bracket. A working criterion is introduced in terms of Lie-admissible algebras and the pseudo-Hamiltonian mechanics introduced by R. J. Duffin for discrete dissipative systems is considered as an explicit choice able to reduce to the Hamiltonian mechanics when the systems become conservative. An example of dissipative plasma is explicitly investigated. Furthermore the procedure is extended to continuous systems and classical interpolating dissipative fields induced by Lie-admissible structures are constructed.

On a class of nodal noncommutative Jordan algebras

where f-g is the product of f=f(x1, . . ., xn) and g=g(x1, . . ., xn) in Bn and the = 1[xi, xj] = (xixj -xjxi) are arbitrary except for the proviso that at least one of them is nonsingular. That is, there must exist a cij = aij1 + wij with aij =0. This implies that n ? 2. The class K was constructed by L. Kokoris who proved [2], [3] that every simple nodal noncommutative Jordan algebra is in K. He also proved that not all the algebras of the class K are simple. Two papers have appeared which were concerned with studying derivations of some of the algebras in K. The first of these by R. Schafer [7] described derivation algebras for the cases cij in F and n =2 and demonstrated relationships between certain ideals of these algebras and types of simple Lie algebras of characteristic p. He made use of two properties which were later generalized by R. Oehmke, namely that the algebras in K for the cases cij in F and n =2 are Lie-admissible and that, for n = 2, the generators xl and x2 could be chosen so that c12 = 1 + axp -1*x2-1 for a in F. In his paper [4], Oehmke determined the derivation algebras of all simple Lieadmissible algebras A of K. He proved that generators could be chosen for these A which satisfy a useful Lie-multiplication table, which is a generalization of