Direct Sum Of Simple Ideals Research Articles

Some remarks on semisimple Leibniz algebras

From Levi's Theorem it is known that every finite-dimensional Lie algebra over a field of characteristic zero is decomposed into semidirect sum of its solvable radical and semisimple subalgebra. Moreover, semisimple part is the direct sum of simple ideals. In Barnes (preprint) [6] Levi's Theorem is extended to the case of Leibniz algebras. In the present paper we investigate the semisimple Leibniz algebras and we show that the splitting theorem for semisimple Leibniz algebras is not true. Moreover, we consider some special classes of the semisimple Leibniz algebras and we find a condition under which they decompose into direct sum of simple ideals.

The socle of a nondegenerate Lie algebra

We define the socle of a nondegenerate Lie algebra as the sum of all its minimal inner ideals. The socle turns out to be an ideal which is a direct sum of simple ideals, and satisfies the descending chain condition on principal inner ideals. Every classical finite dimensional Lie algebra coincides with its socle, while relevant examples of infinite dimensional Lie algebras with nonzero socle are the simple finitary Lie algebras and the classical Banach Lie algebras of compact operators on an infinite dimensional Hilbert space. This notion of socle for Lie algebras is compatible with the previous ones for associative algebras and Jordan systems. We conclude with a structure theorem for simple nondegenerate Lie algebras containing abelian minimal inner ideals, and as a consequence we obtain that a simple Lie algebra over an algebraically closed field of characteristic 0 is finitary if and only if it is nondegenerate and contains a rank-one element.

Calculating the structure of a semisimple Lie algebra

First we briefly describe two previously published algorithms: one that constructs a Cartan subalgebra and one that decomposes a semisimple Lie algebra L as a direct sum of simple ideals. Then, by reducing L modulo a prime we derive an algorithm to obtain the type of L (thereby solving the isomorphism problem for semisimple Lie algebras over Q̄ having structure constants in Q).

A structure theory of Freudenthal-Kantor triple systems

The notion of the radical of a Freudenthal-Kantor triple system is introduced. Using this, it is shown that a semisimple Freudenthal-Kantor triple system is decomposed into a direct sum of simple ideals. Also this paper defines and characterizes the structure groups for Freudenthal-Kantor triple systems.

Ein Einfachheitskriterium für nicht-assoziative Algebren

A nonassociative algebra B over a field K is called H-simple for a subgroup H of the group of automorphisms of B if B 2 ≠ {0} and if B has no H-invariant ideals except {0} and B. It is proved that every H-simple, finite-dimensional B is a direct sum of simple ideals which are conjugate under H. This implies that every H-simple, finite-dimensional B is simple if H has no normal subgroup different from H and of finite index in H or if H has no subgroup different from H whose index in H is a divisor of the dimension of B over K.

Finite Cartan matrices and Lie algebras of classical type

The Mills-Seligman result is that a classical Lie algebra L is a direct sum of simple ideals, each of which is again classical; that the simple classical Lie algebras are determined by their Cartan matrices; and that the possible indecomposable Cartan matrices are given by the usual classification, i.e., are of types A,,, B, ,..., G,. The proof of this result, in [3], is surprisingly hard. The difficulty arises from the fact that the root system of L is contained in H*, a vector space over F, while on the other hand, the Cartan matrix must be a matrix over H. Deducing information about the Cartan matrix (over Z) from information about the root system (over F) is difftcult. (See, for example, Lemmas 115.3 and 11.5.4 of [3].) In this paper we give a simpler proof of the Mills-Seligman theorem. Our technique is to associate to L a certain subset, A, of H’ which plays the role of root system and apply standard characteristic zero results (of [ 11) to A. These transference methods were used before by Seligman (see [4]), but only after much more theory in characteristic p was developed.

Jordan Rings with Nonzero Socle

Let $\mathcal {J}$ be a nondegenerate Jordan algebra over a commutative associative ring $\Phi$ containing $\tfrac {1}{2}$. Defining the socle $\mathcal {G}$ of $\mathcal {J}$ to be the sum of all minimal inner ideals of $\mathcal {J}$, we prove that $\mathcal {G}$ is the direct sum of simple ideals of $\mathcal {J}$. Our main result is that if $\mathcal {J}$ is prime with nonzero socle, then either (i) $\mathcal {J}$ is simple unital and satisfies DCC on principal inner ideals, (ii) $\mathcal {J}$ is isomorphic to a Jordan subalgebra $\mathcal {J}’$ of the plus algebra ${A^ + }$ of a primitive associative algebra A with nonzero socle S, and $\mathcal {J}’$ contains ${S^ + }$, or (iii) $\mathcal {J}$ is isomorphic to a Jordan subalgebra $\mathcal {J}''$ of the Jordan algebra of all symmetric elements H of a. primitive associative algebra A with nonzero socle S, and $\mathcal {J}''$ contains $H \cap S$. Conversely, any algebra of type (i), (ii), or (iii) is a prime Jordan algebra with nonzero socle. We also prove that if $\mathcal {J}$ is simple then $\mathcal {J}$ contains a completely primitive idempotent if and only if either $\mathcal {J}$ is unital and satisfies DCC on principal inner ideals or $\mathcal {J}$ is isomorphic to the Jordan algebra of symmetric elements of a $*$-simple associative algebra A with involution $*$ containing a minimal one-sided ideal.

Theorem on semisimple lie p-algebras

The known fact on the decomposability of any semisimple Lie algebra over an algebraically closed field of characteristic zero into a direct sum of simple ideals remains true in the case of characteristic p > 0. It is also shown that a semisimple Lie p-algebra, admitting a faithful p-representation of dimension n < p−1 has such a decomposition. Its direct factors are Lie p-algebras of the classical type with a non-degenerate bilinear form of trace. The restriction n < p−1 is essential. Bibliography has 6 references.