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2004•Communications in AlgebraRequires access

Multiplicative Semiprimeness of Skew Lie Algebras

Juan Carlos Cabello, M. Cabrera, Ginés López-Pérez, Wallace S. Martindale

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Abstract

Let A be a semiprime associative algebra with an involution over a field of characteristic not 2, let K be the Lie algebra of all skew elements of A, and let Z [K, K] denote the annihilator of the Lie algebra [K, K]. We will prove that the multiplication algebra of the semiprime Lie algebra [K, K]/Z [K, K] is also semiprime. As a consequence, the multiplication algebra of [K, K]/Z [K, K] is prime, whenever [K, K]/Z [K, K] is prime. We will obtain similar results for the Lie algebra K/Z K whenever the base field has characteristic zero.

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What this paper is about

Let A be a semiprime associative algebra with an involution over a field of characteristic not 2, let K be the Lie algebra of all skew elements of A, and let Z [K, K] denote the annihilator of the Lie algebra [K, K]. We will prove that the multiplication algebra of the semiprime Lie algebra [K, K]/Z [K, K] is also semiprime. As a consequence, the multiplication algebra of [K, K]/Z [K, K] is prime, whenever [K, K]/Z [K, K] is prime. We will obtain similar results for the Lie algebra K/Z K whenever the base field has characteristic zero.

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Available abstract

Let A be a semiprime associative algebra with an involution over a field of characteristic not 2, let K be the Lie algebra of all skew elements of A, and let Z [K, K] denote the annihilator of the Lie algebra [K, K]. We will prove that the multiplication algebra of the semiprime Lie algebra [K, K]/Z [K, K] is also semiprime. As a consequence, the multiplication algebra of [K, K]/Z [K, K] is prime, whenever [K, K]/Z [K, K] is prime. We will obtain similar results for the Lie algebra K/Z K whenever the base field has characteristic zero.

Key concepts: Mathematics, Associative algebra, Multiplicative function, Lie conformal algebra, Pure mathematics, Annihilator, Semiprime, Semiprime ring

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