Nilpotent algebra

In mathematics, specifically in ring theory, a nilpotent algebra over a commutative ring is an algebra over a commutative ring, in which for some positive integer n every product containing at least n elements of the algebra is zero. The concept of a nilpotent Lie algebra has a different definition, which depends upon the Lie bracket. (There is no Lie bracket for many algebras over commutative rings; a Lie algebra involves its Lie bracket, whereas, there is no Lie bracket defined in the general case of an algebra over a commutative ring.) Another possible source of confusion in terminology is the quantum nilpotent algebra,[1] a concept related to quantum groups and Hopf algebras.

Formal definition

An associative algebra ${\displaystyle A}$ over a commutative ring ${\displaystyle R}$ is defined to be a nilpotent algebra if and only if there exists some positive integer ${\displaystyle n}$ such that ${\displaystyle 0=y_{1}\ y_{2}\ \cdots \ y_{n}}$ for all ${\displaystyle y_{1},\ y_{2},\ \ldots ,\ y_{n}}$ in the algebra ${\displaystyle A}$. The smallest such ${\displaystyle n}$ is called the index of the algebra ${\displaystyle A}$.[2] In the case of a non-associative algebra, the definition is that every different multiplicative association of the ${\displaystyle n}$ elements is zero.

Nil algebra

An algebra in which every element of the algebra is nilpotent is called a nil algebra.

Nilpotent algebras are trivially nil, whereas nil algebras may not be nilpotent, as each element being nilpotent does not force products of distinct elements to vanish.

See also

• Algebraic structure (a much more general term)
• nil-Coxeter algebra
• Lie algebra
• Example of a non-associative algebra

References

1. Goodearl, K. R.; Yakimov, M. T. (1 Nov 2013). "Unipotent and Nakayama automorphisms of quantum nilpotent algebras". arXiv:1311.0278.
2. Albert, A. Adrian (2003) [1939]. "Chapt. 2: Ideals and Nilpotent Algebras". Structure of Algebras. Colloquium Publications, Col. 24. Amer. Math. Soc. p. 22. ISBN 0-8218-1024-3. ISSN 0065-9258; reprint with corrections of revised 1961 edition

• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556

External links

• Nilpotent algebra – Encyclopedia of Mathematics