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 [math]\displaystyle{ A }[/math] over a commutative ring [math]\displaystyle{ R }[/math] is defined to be a nilpotent algebra if and only if there exists some positive integer [math]\displaystyle{ n }[/math] such that [math]\displaystyle{ 0=y_1\ y_2\ \cdots\ y_n }[/math] for all [math]\displaystyle{ y_1, \ y_2, \ \ldots,\ y_n }[/math] in the algebra [math]\displaystyle{ A }[/math]. The smallest such [math]\displaystyle{ n }[/math] is called the index of the algebra [math]\displaystyle{ A }[/math].[2] In the case of a non-associative algebra, the definition is that every different multiplicative association of the [math]\displaystyle{ n }[/math] elements is zero.
Nil algebra
A power associative algebra in which every element of the algebra is nilpotent is called a nil algebra.[3]
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
- ↑ Goodearl, K. R.; Yakimov, M. T. (1 Nov 2013). "Unipotent and Nakayama automorphisms of quantum nilpotent algebras". arXiv:1311.0278 [math.QA].
- ↑ Albert, A. Adrian (2003). "Chapt. 2: Ideals and Nilpotent Algebras". Structure of Algebras. Colloquium Publications, Col. 24. Amer. Math. Soc.. p. 22. ISBN 0-8218-1024-3. https://books.google.com/books?id=1G0HcOcoJ1cC&pg=PA22; reprint with corrections of revised 1961 edition
- ↑ Nil algebra – Encyclopedia of Mathematics
- Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4
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