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

edit

An associative algebra   over a commutative ring   is defined to be a nilpotent algebra if and only if there exists some positive integer   such that   for all   in the algebra  . The smallest such   is called the index of the algebra  .[2] In the case of a non-associative algebra, the definition is that every different multiplicative association of the   elements is zero.

Nil algebra

edit

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

edit

References

edit
  1. ^ Goodearl, K. R.; Yakimov, M. T. (1 Nov 2013). "Unipotent and Nakayama automorphisms of quantum nilpotent algebras". arXiv:1311.0278 [math.QA].
  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{{cite book}}: CS1 maint: postscript (link)
  3. ^ Nil algebra – Encyclopedia of Mathematics
edit