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Nilpotent algebra

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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

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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

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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

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References

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  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
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