View source for Nilpotent algebra

An algebra for which there is a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066700/n0667001.png" /> such that any product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066700/n0667002.png" /> elements of the algebra is zero. If there is a non-zero product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066700/n0667003.png" /> elements, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066700/n0667004.png" /> is called the index of nilpotency of the algebra. Examples of nilpotent algebras are: an algebra with zero multiplication; an algebra of strictly upper-triangular matrices; direct sums of nilpotent algebras the indices of nilpotency of which are uniformly bounded; and the tensor product of two algebras one of which is nilpotent.

The class of nilpotent algebras is closed under taking homomorphic images and subalgebras. In an associative algebra (cf. [[Associative rings and algebras|Associative rings and algebras]]) the sum of finitely many nilpotent ideals is a [[Nilpotent ideal|nilpotent ideal]], and the sum of an arbitrary set of nilpotent ideals is, generally speaking, locally nilpotent. A finite-dimensional algebra over a field of characteristic zero having a basis consisting of nilpotent elements is nilpotent. If an algebra satisfies a polynomial identity of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066700/n0667005.png" />, then every nilpotent subring of it of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066700/n0667006.png" /> belongs to a sum of nilpotent ideals. The derived algebra of a finite-dimensional [[Lie algebra|Lie algebra]] over a field of characteristic zero is nilpotent. Nilpotent subalgebras that coincide with their normalizer (Cartan subalgebras) play an essential role in the classification of simple Lie algebras of finite dimension. A nilpotent Lie algebra has an outer automorphism. A Lie algebra with a regular automorphism (that is, one having no fixed point except zero) of prime period is nilpotent.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,   "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Jacobson,   "Lie algebras" , Interscience  (1962)  ((also: Dover, reprint, 1979))</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Albert,   "Structure of algebras" , Amer. Math. Soc.  (1939)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N. Jacobson,   "A note on automorphisms and derivations of Lie algebras"  ''Proc. Amer. Math. Soc.'' , '''6''' :  2  (1955)  pp. 281–283</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G. Higman,   "Groups and rings having automorphisms without non-trivial fixed elements"  ''J. London Math. Soc.'' , '''32''' :  3  (1957)  pp. 321–334</TD></TR></table>