# Difference between revisions of "Nilpotent algebra"

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− | An algebra for which there is a natural number | + | {{TEX|done}} |

+ | An algebra for which there is a natural number $n$ such that any product of $n$ elements of the algebra is zero. If there is a non-zero product of $n-1$ elements, then $n$ 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 | + | 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 $d$, then every nilpotent subring of it of degree $[d/2]$ 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==== | ====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> | <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> |

## Latest revision as of 13:19, 10 April 2014

An algebra for which there is a natural number $n$ such that any product of $n$ elements of the algebra is zero. If there is a non-zero product of $n-1$ elements, then $n$ 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) the sum of finitely many nilpotent ideals is a 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 $d$, then every nilpotent subring of it of degree $[d/2]$ belongs to a sum of nilpotent ideals. The derived algebra of a finite-dimensional 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

[1] | N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) |

[2] | N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) |

[3] | A.A. Albert, "Structure of algebras" , Amer. Math. Soc. (1939) |

[4] | N. Jacobson, "A note on automorphisms and derivations of Lie algebras" Proc. Amer. Math. Soc. , 6 : 2 (1955) pp. 281–283 |

[5] | G. Higman, "Groups and rings having automorphisms without non-trivial fixed elements" J. London Math. Soc. , 32 : 3 (1957) pp. 321–334 |

**How to Cite This Entry:**

Nilpotent algebra.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Nilpotent_algebra&oldid=18078