Identité polynomiale

En mathématiques, une identité polynomiale sur une algèbre associative est définie par l’annulation d’un polynôme non commutatif sur toute famille d’éléments de l’algèbre. L’ensemble des identités polynomiales sur une telle algèbre forme un T-idéal de l’algèbre associative libre engendrée par une famille dénombrable de variables formelles, c’est-à-dire un idéal stable par substitution des variables par d’autres polynômes non commutatifs.

Par exemple, toute algèbre commutative vérifie l’identité $x y - y x = 0$ . De même, une algèbre nilpotente d’ordre $n$ (telle que l’ensemble des matrices triangulaires supérieures strictes d’ordre $n$ ) admet l’identité $x n = 0$ et même $x 1 x 2 \dots x n = 0$ .

D’après le théorème d’Amitsur–Levitski^[1], toute algèbre de matrices à coefficients dans un anneau commutatif satisfait l’identité standard :

\sum _{\sigma \in S_{2n}}\varepsilon (\sigma )x_{\sigma (1)}x_{\sigma (2)}\cdots x_{\sigma (2n)}=0

où $S 2 n$ est le groupe symétrique d’ordre $2 n$ et la fonction $ε$ est la signature associée.

Bibliographie

Eli Aljadeff, Antonio Giambruno, Claudio Procesi, Amitai Regev, Rings with Polynomial Identities and Finite Dimensional Representations of Algebras, American Mathematical Society, Colloquium Publications volume 66, 2020.

Notes et références

↑ Amitsur-Levitzki theorem sur Encyclopedia of maths

Portail de l’algèbre

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