The discriminant controls automorphism groups of noncommutative algebras. (English) Zbl 1337.16032

Let \(A\) be an associative filtered algebra over a field \(k\) such that \(\text{gr\,}A\) is a domain, and \(A\) is affine over its center \(R\). Under some assumption on discriminant of the extension \(A/R\) it is shown that \(A_1\) is stable under every automorphism of \(A\). Any automorphism of the polynomial algebra \(A[t]\) has the form \(h(t)=ct+r\), \(c\in k^*\), \(r\in R\), and \(g(A)=A\). If the base field \(k\) has characteristic zero, then every locally nilpotent derivation of \(A\) is zero and the automorphism group is an extension of \((k^*)\) by a finite group.
In particular let \(W_n\) be the algebra generated by \(x_1,\ldots,x_n\) with defining relations \(x_ix_j+x_jx_i=1\) for all \(i\neq j\). Then the automorphism group of \(W_n\) is a direct product of the symmetric group \(S_n\) and the group \(\{\pm 1\}\).


16W20 Automorphisms and endomorphisms
16W70 Filtered associative rings; filtrational and graded techniques
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