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 $$\{\pm 1\}$$.

MSC:

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

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