The group of automorphisms of the first Weyl algebra in prime characteristic and the restriction map
نویسنده
چکیده
Let K be a perfect field of characteristic p > 0, A1 := K〈x, ∂ | ∂x−x∂ = 1〉 be the first Weyl algebra and Z := K[X := x, Y := ∂] be its centre. It is proved that (i) the restriction map res : AutK(A1) → AutK(Z), σ 7→ σ|Z , is a monomorphism with im(res) = Γ := {τ ∈ AutK(Z) | J (τ) = 1} where J (τ) is the Jacobian of τ (note that AutK(Z) = K ∗ ⋉ Γ and if K is not perfect then im(res) 6= Γ); (ii) the bijection res : AutK(A1) → Γ is a monomorphism of infinite dimensional algebraic groups which is not an isomorphism (even if K is algebraically closed); (iii) an explicit formula for res−1 is found via differential operators D(Z) on Z and negative powers of the Fronenius map F . Proofs are based on the following (non-obvious) equality proved in the paper: ( d dx + f) = ( d dx ) + dp−1f dxp−1 + f, f ∈ K[x].
منابع مشابه
OD-characterization of $S_4(4)$ and its group of automorphisms
Let $G$ be a finite group and $pi(G)$ be the set of all prime divisors of $|G|$. The prime graph of $G$ is a simple graph $Gamma(G)$ with vertex set $pi(G)$ and two distinct vertices $p$ and $q$ in $pi(G)$ are adjacent by an edge if an only if $G$ has an element of order $pq$. In this case, we write $psim q$. Let $|G= p_1^{alpha_1}cdot p_2^{alpha_2}cdots p_k^{alpha_k}$, where $p_1
متن کاملSome commutativity theorems for $*$-prime rings with $(sigma,tau)$-derivation
Let $R$ be a $*$-prime ring with center $Z(R)$, $d$ a non-zero $(sigma,tau)$-derivation of $R$ with associated automorphisms $sigma$ and $tau$ of $R$, such that $sigma$, $tau$ and $d$ commute with $'*'$. Suppose that $U$ is an ideal of $R$ such that $U^*=U$, and $C_{sigma,tau}={cin R~|~csigma(x)=tau(x)c~mbox{for~all}~xin R}.$ In the present paper, it is shown that if charac...
متن کاملThe inversion formulae for automorphisms of polynomial algebras and differential operators in prime characteristic
Let K be an arbitrary field of characteristic p > 0, let A be one of the following algebras: Pn := K[x1, . . . , xn] is a polynomial algebra, D(Pn) is the ring of differential operators on Pn, D(Pn) ⊗ Pm, the n’th Weyl algebra An, the n’th Weyl algebra An ⊗ Pm with polynomial coefficients Pm, the power series algebra K[[x1, . . . , xn]], Tk1,...,kn is the subalgebra of D(Pn) generated by Pn and...
متن کاملAutomorphism Group of a Possible 2-(121, 16, 2) Symmetric Design
Let D be a symmetric 2-(121, 16, 2) design with the automorphism group of Aut(D). In this paper the order of automorphism of prime order of Aut(D) is studied, and some results are obtained about the number of fixed points of these automorphisms. Also we will show that |Aut(D)|=2p 3q 5r 7s 11t 13u, where p, q, r, s, t and u are non-negative integers such that r, s, t, u ? 1. In addition we prese...
متن کامل. R A ] 8 D ec 2 00 5 Automorphisms of the Weyl algebra Alexei Belov - Kanel and Maxim Kontsevich February 2 , 2008
We discuss a conjecture which says that the automorphism group of the Weyl algebra in characteristic zero is canonically isomorphic to the automorphism group of the corresponding Poisson algebra of classical polynomial symbols. Several arguments in favor of this conjecture are presented, all based on the consideration of the reduction of the Weyl algebra to positive characteristic.
متن کامل