Dears,
I have question related to first Weyl Algebra $A_1$ (over a field $F$). Firstly, In some sources it is said to be an algebra of differential operators $d/dx$, which can be multiplied by polynomials of $x$ with coefficients in $F$.
Secondly, in other places it is written as free algebra generated by $x,y$ and divided by ideal generated by $(yx-xy-1)$, so commutator $\left[y,x\right]=1$;
I understand it is exactly the same, just instead of writing $d/dx$ we write $y$.
Thirdly, it is written like here on Page 4th http://www-personal.umich.edu/~equinlan/Rings%20of%20Differential%20Operators.pdf
We have generators $\{x,y\}$ and operator $d/dx$; and $\left[d/dx,x\right]=1$.
It is also then denoted $F\left[x\right]\left[y,d/dx\right]$.
Is it the same Weyl Algebra? What is $y$ here, how elements look like, how it all works? Where I can read more about that 3rd form, which seems to be the least common?
For an arbitrary field $F$ the correct definition of the first Weyl algebra is the second one: it is the quotient of the free $F$-algebra $F \langle x,y \rangle$ on two generators by the relation $yx-xy=1$. This algebra has a representation on $F[x]$ in which $x$ acts by multiplication and $y$ acts by differentiation, $$y \cdot f=\frac{d f}{dx}.$$ However, if the characteristic of $F$ is $p>0$ then this representation is not faithful: the element $y^p$ acts as zero. Thus the first and second definitions do not agree in this case.
The reason for preferring this definition is precisely that it is independent of the field $F$, and actually works integrally: in fact if you define the integral Weyl algebra to be the free $\mathbf{Z}$-algebra (i.e., ring) $\mathbf{Z}\langle x,y \rangle$ modulo the relation $yx-xy=1$, then tensoring by $F$ gives the previous definition, for any $F$, so we may think of the Weyl algebra as a flat family of algebras that varies continuously with the choice of field (equivalently, the analog of the PBW theorem in this context is the fact that the monomials $x^m y^n$ are a $\mathbf{Z}$-basis of the integral Weyl algebra, and hence an $F$-basis of the version over $F$).