Heisenberg Lie Algebras and Algebraic Weyl Algebra: Examples and Counterexamples

Define $D(f)=f^{\mathrm{\prime }}$ by $D(t^m)=m{t}^{m-1}$ and extend $k$-linearly. For each $m\ge 0$, $D(t^p{t}^m)=(p+m)t^{p+m-1}=m{t}^{p+m-1}=t^{p}D(t^m)$ because $p=0$ in $k$. By [F4], every polynomial is a finite linear combination of monomials, so linearity gives $D(t^{p}f)=t^{p}D(f)$ for every $f$; hence $D$ preserves $(t^p)$. Multiplication $X[f]=[tf]$ also descends because $t(t^{p}f)\in (t^p)$.

For each monomial $t^m$, $D(t{t}^m)-tD(t^m)=(m+1)t^m-m{t}^m=t^m$. By [F4] and linearity the identity holds for every polynomial, hence on $M$ it gives $DX-XD=I_M$.

Pointwise addition and scalar multiplication and composition make ${\mathrm{End}}_k(M)$ a unital associative $k$-algebra: composition is associative, $I_M$ is its unit, and $k$-bilinearity follows by evaluating on vectors. Send the generators $x_1,p_1$ to $X,D$. By [F5] this linear assignment extends to a unital algebra map from $T(E_1)$. Step 2.1 and $[X,X]=[D,D]=0$ show that each defining relator in [F1] maps to zero. The map's kernel is a two-sided ideal because it is an algebra homomorphism, so it contains their generated ideal by [F2]. By [F6] the map factors through the quotient $A_1(k)$; write $\rho$ for this factor map, which is $k$-linear because the quotient map is surjective. Its evaluation $a\cdot m:=\rho (a)m$ is a unital left module action by [F7].

By [F4], every polynomial has finitely many monomials; subtracting suitable multiples of $t^p$ removes every term whose exponent is at least $p$, so each coset is represented by a linear combination of $1,t,\dots ,t^{p-1}$. A polynomial multiple of $t^p$ has only monomials with exponent at least $p$, so no nonzero linear combination of those $p$ lower monomials lies in $(t^p)$. Hence $1,[t],\dots ,[t^{p-1}]$ is a basis of $M$, and $M$ is nonzero of dimension $p$. This is the positive-characteristic exception to the characteristic-zero obstruction.

On a monomial $t^{\gamma }=t_1^{{\gamma }_1}\cdots t_n^{{\gamma }_n}$ define $X_i(t^{\gamma })=t^{\gamma +e_i}$, and define $D_i(t^{\gamma })={\gamma }_i{t}^{\gamma -e_i}$ when ${\gamma }_i>0$ and $0$ when ${\gamma }_i=0$; extend both maps $k$-linearly using [F1].

The $X_i$ commute because multiplication in the polynomial ring commutes; for distinct $i,j$, the $D_i$ commute because applying the two exponent-lowering maps multiplies by ${\gamma }_i{\gamma }_j$ in either order, and $D_i$ commutes with itself. For each $i,j$, applying $D_i{X}_j-X_j{D}_i$ to $t^{\gamma }$ gives ${\delta }_{ij}{t}^{\gamma }$: when $i=j$ the coefficient difference is $({\gamma }_i+1)-{\gamma }_i=1$, and when $i\mathrm{\ne }j$ the two terms agree. Hence the generator assignments satisfy [F2].

Pointwise addition and scalar multiplication and composition make ${\mathrm{End}}_k(k[t_1,\dots ,t_n])$ a unital associative $k$-algebra: composition is associative, the identity map is its unit, and $k$-bilinearity follows by evaluating on polynomials. Send each generator $x_i,p_i$ to $X_i,D_i$. By [F4] this linear assignment extends to a unital algebra map from $T(E_n)$. Step 2.1 shows that it kills every defining relator in [F2]. Its kernel is a two-sided ideal because it is an algebra homomorphism, so it contains their generated ideal by [F7]. By [F5] it factors through the quotient $A_n(k)$; write $\rho$ for this factor, which is $k$-linear because the quotient map is surjective. Evaluation $a\cdot f:=\rho (a)f$ gives a unital left module by [F6]. If $\mathrm{char}(k)=p>0$, then for every monomial, $D_i^p(t^{\gamma })=0$: it vanishes when ${\gamma }_i<p$, and otherwise its coefficient is a product of $p$ consecutive integers, hence divisible by $p$. Thus $D_i^p=0$.

By [F3], $p_i^p$ is a nonzero normal-ordered basis monomial in $A_n(k)$, while step 3.1 shows that the representation sends it to zero. Therefore the representation is nonfaithful in positive characteristic.