Category O, KLR, and Hecke Categorification Route: Examples and Counterexamples

The adjoint action of $h$ has distinct eigenvalues $2,-2,0$ on the basis $e,f,h$ [F9]; for any nonzero ideal $I$, Lagrange projection polynomials in $\mathrm{ad}h$ preserve $I$ and isolate a nonzero basis component. If $e\in I$, then $h=[e,f]\in I$ and $-2f=[h,f]\in I$; if $f\in I$, then $h=[e,f]\in I$ and $2e=[h,e]\in I$; if $h\in I$, then $2e=[h,e]\in I$ and $-2f=[h,f]\in I$. Thus $I={\mathfrak{s}\mathfrak{l}}_2$, which is nonabelian since $[e,f]=h\mathrm{\ne }0$, so it is simple [F10]. Its derived algebra equals itself by [F9], so it is not solvable [F13]; the solvable radical is an ideal [F11,F12], hence simplicity forces it to be zero, and [F11] makes ${\mathfrak{s}\mathfrak{l}}_2$ semisimple. The restricted module $M$ is one-dimensional and finite-dimensional, and $\mathfrak{h}=\mathbb{C}h$ acts by zero, so it is $\mathfrak{h}$-semisimple; [F6] puts it in $\mathcal{O}$ as defined by [F7].

For $d=0$, $M\otimes V^{\otimes 0}=M$ and $H_0=\mathbb{C}$, so the action is scalar [F8]. For $d=1$, [F2] and [F4] give $x_1={\mathrm{\Omega }}_{01}=0$ on the trivial factor, while $z_M=0$; the normalization [F3] at $N=2$ therefore gives $y_1=\frac{1}{2}{\mathrm{id}}_{M\otimes V}$. The action exists by [F5].

By [F8], $H_1=\mathbb{C}[x_1]$; the source-normalized route [F1] has $y_1=x_1+\frac{1}{2}$ by [F3]. Since [F2] and [F4] give $x_1=z_M=0$, evaluation at $y_1=\frac{1}{2}$ is the action, with kernel $(y_1-\frac{1}{2})$ by polynomial division. Thus the action factors through $\mathbb{C}[y_1]\mathrm{/}(y_1-\frac{1}{2})\cong \mathbb{C}$.

In $\mathbb{C}$, $\xi =-1$ has order two, so $e=2$ and $\xi \mathrm{\ne }1$ is the nondegenerate branch [F1,F2]. By [F2], the $q$ in the cyclotomic presentation [F3] is $\xi =-1$. Since $⟨h_0,{\mathrm{\Lambda }}_0⟩=1$ and the other level-one pairing is zero, the height-one cyclotomic relation is $X_1-1=0$. At $d=1$ the only affine generators are $X_1^{\pm 1}$ [F3], so both become $1$ in the quotient; the scalar map sending its unit to $1$ is inverse to the scalar inclusion, hence $H_1^{{\mathrm{\Lambda }}_0}\cong \mathbb{C}$. The residue values ${\xi }^0=1$ and ${\xi }^1=-1$ are distinct; thus [F7] gives sole projector $e(0)=1$ and $e_{{\alpha }_0}=1\mathrm{\ne }0$.

Since ${\alpha }_0$ has height one, its only sequence is $(0)$; [F4] gives the idempotent $e(0)=1$, the dot $x_1$, and no crossing. The sole dot commutes with itself, so the presented algebra is $\mathbb{C}[x_1]e(0)$. The standard first-strand relation [F5] is $x_{1}e(0)=0$, and evaluation at zero maps $e(0)$ to $1$, so the quotient is nonzero and equals $\mathbb{C}e(0)\cong \mathbb{C}$. Its unit has degree zero and its dot is zero [F6], so the quotient is concentrated in degree zero. Every nonzero simple over this field algebra is one-dimensional because any nonzero vector spans a nonzero submodule; a graded simple is concentrated in one degree because the span of a homogeneous vector is a graded submodule. Thus the degree-zero representative and its grading shifts are exactly the graded simples.

In this rank-one data, [F3] sets $X_1=1$ and the distinct spectral values $1,-1$ make [F7] give $e_{{\alpha }_0}=e(0)=1\mathrm{\ne }0$. Also [F4] and [F5] give the nonzero quotient $\mathbb{C}e(0)$ at ${\alpha }_0$. Thus the hypotheses of [F8] hold over $\mathbb{C}$ with $e=2$, and its block isomorphism identifies the two scalar algebras.

The place permutation satisfies $P(e_1\otimes e_1)=e_1\otimes e_1=v$ by [F4], and $v\mathrm{\ne }0$ because tensor products of basis vectors form a basis [F6]; hence $P^{2}v=v$.

The parameter $\xi =-1\mathrm{\ne }1$ has order two in $\mathbb{C}$, so [F2] puts it in the nondegenerate finite-$e$ branch with $e=2$. On the nonzero vector $v=e_1\otimes e_1$, [F4] gives $Pv=v$ and [F6] gives $v\mathrm{\ne }0$. The quadratic relation [F3] at $q=\xi =-1$ is $T_1^2=-2T_1-1$; assigning $T_1=P$ would require $v=-2v-v=-3v$, hence $4v=0$, a contradiction over $\mathbb{C}$. Thus the defining relation fails.

This only refutes the specified assignment: other parameter conversions or homomorphisms remain outside this calculation, and the cited degenerate action [F5] is still valid.