Nonzero C2 symbol of the actual admissible affine sl2 vacuum null state

The actual projected module before any localization. Use the negative affine Borel generated by $E=e(-1)$ and $F=f(0)$. Its action on the vacuum is the usual quotient by the left ideal generated by F. In the vacuum PBW basis, C2 is exactly the span of words containing a current x(-r) with $r\ge 2$. Such modes are (-2) modes of translates of x(-1)1. Commuting a preceding current past such a mode creates only modes of index at most -2, so every such word lies in C2. Conversely the normal-order mode expansion of a(-2)b, for homogeneous current words a,b, has loop energy one greater than the sum of their energies, whereas its number of current factors is no greater than the sum of their numbers of factors. Thus every resulting word has a mode of index at most -2. Commuting two (-1) currents produces only a (-2) current. The PBW basis therefore gives $R_{V^k}=C[e,h,f]$, without extra relations. E acts by multiplication by e. F acts by the finite adjoint derivation $-h{\mathrm{\partial }}_e+2f{\mathrm{\partial }}_h$. Both preserve the ideal(f), so the further quotient is the actual negative-Borel module C[e,h]. Write $z=-h$. Its actions are $E=e$ multiplication and $F=z{\mathrm{\partial }}_e$, with [F,E]=z and central z. Equivalently, let $W=\mathbb{C}⟨E,F,z⟩\mathrm{/}([F,E]-z,[z,E],[z,F])$; this is the Heisenberg enveloping algebra, and this actual projected vacuum module is $W\mathrm{/}WF=C[E,z]$. The affine negative-Borel map phi to W kills the modes of index at most -2 and f(-1); its images of E,F,[F,E] are E,F,z. This is an algebra/module quotient, not a map of fractional powers on a vector.

Which singular state and which maximal ideal. The actual state supplied by F1, Proof 4.1–5.1, is the polynomial negative-Borel element with middle integer $m=p-1$ and successive shifts $t=p\mathrm{/}q$. No polynomiality is inferred here from a total exponent. Its agreement with the extra maximal-ideal generator can also be read directly from the stated BGG interface used by that supplier. For $\lambda =k{\mathrm{\Lambda }}_0$, the positive integral rank-one root system has simple roots alpha and $\beta =-\alpha +q\delta$: their shifted pairings are respectively 1 and p-1. Indeed the pairings of alpha+n delta and -alpha+n delta are 1+nt and -1+nt, and integrality is equivalent to q dividing n. Both simple pairings are positive, so the supplied actual BGG complex applies. Its degree-one differential has precisely the two singular inclusions $M(s_{\alpha }\cdot \lambda )$ and $M(s_{\beta }\cdot \lambda )$ into M(lambda), and its H0 is L(lambda). Killing the first inclusion kills F1 and gives exactly the affine vacuum module by PBW. The second inclusion therefore generates the maximal affine vacuum submodule. Its image is nonzero: otherwise its highest generator, paired with a suitable vector in the first summand, would lie in $\ker d_1=\mathrm{im}{d}_2$; but every component of d2 into the second summand is a strict lower-weight singular inclusion and cannot contain that summand's highest line. Its weight is lambda-m beta, hence finite h-weight 2m and loop degree mq. The one-dimensional singular Hom space and the genuine polynomial Ore transport in F1 identify its line with A1, up to a nonzero scalar, not with a highest-top annihilation polynomial. There is no extra classification theorem in this identification.

Affine submodule versus vertex ideal in this use. Here $t=k+2$ is nonzero. With a B-dual basis $x_a,x^a$ of sl2, set $S_{-1}={\sum }_{a,n}:x_a(n)x^a(-1-n):$. Its action is locally finite on each vacuum PBW vector. The affine current commutators give $[S_{-1},x(r)]=-2(k+2)r$ x(r-1): the two central contributions give -2kr x(r-1), and commuting the remaining brackets into normal order gives -4r x(r-1), using ${\sum }_a[x_a,[x^a,x]]=4x$. The latter identity is checked on e,h,f with dual basis f,h/2,e. Consequently $S_{-1}\mathrm{/}(2(k+2))$ has the same current commutators as translation T and kills the vacuum. Their difference commutes with every current and kills the cyclic vacuum, so they coincide on $V^k$. Every affine submodule is therefore T-stable. The reconstruction of fields from currents expresses each field mode as a locally finite sum of products of current modes and translations, so an affine submodule is a vertex ideal here. Conversely a vertex ideal is current-mode stable. Thus the maximal submodule in step 2.1 is exactly the maximal vertex ideal used to define $L_k$. This binds the polynomial A to the actual ideal whose C2 image is used below.

Functoriality of the precise Ore transport. The quotient phi in step 1.1 is compatible with each transport, not merely with integral words. For $X=E$ or F and u in the enveloping algebra, its finite Theta identity is $\varphi ({\mathrm{\Theta }}_b^X(u))={\sum }_j\left(\genfrac{}{}{0px}{}{b}{j}\right)(\mathrm{ad}\varphi (X){)}^j(\varphi (u))\varphi (X{)}^{-j}$. This follows term by term from phi preserving products and brackets. The target E and F are nonzero Ore elements of W; one can work in its fraction skew field, obtained from C(z,E)[F; z ${\mathrm{\partial }}_E]$. The corresponding target automorphisms are explicitly E fixed, F maps to F-bz $E^{-1}$, and F fixed, E maps to E+bz $F^{-1}$. Their inverses have parameter -b. They extend over every inverse occurring in the finite transport diagram. In particular a transported denominator, being the image under these automorphisms of E or F, remains nonzero. The universal property of localization and these termwise identities give a commuting map at each node of the supplier's Ore-Theta diagram. Its middle integral singular embedding maps by its ordinary polynomial action. The final polynomial element A in that diagram must consequently map to the result of the same target transport calculation. This argument localizes algebras to compute a polynomial; it never inverts F on the vacuum, where $F1=0$.

Finite polynomial identity for the projected palindrome. Put $N=EF$. For any positive integer m, integer $Q\ge 1$ and scalar t, let $P_Q^E$ and $P_Q^F$ denote the alternating Ore-Theta palindromes with 2Q-1 factors, beginning and ending in E and F respectively, and exponents m+(Q-1)t,m+(Q-2)t,...,m-(Q-1)t. Their meaning is the same recursive transported expression as in step 3.2; $Q=1$ is the ordinary power. In W their exact values are $P_Q^E$ = $E^m$ ${\prod }_{j=1}^{Q-1}$ ${\prod }_{r=1}^m$ (N+(r-jt)z), $P_Q^F$ = $F^m$ ${\prod }_{j=1}^{Q-1}$ ${\prod }_{r=1}^m$ (N+(1-r+jt)z). Here every displayed product is finite, its factors commute, and both answers are integral polynomials. Prove both formulas simultaneously. For $Q=1$ they are tautologies. Write $a=m+(Q-1)t$ and $b=m-(Q-1)t$. The recursive E palindrome is $E^a$ $P_{Q-1}^F$ $E^b$. Substitute the F formula from the induction hypothesis. The identities $E^a$ $F^m=E^{a-m}$ ${\prod }_{s=0}^{m-1}(N-sz)$ and $P(N)E^b=E^b$ P(N+bz) follow respectively from [F,E]=z and the finite ${\mathrm{\Theta }}_E$ formula; they hold for the indicated transported powers without applying them to a vector. Therefore the E palindrome becomes $E^m$ ${\prod }_{s=0}^{m-1}(N+(b-s)z)$ ${\prod }_{j=1}^{Q-2}$ ${\prod }_{r=1}^m$ (N+(1-r+jt+b)z). The first product has $b-s=r^{\mathrm{\prime }}-(Q-1)t$ with r'=m-s in 1,...,m. In the second put r'=m+1-r and j'=Q-1-j; its coefficient is r'-j't, with j'=1,...,Q-2. These are exactly all factors of the asserted E formula. Interchanging E,F and replacing z by -z, so $FE=N+z$, proves the F formula from the same calculation. This completes the simultaneous induction with every reindexing specified.

Actual vacuum specialization and nonzero coefficient. Set $Q=q$ and $m=p-1$. The supplier's exponents m+(q-1)t,...,m-(q-1)t are exactly the palindrome A of F1. By step 3.2, step 4.1 computes phi(A), not a different rational-power representation. Now, and only now, act with its final integral polynomial on the genuine projected vacuum 1 in W/WF. Since $N1=0$, it gives $\pi (vbar)=C_N$ $z^{m(q-1)}$ $e^m$ ${\prod }_{j=1}^{q-1}$ ${\prod }_{r=1}^m$ (r-jp/q), where $C_N$ is the nonzero scalar expressing the chosen normalization of the actual singular line relative to A1. No factor vanishes: $r=jp\mathrm{/}q$ would imply q divides j, which is impossible for $1\le j<q$. For $q=1$ the product is empty, $A=E^m$ and its value is $e^m$. Coprimality makes $B=m(q-1)$ even, so $z^B=(-h{)}^B=h^B$. Thus $C0={\prod }_{j=1}^{q-1}$ ${\prod }_{r=1}^{p-1}(r-jp\mathrm{/}q)$ is an explicitly given nonzero rational number, and $\pi (vbar)=C_N$ C0 $e^m$ $h^B$. This proves the required nonzero C2 symbol of the actual maximal-ideal generator directly. No epsilon limit, fractional action on an annihilated vacuum, P1/top inference or special-function theorem enters.

Recover the entire highest C2 symbol. The actual v is killed by e(0), has h(0)-weight2m, and has loop degree $d=mq$. Its image in C[e,h,f] is therefore a finite-adjoint highest polynomial of weight2m and ordinary polynomial degree d. The last assertion holds because every surviving current is a (−1) current of loop degree1. For completeness the kernel of $D_e$ on C[e,h,f] is C[e,Omega], where $D_e=-2e{\mathrm{\partial }}_h+h{\mathrm{\partial }}_f$, $\mathrm{\Omega }=h^2\mathrm{/}2+2ef$. Localize at e and use coordinates e, $u=h\mathrm{/}(2e)$, Omega; then $D_e(u)=-1$ and $D_e$ kills e,Omega. Hence the localized kernel is $C[e,e^{-1},\mathrm{\Omega }]$. Its intersection with C[e,h,f] is C[e,Omega]: if a term has a negative least e power, its coefficient's highest h-degree after setting $f=0$ cannot cancel with any different Omega-degree term. Thus no negative e power can occur in a polynomial in the original variables. Since Omega has h-weight0 and e has weight2, weight2m forces $vbar=e^m$ P(Omega). Homogeneity forces P to be a scalar times ${\mathrm{\Omega }}^a$ with $a=(d-m)\mathrm{/}2=m(q-1)\mathrm{/}2$. Its scalar is nonzero by the projected computation. More precisely comparison at $f=0$ gives $C=2^a$ C0, up to the supplied normalization. This proves the full asserted symbol.

Explicit Casimir nilpotence. Let I be the C2 image of the actual maximal vacuum ideal. It is stable under $D_e,D_f$ and contains $e^m$ ${\mathrm{\Omega }}^a$. Write $v_r=D_f^r(e^m)$, $0\le r\le 2m$, $Q={\sum }_{r=0}^{2m}$ $(-1{)}^r$ $v_r$ $v_{2m-r}$. The ordinary finite sl2 commutator identity gives D_e v_r=r(2m−r+1)v_(r−1) and D_f v_r=v_(r+1), with $v_{2m+1}=0$. In $D_e$ Q the two coefficients of $v_s$ $v_{2m-1-s}$ cancel: both have magnitude(s+1)(2m−s) and opposite signs. In $D_f$ Q adjacent summands cancel likewise. Its h-weight is0. The invariant-polynomial calculation above therefore forces $Q=c$ ${\mathrm{\Omega }}^m$, since its degree is2m. At $e=f=0$ only the $r=m$ summand survives. Indeed weight or direct differentiation shows $v_r$ restricts to0 there unless $r=m$, and $v_m$ restricts to $(-1{)}^m$ m! $h^m$. Hence $Q{\mathrm{\mid }}_{e=f=0}=(-1{)}^m(m!{)}^2$ $h^{2m}$, $Q=(-1{)}^m$ $2^m$ $(m!{)}^2$ ${\mathrm{\Omega }}^m$ $\mathrm{\ne }0$. Every ${\mathrm{\Omega }}^a$ $v_r$ lies in I by its adjoint stability. Multiplying each by $v_{2m-r}$ and adding yields ${\mathrm{\Omega }}^a$ Q in I. Consequently ${\mathrm{\Omega }}^{a+m}=0$ in $R_{L_k(sl2)}$, $a+m=(p-1)(q+1)\mathrm{/}2$. This is an actual explicit nilpotence bound, obtained algebraically without a Nullstellensatz input. In particular every complex point of the associated variety satisfies $\mathrm{\Omega }=0$, the sl2 nilpotent-cone equation.