Ordered monomial PBW basis for the shifted Yangian

RTT spanning and the actual coproduct. In the unshifted RTT algebra the commutator of modes $r,s$ has canonical degree at most $r+s-1$, so induction on total degree and the number of unordered pairs rewrites every word into ordered monomials. Matrix-unit brackets verify RTT for $L_a(u)=I+E^{(a)}\mathrm{/}u$ in the $a$th factor of $U({\mathfrak{g}\mathfrak{l}}_n{)}^{\otimes l}$. Multiplication of RTT matrices with commuting coefficient algebras preserves RTT: moving the two middle commuting tensor factors past each other allows the first RTT identity, then the second. Thus the iterated evaluation map is $T(u)\mapsto L_1(u)\cdots L_l(u)$. More generally shifted evaluation points give the ordered product ${\prod }_{a=1}^l(I+E^{(a)}\mathrm{/}(u-c_a))$, in increasing factor order. Its $u^{-r}$ coefficient when all $c_a=0$ is ${\sum }_{a_1<\cdots <a_r}{E}^{(a_1)}\cdots E^{(a_r)}$, homogeneous of enveloping PBW degree $r$.

We first prove invariance of the actual path generators for both column arrangements, independently of generation. For a box a write c(a) for its column, ${\rho }_c=n-{\sum }_{d\ge c}{q}_d$ and ${\tilde{E}}_{ab}=(-1{)}^{c(b)-c(a)}(E_{ab}+{\delta }_{ab}{\rho }_{c(a)})$. The negative PBW projection $\mathrm{pr}:U(\mathfrak{g})=U(\mathfrak{p})U(\mathfrak{m})\to U(\mathfrak{p})$ applies χ to the right factor. Its elementary rules are $\mathrm{pr}(ZB)=\chi (Z)B+\mathrm{pr}([Z,B])$ and $\mathrm{pr}(BZ)=B\chi (Z)$ for a negative matrix unit Z and a p-factor B. The bracket is $[{\tilde{E}}_{ab},{\tilde{E}}_{hk}]={\delta }_{bh}({\tilde{E}}_{ak}-{\delta }_{ak}{\rho }_{c(a)})-{\delta }_{ak}({\tilde{E}}_{hb}-{\delta }_{hb}{\rho }_{c(b)})$, and $\chi ({\tilde{E}}_{ab})=-{\delta }_{\mathrm{row}(a),\mathrm{row}(b)}$ at adjacent negative columns, zero at greater distance. Define $P_{ij;x}^{(r)}$ as the sum of products $E_{a_1{b}_1}\cdots E_{a_s{b}_s}$ with total degree $\sum (c(b_t)-c(a_t)+1)=r$, p-factors, shared intervening rows, starting/ending rows i,j, strict forward joins when the intervening row exceeds x and weak reverse joins otherwise, and sign $(-1{)}^{r-s}$ times one minus for every intervening row at most x. Its ${\eta }_{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}$ image replaces each factor by $E_{ab}+{\delta }_{ab}{\rho }_{c(a)}$. Equivalently, in the $\tilde{E}$ factors its only external sign is one minus for every intervening row at most x, since their product already contributes $(-1{)}^{r-s}$. Invariance means $\mathrm{pr}([z,{\eta }_{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}(P)])=0$. If an element B is invariant, then $(z-\chi (z))B=B(z-\chi (z))+[z,B]$ proves $U(\mathfrak{g})I_{\chi }B\subseteq U(\mathfrak{g})I_{\chi }$; hence invariant elements form a subalgebra, and products, commutators and coefficients of inverse series with identity constant are invariant.

The local Gan–Ginzburg supplier is stated for the original left-justified diagram; we derive the reflected version by an actual graded Lie isomorphism. Let f reflect boxes, $\tau (E_{ab})=E_{f(b),f(a)}^{\mathrm{\prime }}$ reverse products, and let S on the reflected space be diagonal with $S_{bb}=(-1{)}^{{\mathrm{col}}^{\mathrm{\prime }}(b)}$. The linear map $\varphi (x)=S(-\tau (x))S^{-1}$ is a Lie isomorphism, because $[\tau (x),\tau (y)]=-\tau ([x,y])$. It preserves grading degrees, p and m, and the trace pairing. The diagonal-conjugation sign on every positive adjacent arrow is −1, so $\varphi (e)=e^{\mathrm{\prime }}$ and ${\chi }^{\mathrm{\prime }}\varphi =\chi$. It extends to an ALGEBRA isomorphism of enveloping algebras and takes $U(\mathfrak{g})I_{\chi }$ to $U({\mathfrak{g}}^{\mathrm{\prime }})I_{{\chi }^{\mathrm{\prime }}}$. Thus it takes the entire untwisted left reduction $A={\eta }_{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}(W)$ to $A^{\mathrm{\prime }}$: each defining commutator condition is equivalent under φ. The chart map is the explicit algebra isomorphism $\beta ={\eta }_{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}^{\mathrm{\prime }-1}\varphi {\eta }_{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}:W\to W^{\mathrm{\prime }}$. Scalar translations have smaller positive Kazhdan degree, so $\mathrm{gr}\beta =\mathrm{gr}\varphi$. The doubled good grading, sl2 triple, trace-orthogonal slice and transverse product transport under φ. Consequently the proved original polynomial-slice and cohomological statements imply the reflected statements in its actual grading, including the homogeneous coordinate weights. The polynomial-slice part used here is choice-free; for consumers assuming AC as in the cohomological supplier, its left Skryabin quotient, local-nilpotence category, invariant space and equivalence maps also transport first through φ and then through β. This uses the original proved local supplier without enlarging its stated domain, and does not assert that a transpose subalgebra is the whole reflected reduction or that two chart twists act equally on a fixed module.

Index and inverse sign in the abstract quotient. Write $A_i(u)=D_i(u{)}^{-1}={\sum }_{t\ge 0}{A}_i^{(t)}{u}^{-t}$, so $A_i^{(0)}=1$, while the defining relation uses ${\tilde{D}}_i(u)=-A_i(u)$ with constant $-1$. Thus $[E_i^{(a)},F_i^{(b)}]=-D_{i+1}^{(a+b-1)}-{\sum }_{t=1}^{a+b-1}{A}_i^{(t)}{D}_{i+1}^{(a+b-1-t)}$. Put $i=c-1$, $u=s_{c-1,c}$, $v=s_{c,c-1}$, $a=r-v$, $b=v+1$. Then $a+b-1=r$ and $D_c^{(r)}=-{\sum }_{t=1}^r{A}_{c-1}^{(t)}{D}_c^{(r-t)}-[E_{c-1}^{(r-v)},F_{c-1}^{(v+1)}]$. We use this only for $r>p_c=p_{c-1}+u+v$, so $a>p_{c-1}+u$ is an allowed but HIGH E mode, to be eliminated before the diagonal formula is used. Also $A_i^{(r)}=-{\sum }_{t=1}^r{D}_i^{(t)}{A}_i^{(r-t)}$, giving polynomial inverse coefficients with the same canonical degree bound. This derives the sign and indices, rather than treating a negative inverse as a positive one.

Product Jacobian and independence. Let $l$ be at least the largest coefficient index in a proposed relation. In the commutative symbol algebra of $U({\mathfrak{g}\mathfrak{l}}_n{)}^{\otimes l}$ write $P_r(X_1,\dots ,X_l)={\sum }_{a_1<\cdots <a_r}{X}_{a_1}\cdots X_{a_r}$. At $X_a=h_{a}I$, with nonzero pairwise distinct $h_a$, the differential on every matrix entry is $d{P}_r={\sum }_a{e}_{r-1}(h_1,\dots ,\widehat{h_a},\dots ,h_l)d{X}_a$. The column coefficient polynomials ${\prod }_{b\mathrm{\ne }a}(1+h_{b}z)$ are linearly independent, since evaluating at $z=-1\mathrm{/}{h}_c$ leaves only column $c$, with a nonzero value. Hence the Jacobian is invertible. A polynomial relation of minimal total degree, differentiated and multiplied by the inverse Jacobian over the rational function field, makes each of its partial derivatives a lower-degree relation; minimality and characteristic zero force the relation to be constant and zero. Thus the coordinates $P_{r,ij}$, $1\le r\le l$, are algebraically independent. A highest weighted-degree part of an enveloping relation among their ordered lifts would contradict this polynomial independence by [F2]. This proves RTT PBW for all modes. The $h_a$ are scalar-matrix points in the symbol space, independently of spectral evaluation parameters.

Group equal row lengths into the minimal admissible parabolic blocks ν, with final nonempty block m of size t. D/E/F below are its actual Gauss path coefficients. Deleting the last column embeds the smaller shifted matrix units into the current ones; on unshifted diagonal units this compensates the rho difference by a scalar translation, and on negative units it preserves χ. A dot denotes the smaller coefficients. With final column A, preceding column B, $L_{kj}={\tilde{E}}_{(A,k),(A,j)}$ and $C_{hj}={\tilde{E}}_{(B,h),(A,j)}$, the right end recursions, for any fixed bottom row h, are $D_{a;ij}^{(r)}={\dot{D}}_{a;ij}^{(r)}+{\delta }_{a,m}({\sum }_k{\dot{D}}_{a;ik}^{(r-1)}{L}_{kj}+[{\dot{D}}_{a;ih}^{(r-1)},C_{hj}])$, $E_{a;ij}^{(r)}={\dot{E}}_{a;ij}^{(r)}+{\delta }_{a,m-1}({\sum }_k{\dot{E}}_{a;ik}^{(r-1)}{L}_{kj}+[{\dot{E}}_{a;ih}^{(r-1)},C_{hj}])$, and $F_{a;ij}^{(r)}={\dot{F}}_{a;ij}^{(r)}$. The E boundary formula is used for r>1. Deleting the FIRST column of the reflected arrangement has the ordinary embedding on the remaining shifted factors. With first column A, next B, $L_{ik}={\tilde{E}}_{(A,i),(A,k)}$ and $C_{ih}={\tilde{E}}_{(A,i),(B,h)}$, its LEFT recursions are $D_{a;ij}^{(r)}={\dot{D}}_{a;ij}^{(r)}+{\delta }_{a,m}({\sum }_k{L}_{ik}{\dot{D}}_{a;kj}^{(r-1)}+[C_{ih},{\dot{D}}_{a;hj}^{(r-1)}])$, $E_{a;ij}^{(r)}={\dot{E}}_{a;ij}^{(r)}$, and $F_{a;ij}^{(r)}={\dot{F}}_{a;ij}^{(r)}+{\delta }_{a,m-1}({\sum }_k{L}_{ik}{\dot{F}}_{a;kj}^{(r-1)}+[C_{ih},{\dot{F}}_{a;hj}^{(r-1)}])$, with r>1 in the F boundary formula. To verify these identities, cut each path at its final or initial visit to the added column. Paths missing that column give the dotted coefficient; paths ending or starting in its block give the product sum. For a crossing path the two neighboring cut positions at the common bottom row h have opposite signs and give the two orders in the commutator; further visits pair by moving to the neighboring cut. The identical cut rule in a changed row interval proves the Gauss identities: paths without an intermediate visit occur once; paths with such a visit have the two neighboring cuts with opposite sign. Thus these are identities for the actual Gauss coefficients. In particular LEFT deletion corrects F and leaves E unchanged; it does not use the interchanged E/F display in the source.

Finite spanning: earlier blocks and boundary E/F. In the abstract quotient by $D_1^{(r)}$, $r>p_1$, let ${\mathrm{\Omega }}_c$ contain D modes $1\le a\le c$, $1\le r\le p_a$, E roots $a<b\le c$, $s_{ab}<r\le s_{ab}+p_a$, and F roots $a<b\le c$, $s_{ba}<r\le s_{ba}+p_a$; let $B_c$ be their algebra. Induct on c to show every allowed mode with endpoints at most c belongs to $B_c$ with total canonical degree at most its index. The case c=1 is the imposed truncation, including $p_1=0$. For the inductive step put $u=s_{c-1,c}$ and $v=s_{c,c-1}$. For $r>u+p_{c-1}$, the D–E relation gives $E_{c-1}^{(r)}=[D_{c-1}^{(r-u)},E_{c-1}^{(u+1)}]-{\sum }_{t=1}^{r-u-1}{D}_{c-1}^{(t)}{E}_{c-1}^{(r-t)}$. Replace every earlier D by its block-inductive finite expression and expand its commutator by Leibniz. The bracket of an earlier finite diagonal mode of index a with the lowest boundary E gives terms $D^{(t)}{E}_{c-1}^{(a+u-t)}$ with $a\le p_{c-1}$ in the only nonzero diagonal block, hence new E indices at most $u+p_{c-1}$. An earlier finite root $E_{b,c-1}^{(a)}$ gives $E_{b,c}^{(a+u)}$ and any block-refinement split terms, all with $a+u\le s_{b,c-1}+p_b+u=s_{b,c}+p_b$. Earlier negative roots and earlier positive roots without endpoint $c-1$ commute with the lowest boundary E. Thus the expanded commutator is in $B_c$. In the sum $u+1\le r-t<r$, so induction on r reduces those E modes; all products have degree at most r. The reversed D–F relation and reversed product orders give the same argument with boundary bound $v+p_{c-1}$.

Loop symbols and shifted PBW. Give an RTT mode of index $r$ loop degree $r-1$. In its commutator relation the term with a zero-index factor is the current matrix-unit bracket; every term with two positive-index factors has smaller loop degree. Thus the straightening in step 1.1 preserves loop degree, and the RTT basis of step 2.1 identifies each loop-filtered space with the span of its ordered monomials of bounded loop degree. Their symbols are therefore a basis, and their brackets are $[e_{ij}{t}^a,e_{hk}{t}^b]={\delta }_{jh}{e}_{ik}{t}^{a+b}-{\delta }_{ki}{e}_{hj}{t}^{a+b}$. This identifies the loop graded algebra with $U({\mathfrak{g}\mathfrak{l}}_n[t])$. The required current-algebra PBW uses only [F2]: any finite relation among ordered current monomials maps to a jet quotient ${\mathfrak{g}\mathfrak{l}}_n[t]\mathrm{/}{t}^M{\mathfrak{g}\mathfrak{l}}_n[t]$ with $M$ larger than every exponent in it, where the relevant modes are distinct basis vectors and their ordered monomials are independent. Coefficientwise Gauss inversion changes an RTT mode of index $r$ only by products of at least two positive-index modes with total index $r$, whose loop degree is at most $r-2$. Hence the unshifted Gauss modes have these same current symbols. Sending each simple shifted Drinfeld generator to its unshifted namesake preserves the defining relations and is loop-filtered. For higher roots use their actual shifted commutator recursions: the upper recursion brackets $e_{i,j-1}{t}^{r-s_{j-1,j}-1}$ with $e_{j-1,j}{t}^{s_{j-1,j}}$, giving $e_{ij}{t}^{r-1}$; the reversed lower recursion gives $e_{ji}{t}^{r-1}$ in the same way. Thus the images of all allowed shifted root modes have the distinct current symbols of the unshifted loop algebra just established. Their ordered products are independent there by the jet-PBW argument, so a relation among their lifts, taken in its highest loop degree, is impossible. The shifted coefficient relations, adjacent-root recursions and Serre relations straighten words into these ordered root monomials, with commutator corrections of total canonical index at most $r+s-1$ for modes of indices $r,s$; induction on canonical degree and unordered pairs terminates the straightening. It preserves the canonical bound and the loop bound. Independence now proves both the embedding and the filtered ordered basis. Consequently the intrinsic canonical associated graded is the polynomial algebra on the allowed shifted root symbols. This does not identify those canonical symbols with the unshifted Gauss symbols: for $s_{12}=0,s_{23}=1$, the adjacent relation gives $E_{13}^{(2)}(\sigma )=E_{13}^{(2)}(0)-E_{12}^{(1)}{E}_{23}^{(1)}$, whose correction has canonical degree two but loop degree zero. The independence argument uses loop degree one in this example. No independence after truncation is inferred here.

Induct on the number of columns, deleting the last column in the original left-justified diagram and the first in the reflected diagram. A one-column diagram has no negative algebra, so every mode is invariant. Empty rows have only formal constants and zero positive coefficients. All earlier D blocks, all nonboundary E/F blocks, and F in right deletion or E in left deletion are unchanged by step 2.2. Their path monomials have no forbidden preceding-bottom target in right deletion, or following-bottom source in left deletion; commuting a new negative into such a monomial produces a negative unit of greater column distance with zero χ, and moving it to the right keeps this absent endpoint. Thus their projected brackets vanish. Induction supplies old-negative invariance of every dotted coefficient. The old negative algebra commutes with L and C in step 2.2, so its character ideal is preserved by those factors, proving old-negative invariance of the changed coefficients. The complete negative Lie algebra is generated by the old one and the new adjacent-column matrices: brackets through a row present in intervening bottom rows produce each longer negative matrix. Jacobi and the normalizer identity in step 1.2 propagate invariance from these generators. Only the new adjacent matrices need be checked below.

Higher roots and the final diagonal elimination. For $b<c-1$, the defining root recursions are $E_{b,c}^{(r)}=[E_{b,c-1}^{(r-u)},E_{c-1}^{(u+1)}]$ and $F_{c,b}^{(r)}=[F_{c-1}^{(v+1)},F_{c-1,b}^{(r-v)}]$. Reduce the earlier endpoint factors by the c−1 hypothesis; the same Leibniz bounds in step 2.3 place every new root in $B_c$, including high modes, using the finite allowed new roots in ${\mathrm{\Omega }}_c$. Now use step 1.4 for $D_c^{(r)}$, $r>p_c$. Every inverse earlier coefficient is in $B_{c-1}$ and every $D_c^{(r-t)}$ has smaller index, so induction on r treats its sum. First replace the upper $E_{c-1}^{(r-v)}$ by the finite expression just constructed, then expand the bracket with the lowest boundary F. Earlier D/F factors give bounded new F roots and earlier factors; earlier E factors commute with that boundary F. A finite new E root starting at c−1 has index $a\le u+p_{c-1}$, so its E–F convolution has only $D_c$ indices at most $a+v\le p_c$. A new E root starting at b<c−1 gives earlier-endpoint roots and earlier factors, reduced by the block hypothesis, and cannot create a high $D_c$ mode. Thus this commutator uses only $D_c$ indices at most $p_c$, and the induction closes with the asserted degree control. In particular higher diagonal modes are polynomial expressions, not zero modes.

The changed D^(1) is a sum of the row-pair diagonal-column blocks; its two adjacent-column projected commutators cancel. Let the neighboring column have height u≥t, with b=u−t shorter rows and cutoff x=n−t. In RIGHT deletion put $z={\tilde{E}}_{(A,f),(B,g)}$, f bottom and g arbitrary. In D^(2), and in boundary E^(2) when the shift is1, the single bridge contributes scalar $-u{\delta }_{gi}{\delta }_{fj}$. The bottom plus paths ${\sum }_k{B}_{ik}{A}_{kj}$ contribute $+t{\delta }_{gi}{\delta }_{fj}$ and matrix terms $-{\delta }_{gi}{A}_{fj}+{\delta }_{fj}{B}_{ig}$ for g bottom. The shorter minus paths in B contribute $+b{\delta }_{gi}{\delta }_{fj}$ and $+{\delta }_{fj}{B}_{ig}$ for g shorter. Moving their negative factor through the following block yields one additional χ correction for each shorter row; this is the scalar b term. The single bridge gives the negative of all the matrix terms, and $-u+t+b=0$. In LEFT deletion put $z={\tilde{E}}_{(B,f),(A,g)}$, f arbitrary and g bottom. The bridge scalar is $-t{\delta }_{gi}{\delta }_{fj}$; bottom plus paths ${\sum }_k{A}_{ik}{B}_{kj}$ give $+t{\delta }_{gi}{\delta }_{fj}$ and $-{\delta }_{gi}{B}_{fj}+{\delta }_{fj}{A}_{ig}$ for f bottom. Shorter minus paths $-{\sum }_h{B}_{ih}{B}_{hj}$ give $-{\delta }_{gi}{B}_{fj}$ for f shorter and NO scalar: the extra negative χ involves a shorter row h and bottom row g. The bridge has the opposite matrix terms. Therefore $-t+t=0$ and all matrices cancel, for D^(2) and boundary F^(2). Paths outside these adjacent columns have zero projection by the support argument in step 3.2. This proves the exact jump-one base cases in both actual LEFT reductions.

For a larger boundary shift, or a rectangle, the first two end columns have equal height t. A double dot deletes both and $K_{s;ij}={\ddot{D}}_{m;ij}^{(s)}$; replacing K with the corresponding boundary $\ddot{E}$ in RIGHT deletion or $\ddot{F}$ in LEFT deletion gives the same calculation. In RIGHT deletion call the last positions C,B,A, and write $B_{ck}={\tilde{E}}_{(B,c),(B,k)}$, $A_{kj}={\tilde{E}}_{(A,k),(A,j)}$, $C_{hk}={\tilde{E}}_{(C,h),(B,k)}$, $J_{cj}={\tilde{E}}_{(B,c),(A,j)}$ and $H_{hj}={\tilde{E}}_{(C,h),(A,j)}$. Twice expanding step 2.2 expresses the changed r-mode as the sum of eight groups $A_0=K_{r;ij}$, $B_0={\sum }_c{K}_{r-1;ic}{B}_{cj}$, $C_0=[K_{r-1;ih},C_{hj}]$, $D_0={\sum }_k{K}_{r-1;ik}{A}_{kj}$, $E_0={\sum }_{k,c}{K}_{r-2;ic}{B}_{ck}{A}_{kj}$, $F_0={\sum }_k[K_{r-2;ih},C_{hk}]A_{kj}$, $G_0={\sum }_c{K}_{r-2;ic}{J}_{cj}$, $H_0=[K_{r-2;ih},H_{hj}]$. For $z={\tilde{E}}_{(A,f),(B,g)}$ their projected brackets, in that order, are $0,-{\delta }_{fj}{K}_{r-1;ig},0,+{\delta }_{fj}{K}_{r-1;ig},R,S,-R,-S$, where $R=t{\delta }_{fj}{K}_{r-2;ig}-K_{r-2;ig}{A}_{fj}+{\delta }_{fj}{\sum }_c{K}_{r-2;ic}{B}_{cg}$ and $S={\delta }_{fj}[K_{r-2;ih},C_{hg}]$. The bracket in step 1.2 proves this table by moving each negative factor to the RIGHT. The t summation rows give the scalar in R; the bridge J gives its opposite because ${\rho }_A-{\rho }_B=t$. K lies in the twice-deleted p and commutes with z. The eight expressions cancel for EVERY r>1.

At a RIGHT jump-one boundary, for a fixed earlier-block row g and r>1 the coefficient recursions in U(p) are $D_{m;ij}^{(r+1)}=[F_{m-1;ig}^{(2)},E_{m-1;gj}^{(r)}]+{\sum }_{s=0}^r{\tilde{D}}_{m-1;gg}^{(r+1-s)}{D}_{m;ij}^{(s)}$ and $E_{m-1;ij}^{(r+1)}=[D_{m-1;ig}^{(2)},E_{m-1;gj}^{(r)}]-{\sum }_f{D}_{m-1;if}^{(1)}{E}_{m-1;fj}^{(r)}$. Here $\tilde{D}(u)=-D(u{)}^{-1}$, with order-zero coefficient −I. The corresponding LEFT identities are $D_{m;ij}^{(r+1)}=[F_{m-1;ig}^{(r)},E_{m-1;gj}^{(2)}]+{\sum }_{s=0}^r{D}_{m;ij}^{(s)}{\tilde{D}}_{m-1;gg}^{(r+1-s)}$ and $F_{m-1;ij}^{(r+1)}=[F_{m-1;ig}^{(r)},D_{m-1;gj}^{(2)}]-{\sum }_f{F}_{m-1;if}^{(r)}{D}_{m-1;fj}^{(1)}$. These are algebraic coefficient identities, independent of invariance and generation. Expanding the E identity by the right end recursion gives ${\dot{E}}^{(r+1)}+\sum {\dot{D}}^{(1)}{\dot{E}}^{(r)}$, its product with L, and its commutator with C; collecting with step 2.2 gives $E^{(r+1)}+\sum D^{(1)}{E}^{(r)}$. The left F expansion instead gives ${\dot{F}}^{(r+1)}+\sum {\dot{F}}^{(r)}{\dot{D}}^{(1)}$, its LEFT product with L and $[C,\cdot ]$, which collect to $F^{(r+1)}+\sum F^{(r)}{D}^{(1)}$. The unchanged earlier D^(2) has no boundary endpoint and commutes with L,C. Expanding the boundary E–F bracket similarly gives the D identity: the coefficient convolution $D\tilde{D}=-I$ cancels paired split paths with opposite cut signs, leaving the changed D coefficient. Thus the recursions hold in both product orders. Every earlier D/inverse-D coefficient and the unaffected E or F coefficient is invariant by step 3.2; the changed base coefficients are invariant by step 4.1. Since the sum uses s≤r, induction on r proves invariance of ALL changed jump-one modes.

The REFLECTED calculation is a direct LEFT-ideal calculation, not an anti-map reversal of step 4.2. Call its first positions A,B,C, and write $L_{ik}={\tilde{E}}_{(A,i),(A,k)}$, $B_{kc}={\tilde{E}}_{(B,k),(B,c)}$, $C_{kh}={\tilde{E}}_{(B,k),(C,h)}$, $J_{ic}={\tilde{E}}_{(A,i),(B,c)}$, $H_{ih}={\tilde{E}}_{(A,i),(C,h)}$. The eight groups are $A_0=K_{r;ij}$, $B_0={\sum }_k{B}_{ik}{K}_{r-1;kj}$, $C_0=[C_{ih},K_{r-1;hj}]$, $D_0={\sum }_k{L}_{ik}{K}_{r-1;kj}$, $E_0={\sum }_{k,c}{L}_{ik}{B}_{kc}{K}_{r-2;cj}$, $F_0={\sum }_k{L}_{ik}[C_{kh},K_{r-2;hj}]$, $G_0={\sum }_c{J}_{ic}{K}_{r-2;cj}$, $H_0=[H_{ih},K_{r-2;hj}]$. For $z={\tilde{E}}_{(B,f),(A,g)}$ their projected brackets are $0,+{\delta }_{ig}{K}_{r-1;fj},0,-{\delta }_{ig}{K}_{r-1;fj},R^{\mathrm{\prime }},S^{\mathrm{\prime }},-R^{\mathrm{\prime }},-S^{\mathrm{\prime }}$, with $R^{\mathrm{\prime }}=t{\delta }_{ig}{K}_{r-2;fj}-{\delta }_{ig}{\sum }_c{B}_{fc}{K}_{r-2;cj}+L_{ig}{K}_{r-2;fj}$ and $S^{\mathrm{\prime }}=-{\delta }_{ig}[C_{fh},K_{r-2;hj}]$. For example $[z,L_{ik}]={\delta }_{gi}{\tilde{E}}_{(B,f),(A,k)}$, and moving this negative through B gives $-{\delta }_{fk}{B}_{kc}+{\delta }_{fc}$; summing k gives the t term. The other part of the product bracket gives the L term. The bridge J has the opposite matrix terms and scalar ${\rho }_A-{\rho }_B=-t$; the F/H terms cancel because $[z,H_{ih}]={\delta }_{gi}{C}_{fh}$ and z commutes with K. Thus every group is accounted for and cancels in the reflected LEFT reduction for EVERY r>1. If there are only two columns, K has only its formal constant coefficients and terms involving the third column vanish, so the same table applies.

Steps 3.2, 4.1, 5.1, 4.2 and 5.2 cover every allowed parabolic D/E/F mode for the original and reflected arrangements. Refining a minimal block is its formal Gauss decomposition: coefficients of its inverse and resulting factors are polynomials in its previous coefficients, hence invariant by step 1.2. This gives all ordinary simple modes; higher-root commutator definitions give all higher roots. Therefore the actual untwisted-chart images $P_{ii;i-1}^{(r)},P_{i,i+1;i}^{(r)},P_{i+1,i;i}^{(r)}$ and all their higher-root polynomials belong to their actual left reductions. This conclusion covers the infinite allowed mode families and does not rely on finite generation, finite truncation or a reflected W identification.

Finite Levi Jacobian. Project $U(\mathfrak{p})$ to its column Levi enveloping algebra and take canonical symbols. At its diagonal point $X_c=h_{c}I$, let $Q_i(z)={\prod }_{c\le p_i}(1+h_{c}z)$. The diagonal D first variations are $z{\prod }_{a\le p_i,a\mathrm{\ne }c}(1+h_{a}z)$. For $i<j$, set $m=p_i$, $M=p_j$. A first variation in a common column $c\le m$ has just one off-diagonal factor. Summing the same-column negative-cut repeats and the strictly increasing positive-cut visits in the path formula gives $d{E}_{ij}(z)=z{\prod }_{a=m+1}^M(1+h_{a}z)\mathrm{/}(1+h_{c}z)$ and $d{F}_{ji}(z)=z\mathrm{/}(1+h_{c}z)$ on that entry. These are identities for the allowed coefficient derivatives, not an identification of excluded low product coefficients with shifted generators. Their coefficients in modes $M-m+k$ and $k$, $1\le k\le m$, are respectively $(-h_c{)}^{k-1}{\prod }_{a=m+1}^M(h_a-h_c)$ and $(-h_c{)}^{k-1}$. For higher roots, the canonical symbol of the adjacent root commutator is the Levi Lie–Poisson bracket. At $X_c=h_{c}I$ this Poisson tensor is zero, so the differential of a bracket uses only the two first differentials and the differential of the Poisson tensor. With $i<k<j$ and $p_k=t$, the coefficients from $E_{ik}^{(r-s_{kj})}$ and $E_{kj}^{(s_{kj}+1)}$ multiply to $(-h_c{)}^{a-1}{\prod }_{m<b\le t}(h_b-h_c){\prod }_{t<b\le M}(h_b-h_c)$ when $r=M-m+a$, since $\{X_{c,ik},X_{c,kj}\}=X_{c,ij}$. For F the reversed recursion uses $\{X_{c,jk},X_{c,ki}\}=X_{c,ji}$, with coefficients one and $(-h_c{)}^{r-1}$. Thus induction gives the displayed first variations for all higher roots as well. Choose the $h_c$ nonzero and pairwise distinct. Each row-pair block is Vandermonde with nonzero column factors; each diagonal block is the elementary-symmetric Jacobian from step 2.1. Different row pairs use disjoint entry variables, so the full Jacobian has rank ${\sum }_{ij}\min (p_i,p_j)$. The corresponding finite symbols are algebraically independent by step 2.1's differentiation argument. Reflection reverses column order and interchanges E and F, giving the same invertible blocks with transposed shifts. Empty rows contribute zero-sized blocks.

The whole reduction is generated. Write $\mathcal{W}$ for the entire quantum Hamiltonian reduction, before any identification with a Yangian quotient. By [F3] its homogeneous Hilbert series is $H(t)={\prod }_{i,j}{\prod }_{r=s_{ij}+1}^{s_{ij}+\min (p_i,p_j)}(1-t^r{)}^{-1}$. Each weight is positive, so every coefficient is finite. Step 6.1 places every ordered finite-generator monomial of total degree at most $d$ inside $F_d^K\mathcal{W}$. Step 7.1 and enveloping PBW make these monomials independent: a highest-degree relation after Levi projection would give a polynomial relation among the independent symbols. Their number is exactly ${\sum }_{k\le d}[t^k]H(t)$, which is also $\dim F_d^K\mathcal{W}$ by [F3] for the original pair and step 1.3 for the reflected pair. Thus their span equals $F_d^K\mathcal{W}$ for every $d$. Exhaustivity proves generation of the entire reduction and its ordered filtered basis, and also injectivity of the Levi projection on that entire algebra. The same homogeneous dimension equality proves full generation for the reflected reduction using its own allowed modes and transposed shifts. This uses the Gan–Ginzburg slice theorem independently of W-PBW, not just generic independence in a restricted subalgebra.

Relations, finite order and the whole-algebra identification. The end path identities prove the shifted coefficient relations and the $D_1$ cutoff on the invariant representatives. One can check them after Levi projection, now injective on the entire reduction by step 8.1: the right end images are $D_a^{(r)}\mapsto {\dot{D}}_a^{(r)}+{\delta }_{a,m}{\sum }_k{\dot{D}}_{a;ik}^{(r-1)}{L}_{kj}$, $E_a^{(r)}\mapsto {\dot{E}}_a^{(r)}+{\delta }_{a,m-1}{\sum }_k{\dot{E}}_{a;ik}^{(r-1)}{L}_{kj}$, and $F_a^{(r)}\mapsto {\dot{F}}_a^{(r)}$; the left versions place L before the dotted coefficients and correct F instead. Substitute these finite coefficient expressions in each D–D, D–E, D–F, E–F, equal-root and adjacent-root relation. Terms with no L give the smaller relation. Terms with one L give the smaller D-weight relation; terms with two L use $[L_{ij},L_{hk}]={\delta }_{jh}{L}_{ik}-{\delta }_{ki}{L}_{hj}$. The inverse-D convolution cancels paired split coefficients, and the adjacent triple relations group into those same terms and the matrix Jacobi identity. Hence the relations hold by column induction, starting with matrix-unit brackets at one column; injectivity lifts them back to the actual reduction. For D1 the cutoff row interval is empty, so every join is strictly forward; a path beginning and ending in row1 satisfies $r=\sum (c(b_t)-c(a_t)+1)\le c(b_s)-c(a_1)+1\le p_1$. Thus the smallest-row path has no coefficient above $p_1$, giving the actual $D_1$ cutoff in either arrangement. There is therefore a homomorphism from the abstract truncated quotient onto the invariant-generated whole algebra. Steps 2.3 and 3.3 give finite spanning; commutator swapping lowers canonical degree, so induction on that degree and the number of unordered pairs straightens these finite expressions into ordered ${\mathrm{\Omega }}_n$ monomials. Their images are the independent ordered reduction basis of step 8.1, so the homomorphism is injective as well. This proves the quotient presentation and the finite Gauss PBW basis without restricting an infinite basis through a quotient.

The full generator theorem just proved for BOTH arrangements now identifies the original head/right model. With the reflected-box anti-map τ from step 1.3, send a path $(a_1,b_1),\dots ,(a_s,b_s)$ to $(f(b_s),f(a_s)),\dots ,(f(b_1),f(a_1))$. Each displacement is unchanged. A reversed join compares $l+1-c(a_{k+1})$ with $l+1-c(b_k)$, so its strict-forward or weak-reverse rule is exactly the original one; its row is $\mathrm{row}(a_{k+1})=\mathrm{row}(b_k)$, preserving its sign. Total degree, number of factors and total sign are identical; the product order is precisely τ's reversal. This involutive bijection proves $\tau (P_{ij;x}^{(r)})=P_{ji;x}^{\mathrm{\prime }(r)}$ for ALL i,j,x and modes r, including the diagonal constants at r=0. In particular τ exchanges simple E/F images and fixes the corresponding D images, transposing the shift matrix and preserving mode number. Since these explicit images generate each WHOLE reduction by step 8.1, both inclusions give $\tau (W)=W^{\mathrm{\prime }}$, rather than only the image of a path subalgebra.

The preceding construction applies to both directional shift matrices, using the corresponding left or right end formulas in step 9.1. Thus it proves the full finite quotient basis, equality with the whole reduction, and strict canonical/Kazhdan filtration in both arrangements. Its degree-r homogeneous basis is the monomial basis counted by the exact slice Hilbert series, and the projection used for the relation check is injective on the entire reduction. These are the prerequisites for the original right-normalizer identification below.

Actual original prefix coefficient identity. Let ${\lambda }_m$ be the first m columns of the decreasing-column pyramid, $I=I_{m-1}:U({\mathfrak{g}}_{m-1})\hookrightarrow U({\mathfrak{g}}_m)$ the ordinary UNSHIFTED box embedding, and $b_a(j)$ the box in row j and column a when it exists. Put negative-index coefficients zero and $T_{ij}^{(0)}={\delta }_{ij}$. The original coefficient recursion, for r>0, is $T_{ij}^{(r)}({\lambda }_m)=I(T_{ij}^{(r)})-{\sum }_{h\le n-q_m,s_{hj}\le r}I(T_{ih}^{(r-s_{hj})})I(T_{hj}^{(s_{hj})})+{\sum }_{h>n-q_m}I(T_{ih}^{(r-1)})E_{b_m(h),b_m(j)}-[I(T_{ij}^{(r-1)}),E_{b_{m-1}(j),b_m(j)}]$, where coefficients inside I belong to ${\lambda }_{m-1}$ and shifts in the sum belong to ${\lambda }_m$. The final THREE terms are omitted for $j\le n-q_m$, and a nonexistent box omits its term. To derive the identity before any cutoff is known, apply the right end identities to the last Gauss block in the ordinary chart. D has its smaller D plus its last-column Levi product and minus its previous-to-last bridge commutator; E has the same two corrections and starts with the smaller $\bar{E}$. The precise smaller-block subtraction is ${\bar{E}}_{a,b}(u)=E_{a,b}(u)-{\sum }_{c=a}^{b-1}{E}_{a,c}(u)E_{c,b}^{({\dot{s}}_{cb}+1)}{u}^{-{\dot{s}}_{cb}-1}$, all in the smaller diagram. For a shorter block c and last block b, ${\dot{s}}_{cb}+1=s_{cb}$. Insert these expressions into $T_{ij}(u)=\sum (F_{c,a}(u)D_c(u)E_{c,b}(u){)}_{ij}$. Unchanged terms give the first embedded T; the $\bar{E}$ subtractions group by shorter terminal row h into the product with split $(r-s_{hj},s_{hj})$; the L products group into the Levi sum. Leibniz groups the common bridge into the displayed single commutator. Higher-root end identities follow by their adjacent recursion; its earlier-block factor has no terminal bridge endpoint, so its bracket with that bridge is zero. This proves the prefix identity itself and retains its actual coefficient index.

The two graded algebras. The shifted root relations give the current brackets on the allowed modes $e_{ij}{t}^a$, $a\ge s_{ij}$; shift additivity makes their span a Lie subalgebra. Step 3.1 identifies its ordered enveloping basis with the intrinsic loop-filtered basis, proving ${\mathrm{gr}}^L{Y}_n(\sigma )=U({\mathfrak{g}\mathfrak{l}}_n[t](\sigma ))$. Canonical degree instead gives the commutative polynomial algebra of step 3.1. For the finite reduction the Kazhdan associated graded is the commutative slice algebra from step 8.1, while the leading single-factor terms of the finite generators lie in the actual centralizer ${\mathfrak{g}}^e\subseteq \mathfrak{p}$; the next theorem identifies the loop graded algebra by ambient enveloping PBW. Noncommutativity distinguishes the two graded algebras whenever the corresponding Lie algebra is nonabelian.

Every coefficient above the bound is zero. For the original decreasing-column diagram $S_{ij}=s_{ij}+\min (p_i,p_j)=p_j$ for EVERY ordered row pair. Induct on the column count m using step 10.2; the one-column T has no coefficient above one directly. An empty row contributes only a diagonal constant. If $j\le n-q_m$, its row length is unchanged on removal, so the only embedded T term vanishes for $r>p_j$ by induction. If j is a bottom row, $p_j=m$, and the first embedded T vanishes for $r>m$ because its smaller target-row length is m−1. In each shorter-row product, $p_h$ is unchanged and $s_{hj}=m-p_h$, so $r-s_{hj}>p_h=S_{ih}({\lambda }_{m-1})$ for ANY i, killing its first factor. In the Levi sum every h has smaller row length m−1, and $r-1>m-1=S_{ih}({\lambda }_{m-1})$ kills its coefficient. The same inequality kills the T coefficient inside the final commutator. This covers every term, including the boundary $r=m+1$, and proves $T_{ij}^{(r)}=0$ for $r>S_{ij}$. The lower zero range $0<r\le s_{ij}$ follows coefficientwise from FDE and shift additivity. No truncation fact was assumed to prove itself.

Finite T coordinates and loop symbols. Coefficientwise FDE and Gauss inversion are weighted triangular in the original index r: the mode equals its corresponding Gauss coordinate plus products of smaller mode indices of total canonical degree r. The finite Gauss eliminations from steps 2.3 and 3.3 and the actual cutoff in step 11.2 close both inversions on exactly the finite set. Thus these finite canonical symbols are another polynomial coordinate system, and the ordered finite T monomials have the same filtered basis as the Gauss monomials in step 9.1, in any prescribed order. A signed path term with k factors and coefficient index r has ambient good degree at most r−k; scalar replacements lower it further. Only its single-factor part contributes in degree r−1, giving $(-1{)}^{r-1}{\sum }_{\mathrm{row}(a)=i,\mathrm{row}(b)=j,c(b)-c(a)=r-1}{E}_{ab}=(-1{)}^{r-1}{e}_{ij;r-1}$. Products introduced by Gauss inversion have at least two factors and smaller good degree. This proves the symbol dictionary with the existing coefficient index and no spectral rescaling.

Put ${\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}(E_{ab})=E_{ab}+{\delta }_{ab}(n-{\sum }_{d\le c(a)}{q}_d)$. This and ηtail are algebra automorphisms: a matrix bracket can have diagonal part only from $E_{ab},E_{ba}\in \mathfrak{p}$, which forces equal columns and equal diagonal shifts, so the scalar functional kills the commutator; the negative scalar translation is the inverse. Reflection gives ${\sum }_{d\ge c^{\mathrm{\prime }}(f(a))}{q}_d^{\mathrm{\prime }}={\sum }_{d\le c(a)}{q}_d$, hence $\tau {\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}={\eta }_{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}^{\mathrm{\prime }}\tau$. Also $\tau (\mathfrak{m})={\mathfrak{m}}^{\mathrm{\prime }}$, ${\chi }^{\mathrm{\prime }}\tau =\chi$ by the trace calculation, and $\tau (I_{\chi }U(\mathfrak{g}))=U({\mathfrak{g}}^{\mathrm{\prime }})I_{{\chi }^{\mathrm{\prime }}}$, since τ reverses products and sends each shifted negative generator to its reflected one. Therefore $\tau ([{\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}(u),x])=[\tau (x),{\eta }_{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}^{\mathrm{\prime }}(\tau (u))]$. Bijectivity shows that u satisfies the ORIGINAL head/right condition exactly when τ(u) belongs to the reflected LEFT reduction. Step 9.2 identifies its inverse image with the original W, proving BOTH inclusions in $W=\{u:[{\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}(u),\mathfrak{m}]\subseteq I_{\chi }U(\mathfrak{g})\}$. The induced Gauss transpose also satisfies $\tau (T_{ij}^{(r)})=T_{ji}^{\mathrm{\prime }(r)}$: apply the coefficientwise reversed FDE factors and their Gauss cut identities. Thus step 11.2 gives the reflected upper cutoff $r>S_{ji}=S_{ij}^{\mathrm{\prime }}$, with the same finite triangular T basis. This proves right invariance for every original shifted generator, higher-root mode and product, beyond the numerical identity ${\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}-{\eta }_{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}=-{\mathrm{tr}}_{\mathfrak{m}}\mathrm{ad}$.

Write $J_L=U(\mathfrak{g})I_{\chi }$ and $J_R=I_{\chi }U(\mathfrak{g})$. The identities $(x-\chi (x)){\eta }_{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}(w)={\eta }_{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}(w)(x-\chi (x))+[x,{\eta }_{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}(w)]$ and ${\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}(w)(x-\chi (x))=(x-\chi (x)){\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}(w)+[{\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}(w),x]$ prove $J_L{\eta }_{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}(w)\subseteq J_L$ and ${\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}(w)J_R\subseteq J_R$ for every w∈W. Thus $Q_L=U(\mathfrak{g})\mathrm{/}{J}_L$ has its actual right action $\bar{a}\cdot w=\overline{a{\eta }_{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}(w)}$, and $Q_R=U(\mathfrak{g})\mathrm{/}{J}_R$ its actual left action $w\cdot \bar{a}=\overline{{\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}(w)a}$, commuting with the respective enveloping actions by associativity. These are module quotients, not quotient rings. Right Whittaker vectors act by $v\cdot w=v{\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}(w)$ because they annihilate J_R. Turning a right U(g)-module into a left reflected module by $a^{\mathrm{\prime }}v=v{\tau }^{-1}(a^{\mathrm{\prime }})$ identifies its right Whittaker space and W action with the reflected left ones, using ${\tau }^{-1}{\eta }_{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}^{\mathrm{\prime }}\tau ={\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}$. For consumers assuming AC, transporting their proved left Skryabin equivalence through the reflected Lie isomorphism in step 1.3 and then this anti-map gives the right equivalence with inverse $-{\otimes }_W{Q}_R$; that conditional categorical assertion is separate from the choice-free algebra identity. Every action is the explicit original or reflected chart action; no equality of different twists on an arbitrary fixed representation is assumed.