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The shifted Yangian and the pyramid generators of the type-A finite W-algebra

Definition

Work over $C$ and fix $n \geq 1$ together with a shift matrix $σ = (s_{i, j})_{1 \leq i, j \leq n}$ of nonnegative integers satisfying $s_{i, j} + s_{j, k} = s_{i, k}$ whenever $i \leq j \leq k$ or $i \geq j \geq k$ , and $s_{i, i} = 0$ . The shifted Yangian $Y_{n} (σ)$ is the associative unital $C$ -algebra (Algebras over a commutative ring, central structure maps, and algebra homomorphisms) generated by the shifted Drinfeld generators $D_{i}^{(r)}$ ( $1 \leq i \leq n$ , $r \geq 1$ ), $E_{i}^{(r)}$ ( $1 \leq i < n$ , $r > s_{i, i + 1}$ ), and $F_{i}^{(r)}$ ( $1 \leq i < n$ , $r > s_{i + 1, i}$ ). For $i < j$ , the higher-root coefficients $E_{i, j}^{(r)}$ have range $r > s_{i, j}$ and $F_{j, i}^{(r)}$ have range $r > s_{j, i}$ ; they are defined recursively by adjacent-root commutators, not declared as a duplicate independent family. Put $D_{i}^{(0)} = 1$ , $D_{i} (u) = \sum_{r \geq 0} D_{i}^{(r)} u^{- r}$ and ${\tilde{D}}_{i} (u) = - D_{i} (u)^{- 1}$ . The defining relations are the following, for every index for which all displayed generators exist (in particular the $E$ and $F$ indices exceed their respective shifts):

$[D_{i}^{(r)}, D_{j}^{(s)}] = 0, [E_{i}^{(r)}, F_{j}^{(s)}] = δ_{i j} \sum_{t = 0}^{r + s - 1} {\tilde{D}}_{i}^{(t)} D_{i + 1}^{(r + s - 1 - t)} .$

$[D_{i}^{(r)}, E_{j}^{(s)}] = (δ_{i j} - δ_{i, j + 1}) \sum_{t = 0}^{r - 1} D_{i}^{(t)} E_{j}^{(r + s - 1 - t)} .$

$[D_{i}^{(r)}, F_{j}^{(s)}] = (δ_{i, j + 1} - δ_{i j}) \sum_{t = 0}^{r - 1} F_{j}^{(r + s - 1 - t)} D_{i}^{(t)} .$

$[E_{i}^{(r)}, E_{i}^{(s + 1)}] - [E_{i}^{(r + 1)}, E_{i}^{(s)}] = E_{i}^{(r)} E_{i}^{(s)} + E_{i}^{(s)} E_{i}^{(r)} .$

$[F_{i}^{(r + 1)}, F_{i}^{(s)}] - [F_{i}^{(r)}, F_{i}^{(s + 1)}] = F_{i}^{(r)} F_{i}^{(s)} + F_{i}^{(s)} F_{i}^{(r)} .$

$[E_{i}^{(r)}, E_{i + 1}^{(s + 1)}] - [E_{i}^{(r + 1)}, E_{i + 1}^{(s)}] = - E_{i}^{(r)} E_{i + 1}^{(s)} .$

$[F_{i}^{(r + 1)}, F_{i + 1}^{(s)}] - [F_{i}^{(r)}, F_{i + 1}^{(s + 1)}] = - F_{i + 1}^{(s)} F_{i}^{(r)} .$

For $∣ i - j ∣ > 1$ , $[E_{i}^{(r)}, E_{j}^{(s)}] = [F_{i}^{(r)}, F_{j}^{(s)}] = 0$ . For $∣ i - j ∣ = 1$ impose

$[E_{i}^{(r)}, [E_{i}^{(s)}, E_{j}^{(t)}]] + [E_{i}^{(s)}, [E_{i}^{(r)}, E_{j}^{(t)}]] = 0,$

$[F_{i}^{(r)}, [F_{i}^{(s)}, F_{j}^{(t)}]] + [F_{i}^{(s)}, [F_{i}^{(r)}, F_{j}^{(t)}]] = 0.$

The higher roots are specified without new independent generators by $E_{i, i + 1}^{(r)} = E_{i}^{(r)}$ , $F_{i + 1, i}^{(r)} = F_{i}^{(r)}$ , and, for $j > i + 1$ ,

$E_{i, j}^{(r)} = [E_{i, j - 1}^{(r - s_{j - 1, j})}, E_{j - 1}^{(s_{j - 1, j} + 1)}], F_{j, i}^{(r)} = [F_{j - 1}^{(s_{j, j - 1} + 1)}, F_{j - 1, i}^{(r - s_{j, j - 1})}] .$

These are [BK, Section 2.1, (2.7)–(2.18), and Section 2.2, (2.21)–(2.22)], with the negative root written in matrix-row order here. Two filtrations are fixed on it: the canonical filtration, in which each generator of upper index $r$ has degree $r$ , and the loop filtration, in which it has degree $r - 1$ ; their associated graded algebras are different (the canonical one is commutative polynomial, the loop one is a current algebra).

For the pyramid $λ = (0 \leq p_{1} \leq \dots \leq p_{n})$ with conjugate $q_{1} \geq \dots \geq q_{l}$ (The type-A nilpotent pyramid, good grading and Whittaker character) the relevant shift matrix is $s_{i, j} = p_{j} - p_{i}$ for $i \leq j$ and $s_{i, j} = 0$ for $i \geq j$ , and the type-A finite W-algebra is the quotient $W (λ) = Y_{n} (σ) / (D_{1}^{(r)}, r > p_{1})$ of $Y_{n} (σ)$ by the two-sided ideal generated by the higher modes of $D_{1}$ beyond $p_{1}$ .

Write the Gauss decomposition $T (u) = F (u) D (u) E (u)$ of the generating matrix, with $T (u) = \sum_{r \geq 0} T_{i, j}^{(r)} u^{- r}$ , $T_{i, j}^{(0)} = δ_{i, j}$ , and no additional factor $u^{s_{i, j}}$ : the coefficient of $u^{- r}$ is the mode $T_{i, j}^{(r)}$ . Its modes satisfy $T_{i, j}^{(r)} = 0$ for $0 < r \leq s_{i, j}$ , and in the finite quotient also $T_{i, j}^{(r)} = 0$ for $r > S_{i, j} : = s_{i, j} + p_{\min (i, j)}$ , so the finite quotient has the PBW basis of ordered monomials $T_{i, j}^{(r)}, s_{i, j} < r \leq S_{i, j} .$ The auxiliary signed path polynomials used in the source to construct the Gauss generators are not all identified with these shifted $T$ -modes; the root of $T_{i, j}^{(r)}$ is the weight $ε_{i} - ε_{j}$ , and the surviving modes have loop leading term $(- 1)^{r - 1} e_{i, j; r - 1}$ in the centralizer basis of the pyramid.

Facts & Assumptions

Given: $n \geq 1$ , a shift matrix $σ$ as above, the pyramid $λ$ with row lengths $0 \leq p_{1} \leq \dots \leq p_{n}$ and columns $q_{1} \geq \dots \geq q_{l}$ , and the generators and filtrations of the Definition.

[F1]

The pyramid data: $e \in g_{1}$ of Jordan type $λ$ , the centralizer basis $e_{i, j; r}$ with $s_{i, j} \leq r < p_{j}$ , and the character $χ$ ; the finite W-algebra $W (λ)$ is generated by $p$ -matrix coefficients subject to the loop filtration (The type-A nilpotent pyramid, good grading and Whittaker character, The type-A finite W-algebra and the Whittaker category).

[F2]

The RTT presentation: the ordinary Yangian RTT algebra is generated by the matrix coefficients $T_{i, j}^{(r)}$ with the RTT relation $[T_{i, j}^{(r)}, T_{h, k}^{(s)}] = \sum_{t = 0}^{\min (r, s) - 1} (T_{i, k}^{(r + s - 1 - t)} T_{h, j}^{(t)} - T_{i, k}^{(t)} T_{h, j}^{(r + s - 1 - t)})$ and $T^{(0)} = I$ ; the Gauss factorization is formal matrix multiplication and inversion, and the structure constants of the D/E/F presentation are the coefficients of the products of the RTT blocks (Universal enveloping algebra, Classical complex matrix Lie algebras).

[F3]

The local well-definedness suppliers are Ordered monomial PBW basis for the shifted Yangian ↗ and The loop filtration on the type-A finite W-algebra has associated graded U(g^e) ↗, declared in justified_by. Their proofs start from the abstract presentation and the independently defined quantum Hamiltonian reduction: invariant original/reflected path coefficients, finite Levi symbol independence and an exact weighted slice-dimension comparison establish the finite reduction basis; index-complete coefficient reduction establishes the truncated quotient and finite $T$ range. Ambient enveloping PBW establishes loop-symbol independence. These results justify the Definition, rather than taking its asserted finite PBW basis as a premise.

Proof

technique · direct

Given: The abstract RTT/shifted Drinfeld data, the original pyramid shift, and the local well-definedness constructions of [F3].

1.1F1F2algebra

Ranges and shift arithmetic. The monotone shift identities hold by telescoping $p_{k} - p_{i} = (p_{j} - p_{i}) + (p_{k} - p_{j})$ on increasing indices and by zero lower shifts on decreasing indices. The adjacent positive generators have threshold $s_{i, i + 1}$ , the adjacent negative generators threshold $s_{i + 1, i}$ , and their higher-root commutator recursions give respectively $s_{i, j}$ and $s_{j, i}$ . Thus for the asymmetric original pyramid an upper coefficient begins after $p_{j} - p_{i}$ whereas its lower counterpart begins at mode $1$ ; these are distinct ranges. The formal Gauss coefficients are computed by triangular inversion with diagonal constant term $1$ .

2.1F2F3step 1.1algebra

Infinite filtrations. The RTT relation lowers canonical degree under a commutator, and its degree- $r + s - 2$ loop part is the current Lie bracket. The actual ordered-product evaluation and Gauss Jacobian in [F3] prove independence of the ordered root-mode words, while the defining relations give spanning with both degree assignments. Therefore the canonical associated graded is polynomial and the loop associated graded is the enveloping algebra of the shifted current Lie subalgebra. Their distinction follows from their respective multiplication, rather than filtered-element independence alone.

2.2F1F2F3step 1.1algebra

Finite quotient and actual reduction. By [F3], the actual signed path images satisfy the shifted coefficient relations and $D_{1}^{(r)} = 0$ for $r > p_{1}$ . The coefficient reductions give finite spanning words in the abstract quotient. Their images are independent invariant monomials, whose weighted count equals every filtered dimension of the independently constructed whole reduction. Hence the quotient maps isomorphically to that reduction. The finite $T$ coefficient induction gives zero below the shift and above $S_{i, j} = s_{i, j} + p_{\min (i, j)}$ , and the finite Gauss-to- $T$ coordinate change gives the ordered surviving $T$ -monomial basis. This uses finite spanning and independent reduction images; restriction of an infinite PBW basis to a quotient would not prove independence. Only $D_{1}$ is directly truncated: higher diagonal Gauss coefficients remain polynomial expressions.

3.1F1F3step 2.2algebra∎

Weights and actual loop symbols. The path of canonical degree $r$ with $k$ matrix factors has good degree at most $r - k$ , so only its single-factor part contributes in loop degree $r - 1$ . It is $(- 1)^{r - 1} e_{i, j; r - 1}$ , of row weight $ε_{i} - ε_{j}$ . The ambient homogeneous centralizer basis and enveloping PBW in [F3] make these symbols and their ordered products independent, establishing the finite loop filtration. Auxiliary paths use their declared cut and sign conventions and are not automatically all the shifted $T$ coefficients. For $p = (1, 2)$ the upper threshold is $s_{1, 2} = 1$ , so the first upper coefficient is $T_{1, 2}^{(2)}$ with symbol $- e_{1, 2; 1}$ ; the corresponding lower threshold is $s_{2, 1} = 0$ . This verifies the unrescaled coefficient convention and the asymmetric range.

Remarks

The index convention is deliberately unshifted: the coefficient of $u^{- r}$ is $T_{i, j}^{(r)}$ , no power of $u$ is inserted, and the filtrations only change the assignation of degrees. Rescaling the series by $u^{s_{i, j}}$ would shift the ranges and is not used anywhere on this page.
The finite quotient keeps the higher diagonal modes $D_{i}^{(r)}$ as polynomial expressions rather than setting them to zero; only the modes $D_{1}^{(r)}$ with $r > p_{1}$ are truncated.

Depends on

Used by

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Sources

Jonathan Brundan and Alexander Kleshchev, Representations of shifted Yangians and finite W-algebras, Mem. Amer. Math. Soc. 196 (2008), no. 918 (complete 109-page arXiv memoir) (standard reference, not scraped)
Jonathan Brundan and Alexander Kleshchev, Shifted Yangians and finite W-algebras, arXiv:math/0407012 (49 pp.); Parabolic presentations of the Yangian Y(gl_n), arXiv:math/0407011 (28 pp.) (standard reference, not scraped)