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The Fock-space convention dictionary for finite type-A quantum modules

Definition

Throughout this pair the source parameter of the Fock-space literature is written $s$ and the library parameter of the Drinfeld–Jimbo algebra The Drinfeld-Jimbo quantized enveloping algebra by generators and relations is written $t$ ; the dictionary below sets $t = s^{- 1}$ .

Let $I$ be a nonempty consecutive set of simple-root indices and put $I^{+} = I \cup (I + 1)$ , the coordinate alphabet. On a finite coordinate interval $J = [k, r] \subseteq I^{+}$ the roots have indices $[k, r - 1]$ , the coordinate weight lattice is $⨁_{a = k}^{r} Z δ_{a}$ , $α_{a} = δ_{a} - δ_{a + 1}$ and $h_{a} = e_{a} - e_{a + 1}$ . The full central coordinate is retained. The source algebra $U_{s}$ has toral generators $D_{a}^{\pm 1}$ for $a \in J$ and root generators $E_{a}, F_{a}$ for $k \leq a < r$ , acting on natural vectors $v_{b}$ , $b \in J$ , by

$D_{a} v_{b} = s^{δ_{a b}} v_{b}, E_{a} v_{b} = δ_{a + 1, b} v_{a}, F_{a} v_{b} = δ_{a, b} v_{a + 1},$

$Δ E_{a} = 1 \otimes E_{a} + E_{a} \otimes D_{a} D_{a + 1}^{- 1}, Δ F_{a} = F_{a} \otimes 1 + D_{a}^{- 1} D_{a + 1} \otimes F_{a} .$

The library parameter $t$ is identified with $s^{- 1}$ ; the semilinear Hopf dictionary is the coefficient map $t \mapsto s^{- 1}$ together with the generator assignment

$E_{a} ⟼ E_{a}, F_{a} ⟼ F_{a}, K_{h} ⟼ D_{- h},$

where the $E_{a}, F_{a}$ on the source side are the divided generators of the source normalization and on the library side those of The Lusztig divided-power integral form $U_{A}$ (the assignments agree on the undivided generators $E_{a} = E_{a}^{(1)}$ , $F_{a} = F_{a}^{(1)}$ , and divided powers are carried to divided powers because the quantum integers are invariant under $q \mapsto q^{- 1}$ , Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ ). Root-label weights are preserved, while toral numerical eigenvalues use the reciprocal parameter: on a weight vector of weight $μ$ the library generator $K_{h}$ acts by $t^{μ (h)} = s^{- μ (h)}$ , which is the action of $D_{- h}$ because $(- h) (μ) = - μ (h)$ .

The bar involution sends $s \mapsto s^{- 1}$ , fixes $E_{a}$ and $F_{a}$ and inverts each $D_{a}$ ; the corresponding library operation is the involution with $t \mapsto t^{- 1}$ , $E_{a} \mapsto E_{a}$ , $F_{a} \mapsto F_{a}$ and $K_{h} \mapsto K_{h}^{- 1}$ of The bar lattice and the divided-power integral lattice of a based module and The Lusztig divided-power integral form $U_{A}$ .

The natural highest weight attached to an interval $[k, r]$ is $δ_{k} = ε_{k}$ (equivalently the highest weight reached by the leading coordinate vector $v_{k}$ ), its dual highest weight is $- δ_{r}$ , and the fundamental wedge weight is $δ_{k} + δ_{k + 1} + \dots + δ_{m}$ for $k \leq m \leq r$ . Every fundamental factor carries its own highest-vector-normalized bar, integral lattice and divided-string crystal charts (The Kashiwara lattices $L (λ)$ and $L (\infty)$ and their generating sets); its internal antisymmetrizer embedding is primitive and integral but is not asserted to be based in the library's raw lower $q_{0}$ chart. Infinite intervals are defined by the compatible system of their finite subintervals, and Quantum weight modules, weights in $P$ , and integrability with The integrable quantum highest weight module exists and is unique supplies the library integrable modules used on each finite interval.

The source Fock families $L, T, P$ and the library form-dual upper family (The contravariant form on an integrable highest weight module) will be compared below through their actual forms; they are never identified by their names.

Facts & Assumptions

Given: The type- $A$ root datum with full coordinate weight lattice, a finite interval $[k, r] \subseteq I^{+}$ , the source generators $D_{a}, E_{a}, F_{a}$ with the displayed action and comultiplication, and the library algebra with its generators $E_{a}, F_{a}, K_{h}$ and parameter $t$ .

[F1]

Library presentation and toral action (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations, Positive, negative and toral quantum subalgebras and their root gradings): $K_{h} K_{h^{'}} = K_{h + h^{'}}$ , $K_{0} = 1$ , $K_{h} E_{a} K_{h}^{- 1} = q^{⟨ α_{a}, h ⟩} E_{a}$ , $K_{h} F_{a} K_{h}^{- 1} = q^{- ⟨ α_{a}, h ⟩} F_{a}$ , and on a weight vector of weight $μ$ the element $K_{h}$ acts by $q^{μ (h)}$ ; the $Q (q)$ -algebra is generated by the $E_{a}, F_{a}, K_{h}$ . The Hopf operations of The Drinfeld–Jimbo formulas define a Hopf algebra are $Δ E_{a} = E_{a} \otimes K_{a}^{- 1} + 1 \otimes E_{a}$ , $Δ F_{a} = F_{a} \otimes 1 + K_{a} \otimes F_{a}$ and $Δ K_{h} = K_{h} \otimes K_{h}$ , with counit $ε (E_{a}) = ε (F_{a}) = 0$ , $ε (K_{h}) = 1$ and antipode $S (E_{a}) = - E_{a} K_{a}$ , $S (F_{a}) = - K_{a}^{- 1} F_{a}$ , $S (K_{h}) = K_{- h}$ .

[F2]

Quantum integers (Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ ): $[m]_{q} = (q^{m} - q^{- m}) / (q - q^{- 1})$ is invariant under $q \mapsto q^{- 1}$ ; the divided power is $x^{(m)} = x^{m} / [m]_{q_{a}}!$ .

[F3]

Library form and bar (The Lusztig divided-power integral form $U_{A}$ , The bar lattice and the divided-power integral lattice of a based module, The contravariant form on an integrable highest weight module, The Kashiwara lattices $L (λ)$ and $L (\infty)$ and their generating sets, Quantum weight modules, weights in $P$ , and integrability): the bar involution is the semilinear endomorphism with $t \mapsto t^{- 1}$ fixing the divided generators and inverting $K_{h}$ ; integrable highest-weight modules, their normalized highest vectors, bars and lattices are available on finite intervals.

[F4]

Existence and uniqueness of integrable highest-weight modules (The integrable quantum highest weight module exists and is unique) on the finite-interval subdatum.

Proof

technique · Check the dictionary on generators and relations and conclude it is a well-defined semilinear Hopf identification on each finite interval

1.1F1algebra

Toral relations. Under $t \mapsto s^{- 1}$ and $K_{h} \mapsto D_{- h}$ , the library relation $K_{h} K_{h^{'}} = K_{h + h^{'}}$ becomes $D_{- (h + h^{'})} = D_{- h} D_{- h^{'}}$ , and $K_{0} = 1$ becomes $D_{0} = 1$ ; these are the source toral relations. Toral elements commute in both presentations, so $φ$ is well defined on the toral subalgebra $Q (t) [P^{\lor}] \to Q (s) [Z J]$ .

1.2F1algebra

Mixed relations. The library relation $K_{h} E_{a} K_{h}^{- 1} = t^{⟨ α_{a}, h ⟩} E_{a}$ becomes, after $t \mapsto s^{- 1}$ , $D_{- h} E_{a} D_{- h}^{- 1} = s^{- ⟨ α_{a}, h ⟩} E_{a}$ , and interchanging $h$ and $- h$ this is the source relation $D_{h^{'}} E_{a} D_{h^{'}}^{- 1} = s^{⟨ α_{a}, h^{'} ⟩} E_{a}$ ; the $F_{a}$ -relation is the sign-reversed mirror. This matches the displayed action $D_{a} v_{b} = s^{δ_{a b}} v_{b}$ because $⟨ α_{b}, e_{a} ⟩ = δ_{a b} - δ_{a, b + 1}$ ; pairing with $h_{a} = e_{a} - e_{a + 1}$ gives the type- $A$ Cartan matrix.

1.3F1F2algebra

The $E$ - $F$ commutator. The library relation $E_{a} F_{b} - F_{b} E_{a} = δ_{a b} (K_{a} - K_{a}^{- 1}) / (t_{a} - t_{a}^{- 1})$ maps to $δ_{a b} (D_{- a} - D_{a}) / (s_{a}^{- 1} - s_{a})$ where $D_{\pm a} : = D_{\pm d_{a} h_{a}}$ ; both numerator and denominator change sign under the substitution, so the fraction is unchanged and equals the source relation $δ_{a b} (D_{a} - D_{a}^{- 1}) / (s_{a} - s_{a}^{- 1})$ after relabelling. No divided generator is modified, as asserted.

1.4F2algebra

Serre relations. The Gaussian binomials and the divided powers $E_{a}^{(m)} = E_{a}^{m} / [m]_{q_{a}}!$ are unchanged as elements under $q_{a} \mapsto q_{a}^{- 1}$ by [F2]; hence each positive and negative quantum Serre element of the library presentation maps to the corresponding source Serre element, since only the divided powers and the binomial coefficients occur.

1.5F1algebra

Comultiplication. Since $K_{a}$ maps to $D_{- h_{a}} = D_{a}^{- 1} D_{a + 1}$ , its inverse maps to $D_{h_{a}} = D_{a} D_{a + 1}^{- 1}$ . Thus the library formulas map to $Δ E_{a} = 1 \otimes E_{a} + E_{a} \otimes D_{a} D_{a + 1}^{- 1}$ and $Δ F_{a} = F_{a} \otimes 1 + D_{a}^{- 1} D_{a + 1} \otimes F_{a}$ , exactly the source coproduct. The counit and antipode agree on these generators and hence on the generated algebra.

1.6F3algebra

Bar involution. On the source side bar fixes $E_{a}, F_{a}$ and inverts $D_{a}$ , hence it sends $D_{- h} \mapsto D_{h} = D_{- h}^{- 1}$ ; on the library side bar fixes the divided generators and inverts $K_{h}$ . Under the dictionary the two involutions intertwine, and both conjugate the parameter. Consequently bar commutes with the dictionary on the generating set and hence on the generated algebra.

1.7F1algebra

Weights and labels. A weight vector of weight $μ = \sum_{a} m_{a} ε_{a}$ is an eigenvector of $K_{h}$ with eigenvalue $t^{μ (h)} = s^{- μ (h)}$ , which equals the eigenvalue of $D_{- h}$ ; hence root-label weights are preserved while the toral eigenvalues use the reciprocal parameter, as asserted. In particular $δ_{k} = ε_{k}$ and $- δ_{r}$ are genuine coordinate characters, and the fundamental wedge carries $δ_{k} + \dots + δ_{m}$ ; the central coordinate $δ_{k} + \dots + δ_{r}$ is retained.

2.1step 1.1step 1.2step 1.3step 1.4step 1.5step 1.6step 1.7∎

Conclusion. Steps 1.1-1.6 show that the generator assignment preserves every family of defining relations and the Hopf operations on each finite interval, so it extends uniquely to a semilinear Hopf-algebra map $φ : U_{t} \to U_{s}$ with $t \mapsto s^{- 1}$ , and symmetrically for the inverse; it is an isomorphism on the subalgebra generated by the finitely many $D_{a}^{\pm 1}, E_{a}, F_{a}$ with coordinate indices $k \leq a \leq r$ and root indices $k \leq a < r$ . Infinite intervals are covered by the compatible finite subintervals, and the library integrable modules and lattices of [F3], [F4] are transported by $φ$ on each of them. The comparison of the Fock families through their actual forms, not their names, is carried out in the following items.

Depends on

Used by

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Sources

Jonathan Brundan and Alexander Kleshchev, The degenerate analogue of Ariki's categorification theorem, Math. Z. 266 (2010) 877-919 (complete 44-page arXiv article) (standard reference, not scraped)
Jonathan Brundan, Dual canonical bases and Kazhdan-Lusztig polynomials, J. Algebra 306 (2006) 17-46 (complete 29-page arXiv v2 article) (standard reference, not scraped)
Volodymyr Mazorchuk, Lectures on Algebraic Categorification, QGM Master Class Series, European Mathematical Society 2012 (complete 77-page arXiv lecture-note set) (standard reference, not scraped)