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Kashiwara's skew derivations $e_{i}^{'}$ and $e_{i}^{''}$ on the negative half

Definition

Let $U_{q} (g)$ be the Drinfeld–Jimbo algebra of The Drinfeld-Jimbo quantized enveloping algebra by generators and relations, with its generated subalgebras $U_{q}^{-}, U_{q}^{0}, U_{q}^{+}$ and root gradings $U_{q}^{-} = ⨁_{ν \in Q_{+}} U_{q}^{-} [- ν]$ of Positive, negative and toral quantum subalgebras and their root gradings ( $Q_{+} = ⨁_{i} Z_{\geq 0} α_{i}$ , height $ht$ of Kac Moody root lattice height and positive cone). Fix $i \in I$ and write $K_{i} : = K_{d_{i} h_{i}}$ , so that $K_{i} F_{j} K_{i}^{- 1} = q_{i}^{- a_{i j}} F_{j}$ and $q_{i} - q_{i}^{- 1} \neq 0$ .

For every $P \in U_{q}^{-}$ there exist unique elements $Q, R \in U_{q}^{-}$ with

$E_{i} P - P E_{i} = \frac{K_{i} Q - K_{i}^{- 1} R}{q_{i} - q_{i}^{- 1}} .$

Define the Kashiwara skew derivations $e_{i}^{'}, e_{i}^{''} : U_{q}^{-} \to U_{q}^{-}$ by

$e_{i}^{'} (P) : = R and e_{i}^{''} (P) : = Q .$

Thus $e_{i}^{'}$ is the operator whose value is the coefficient of $K_{i}^{- 1}$ and $e_{i}^{''}$ is the operator whose value is the coefficient of $K_{i}$ in the commutator with $E_{i}$ . The maps have the following properties, all proved in the Verification below.

(a) $e_{i}^{'}$ and $e_{i}^{''}$ are $Q (q)$ -linear, vanish on $1$ , and $e_{i}^{'} (F_{j}) = e_{i}^{''} (F_{j}) = δ_{i j}$ for all $j \in I$ .

(b) (Twisted Leibniz rules) For homogeneous $x, y \in U_{q}^{-}$ , with the actual weight $wt (x) \in Q$ of Positive, negative and toral quantum subalgebras and their root gradings, $e_{i}^{'} (x y) = e_{i}^{'} (x) y + q_{i}^{⟨ wt (x), h_{i} ⟩} x e_{i}^{'} (y), e_{i}^{''} (x y) = e_{i}^{''} (x) y + q_{i}^{- ⟨ wt (x), h_{i} ⟩} x e_{i}^{''} (y) .$

(c) (Commutation with the generators) For all $j \in I$ and all $y \in U_{q}^{-}$ , $e_{i}^{'} (F_{j} y) = q_{i}^{- a_{i j}} F_{j} e_{i}^{'} (y) + δ_{i j} y, e_{i}^{''} (F_{j} y) = q_{i}^{a_{i j}} F_{j} e_{i}^{''} (y) + δ_{i j} y,$

equivalently $e_{i}^{'} F_{j} = q_{i}^{- a_{i j}} F_{j} e_{i}^{'} + δ_{i j}$ and $e_{i}^{''} F_{j} = q_{i}^{a_{i j}} F_{j} e_{i}^{''} + δ_{i j}$ as operators on $U_{q}^{-}$ .

(d) (Grading, height and nilpotence) For every $ν \in Q_{+}$ ,

$e_{i}^{'} (U_{q}^{-} [- ν]), e_{i}^{''} (U_{q}^{-} [- ν]) \subseteq U_{q}^{-} [- (ν - α_{i})],$

the right-hand side being $0$ when $ν - α_{i} \notin Q_{+}$ . In particular both maps raise the actual weight by $α_{i}$ , lower the positive height by one, and are locally nilpotent: $(e_{i}^{'})^{k} (P) = (e_{i}^{''})^{k} (P) = 0$ for $k$ larger than the height of the degree of a homogeneous $P$ .

(e) (Bar compatibility) Let $ψ$ be the semilinear algebra involution $q \mapsto q^{- 1}$ , $E_{i} \mapsto E_{i}$ , $F_{i} \mapsto F_{i}$ , $K_{h} \mapsto K_{- h}$ of The Chevalley involution, bar involution, and contravariant anti-involution preserve the Drinfeld–Jimbo ideal. Then $e_{i}^{''} = ψ e_{i}^{'} ψ$ on $U_{q}^{-}$ .

The commutator construction also fixes the orientation: the $K_{i}^{- 1}$ -coefficient is $e_{i}^{'}$ and the $K_{i}$ -coefficient is $e_{i}^{''}$ . The twists $q_{i}^{\pm ⟨ wt (x), h_{i} ⟩}$ in (b) are the twists of the corresponding coefficient maps of the negative-half coproduct in the source's dictionary; the explicit commutator computation used here is the definition of record, and no imaginary-simple-root convention is needed, since every Cartan datum of this library has $a_{i i} = 2$ .

Facts & Assumptions

Given: A fixed $i \in I$ , the Drinfeld–Jimbo algebra $U_{q} (g)$ and its generated subalgebras, the root grading and the height $ht : Q_{+} \to Z_{\geq 0}$ (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations, Positive, negative and toral quantum subalgebras and their root gradings, Kac Moody root lattice height and positive cone), and the finite words $w = F_{j_{1}} \dots F_{j_{m}}$ ( $m \geq 0$ , the empty word for $m = 0$ ) in the negative generators.

[F1]

The multiplication $U_{q}^{-} \otimes_{Q (q)} U_{q}^{0} \otimes_{Q (q)} U_{q}^{+} \to U_{q} (g)$ is a vector-space isomorphism, and the $K_{h}$ ( $h \in P^{\lor}$ ) are linearly independent; every element of $U_{q}^{-}$ is a finite $Q (q)$ -linear combination of words in the $F_{j}$ ; the root grading $U_{q} (g) = ⨁_{β \in Q} U_{q} (g) [β]$ has $U_{q}^{-} [- ν] U_{q}^{0} \subseteq U_{q} (g) [- ν]$ and the images of the subspaces $U_{q}^{-} [- ν] \otimes U_{q}^{0} \otimes Q (q)$ for distinct $ν$ are linearly independent (Triangular decomposition of a quantized enveloping algebra, Positive, negative and toral quantum subalgebras and their root gradings).

[F2]

For homogeneous $x \in U_{q} (g)$ of degree $β$ and $h \in P^{\lor}$ , $K_{h} x K_{h}^{- 1} = q^{β (h)} x$ , hence $K_{h} x = q^{β (h)} x K_{h}$ and $K_{h}^{- 1} x = q^{- β (h)} x K_{h}^{- 1}$ ; in particular $K_{i} F_{j} = q_{i}^{- a_{i j}} F_{j} K_{i}$ and $K_{i}^{- 1} F_{j} = q_{i}^{a_{i j}} F_{j} K_{i}^{- 1}$ ( Positive, negative and toral quantum subalgebras and their root gradings, Kac Moody root lattice height and positive cone for $wt (F_{j}) = - α_{j}$ ).

[F3]

$E_{i} F_{j} - F_{j} E_{i} = δ_{i j} (K_{i} - K_{i}^{- 1}) / (q_{i} - q_{i}^{- 1})$ in $U_{q} (g)$ (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations).

[F4]

$ψ$ is an involutive semilinear algebra automorphism of $U_{q} (g)$ with $ψ (q) = q^{- 1}$ , $ψ (E_{i}) = E_{i}$ , $ψ (F_{i}) = F_{i}$ , $ψ (K_{h}) = K_{- h}$ ; it preserves $U_{q}^{-}$ (The Chevalley involution, bar involution, and contravariant anti-involution preserve the Drinfeld–Jimbo ideal).

[F5]

The height is additive: $ht (ν - α_{i}) = ht (ν) - 1$ for $ν - α_{i} \in Q_{+}$ ; if $ν - k α_{i} \in Q_{+}$ then $ht (ν - k α_{i}) = ht (ν) - k \geq 0$ , so there is no word of that degree once $k > ht (ν)$ ; and the degrees of the direct sum $⨁_{ν \in Q_{+}} U_{q}^{-} [- ν]$ are pairwise distinct (Kac Moody root lattice height and positive cone, Positive, negative and toral quantum subalgebras and their root gradings).

[F6]

$q_{i} = q^{d_{i}}$ with $d_{i} > 0$ , so $q_{i} - q_{i}^{- 1} \neq 0$ in $Q (q)$ (Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ ).

Verification

technique · Construct $Q_w,R_w$ on words by moving $E_i$ to the right, prove the defining identity on words by induction, and obtain existence and uniqueness for all elements from [F1] and the linear independence of the toral coefficients

1.1givenF1F2construct

For each word $w = F_{j_{1}} \dots F_{j_{m}}$ define $Q_{w}, R_{w} \in U_{q}^{-}$ by recursion on $m$ : for the empty word $Q_{1} : = R_{1} : = 0$ , and for $m \geq 1$ , with $w^{'} = F_{j_{2}} \dots F_{j_{m}}$ , set $Q_{w} : = q_{i}^{a_{i j_{1}}} F_{j_{1}} Q_{w^{'}} + δ_{i j_{1}} w^{'}$ and $R_{w} : = q_{i}^{- a_{i j_{1}}} F_{j_{1}} R_{w^{'}} + δ_{i j_{1}} w^{'}$ . Each is a finite $Q (q)$ -linear combination of words. If $w$ has degree $- ν$ , every nonzero term of $Q_{w}, R_{w}$ has degree $- ν + α_{i}$ : inductively a nonzero $Q_{w^{'}}, R_{w^{'}}$ has degree $\deg w^{'} + α_{i}$ , multiplication by $F_{j_{1}}$ gives $\deg w + α_{i}$ , and the term $δ_{i j_{1}} w^{'}$ has that degree when $j_{1} = i$ .

2.1step 1.1F2F3algebra

For every word $w$ one has $E_{i} w - w E_{i} = (K_{i} Q_{w} - K_{i}^{- 1} R_{w}) / (q_{i} - q_{i}^{- 1})$ . For $m = 0$ both sides vanish. For $m \geq 1$ , write $w = F_{j_{1}} w^{'}$ . Multiplying the inductive identity for $w^{'}$ by $F_{j_{1}}$ on the left gives $F_{j_{1}} E_{i} w^{'} = F_{j_{1}} w^{'} E_{i} + F_{j_{1}} K_{i} Q_{w^{'}} / (q_{i} - q_{i}^{- 1}) - F_{j_{1}} K_{i}^{- 1} R_{w^{'}} / (q_{i} - q_{i}^{- 1})$ , and [F3] gives $E_{i} F_{j_{1}} - F_{j_{1}} E_{i} = δ_{i j_{1}} (K_{i} - K_{i}^{- 1}) / (q_{i} - q_{i}^{- 1})$ . Adding, and using [F2] in the form $F_{j_{1}} K_{i} = q_{i}^{a_{i j_{1}}} K_{i} F_{j_{1}}$ , $F_{j_{1}} K_{i}^{- 1} = q_{i}^{- a_{i j_{1}}} K_{i}^{- 1} F_{j_{1}}$ , yields $E_{i} w - w E_{i} = (K_{i} (q_{i}^{a_{i j_{1}}} F_{j_{1}} Q_{w^{'}} + δ_{i j_{1}} w^{'}) - K_{i}^{- 1} (q_{i}^{- a_{i j_{1}}} F_{j_{1}} R_{w^{'}} + δ_{i j_{1}} w^{'})) / (q_{i} - q_{i}^{- 1}) = (K_{i} Q_{w} - K_{i}^{- 1} R_{w}) / (q_{i} - q_{i}^{- 1})$ , which is the claim.

3.1step 2.1F1algebra

Existence follows for every $P \in U_{q}^{-}$ : by [F1] write $P = \sum_{w} c_{w} w$ as a finite linear combination of words, and set $Q : = \sum_{w} c_{w} Q_{w}$ , $R : = \sum_{w} c_{w} R_{w}$ . Summing the identities of step 2.1 with coefficients $c_{w}$ gives $E_{i} P - P E_{i} = (K_{i} Q - K_{i}^{- 1} R) / (q_{i} - q_{i}^{- 1})$ .

4.1step 3.1F1F2F6algebra

For uniqueness suppose $K_{i} Q - K_{i}^{- 1} R = 0$ with $Q, R \in U_{q}^{-}$ . Decompose $Q = \sum_{ν} Q_{ν}$ , $R = \sum_{ν} R_{ν}$ into the finitely many homogeneous components with $Q_{ν}, R_{ν} \in U_{q}^{-} [- ν]$ (Positive, negative and toral quantum subalgebras and their root gradings). By [F2], $(K_{i} Q_{ν} - K_{i}^{- 1} R_{ν}) K_{i} = q_{i}^{- ν (h_{i})} Q_{ν} K_{i}^{2} - q_{i}^{ν (h_{i})} R_{ν}$ after multiplying on the right by $K_{i}$ ; the terms for distinct $ν$ lie in distinct degrees $- ν$ of the direct sum [F1], so for every $ν$ the identity $q_{i}^{- ν (h_{i})} Q_{ν} \otimes K_{i}^{2} = q_{i}^{ν (h_{i})} R_{ν} \otimes 1$ holds in $U_{q}^{-} [- ν] \otimes U_{q}^{0}$ under the injective multiplication map. Applying $i d \otimes λ$ , where $λ$ is the linear functional on $U_{q}^{0}$ with $λ (K_{2 d_{i} h_{i}}) = 1$ and $λ (K_{h}) = 0$ for $h \neq 2 d_{i} h_{i}$ (existence and well-definedness by the linear independence in [F1]), gives $Q_{ν} = 0$ ; then $R_{ν} = 0$ . Hence $Q = R = 0$ , and two pairs $(Q, R)$ , $(Q^{'}, R^{'})$ giving the same commutator have $Q = Q^{'}$ and $R = R^{'}$ .

5.1step 1.1step 3.1step 4.1F4algebra

Consequently $e_{i}^{'}$ and $e_{i}^{''}$ are well defined on all of $U_{q}^{-}$ ; they are $Q (q)$ -linear because the defining identity is linear in $P$ : if $P = c_{1} P_{1} + c_{2} P_{2}$ , the pairs $(c_{1} Q (P_{1}) + c_{2} Q (P_{2}), c_{1} R (P_{1}) + c_{2} R (P_{2}))$ satisfy the defining identity for $P$ , so by uniqueness each map is linear. The recursion gives $Q_{1} = R_{1} = 0$ , hence $e_{i}^{'} (1) = e_{i}^{''} (1) = 0$ ; and for the one-letter word $w = F_{j}$ the recursion gives $Q_{F_{j}} = R_{F_{j}} = δ_{i j}$ , hence $e_{i}^{'} (F_{j}) = e_{i}^{''} (F_{j}) = δ_{i j}$ .

5.2step 4.1F2algebra

For homogeneous $x, y \in U_{q}^{-}$ decompose $E_{i} (x y) - (x y) E_{i} = (E_{i} x - x E_{i}) y + x (E_{i} y - y E_{i})$ and substitute the defining identities for $x$ and $y$ : the right side equals $(K_{i} e_{i}^{''} (x) y - K_{i}^{- 1} e_{i}^{'} (x) y + x K_{i} e_{i}^{''} (y) - x K_{i}^{- 1} e_{i}^{'} (y)) / (q_{i} - q_{i}^{- 1})$ . By [F2] with $wt (x)$ in place of $β$ , $x K_{i} = K_{i} q_{i}^{- ⟨ wt (x), h_{i} ⟩} x$ and $x K_{i}^{- 1} = K_{i}^{- 1} q_{i}^{⟨ wt (x), h_{i} ⟩} x$ . Collecting the $K_{i}$ - and $K_{i}^{- 1}$ -coefficients and using the uniqueness of step 4.1 yields the two twisted Leibniz rules of (b).

6.1step 1.1step 5.1F5algebra

Grading and nilpotence. The degree computation of step 1.1 shows that for a word $w$ of degree $- ν$ , every nonzero term of $Q_{w}, R_{w}$ has degree $- ν + α_{i}$ . These are negative words, so both coefficients vanish if $ν - α_{i} \notin Q_{+}$ ; otherwise they belong to $U_{q}^{-} [- (ν - α_{i})]$ . By linearity the same holds on every homogeneous component. Thus $e_{i}^{'}, e_{i}^{''}$ raise actual degree by $α_{i}$ and lower positive height by one, and their $k$ -th powers vanish on degree $- ν$ for $k > ht (ν)$ . Every vector has finite homogeneous support, so both maps are locally nilpotent.

6.2step 5.1step 5.2algebra

The commutation relations of (c) are the case $x = F_{j}$ of the Leibniz rules: $wt (F_{j}) = - α_{j}$ , so $q_{i}^{⟨ wt (F_{j}), h_{i} ⟩} = q_{i}^{- a_{i j}}$ and $q_{i}^{- ⟨ wt (F_{j}), h_{i} ⟩} = q_{i}^{a_{i j}}$ , while $e_{i}^{'} (F_{j}) = e_{i}^{''} (F_{j}) = δ_{i j}$ by step 5.1. Reading the resulting identities $e_{i}^{'} (F_{j} y) = q_{i}^{- a_{i j}} F_{j} e_{i}^{'} (y) + δ_{i j} y$ and $e_{i}^{''} (F_{j} y) = q_{i}^{a_{i j}} F_{j} e_{i}^{''} (y) + δ_{i j} y$ as operator identities gives (c).

7.1step 4.1step 5.1step 6.1step 5.2step 6.2F4algebra∎

For the bar relation, apply the semilinear algebra automorphism $ψ$ of [F4] to the defining identity of $P$ : since $ψ (E_{i}) = E_{i}$ , $ψ (K_{i}) = K_{i}^{- 1}$ and $ψ (q_{i} - q_{i}^{- 1}) = - (q_{i} - q_{i}^{- 1})$ , one gets $E_{i} ψ (P) - ψ (P) E_{i} = (K_{i} ψ (e_{i}^{'} (P)) - K_{i}^{- 1} ψ (e_{i}^{''} (P))) / (q_{i} - q_{i}^{- 1})$ . Comparing with the defining identity for $ψ (P)$ , namely $E_{i} ψ (P) - ψ (P) E_{i} = (K_{i} e_{i}^{''} (ψ (P)) - K_{i}^{- 1} e_{i}^{'} (ψ (P))) / (q_{i} - q_{i}^{- 1})$ , uniqueness (step 4.1) gives $e_{i}^{''} (ψ (P)) = ψ (e_{i}^{'} (P))$ and $e_{i}^{'} (ψ (P)) = ψ (e_{i}^{''} (P))$ . Applying the second identity to $ψ (P)$ in place of $P$ and using $ψ^{2} = i d$ gives $e_{i}^{''} (P) = ψ (e_{i}^{'} (ψ (P)))$ , that is, $e_{i}^{''} = ψ e_{i}^{'} ψ$ . This proves (a)–(e) and completes the definition.

Depends on

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Sources

Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390 (standard reference, not scraped)
George Lusztig, Introduction to Quantum Groups (Progress in Mathematics 110, Birkhauser, 1993; second printing 1994; author-hosted scan) (standard reference, not scraped)

Kashiwara's skew derivations ei′ei′​ and ei′′ei′′​ on the negative half