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Arakawa-Suzuki Casimir operators on tensor functors

Definition

Work over $C$ (Classical complex matrix Lie algebras, Algebras over a commutative ring, central structure maps, and algebra homomorphisms). Fix $N \geq 1$ , let $g = {g l}_{N} (C)$ with its matrix units $E_{i j}$ and the trace form $B (X, Y) = tr (X Y)$ , which is symmetric, nondegenerate and invariant (Trace forms are symmetric and invariant), and let $z : = \sum_{i = 1}^{N} E_{i i} \in g$ be the central element (the identity matrix). The dual basis of $(E_{i j})$ under $B$ is $(E_{j i})$ , so the invariant Casimir tensor is $Ω = \sum_{i, j = 1}^{N} E_{i j} \otimes E_{j i} \in U (g) \otimes U (g),$ Its action on a tensor product of two $g$ -modules commutes with the diagonal action, as computed in [F2]. Multiplying its two factors gives the central Casimir element of Casimir operator relative to an invariant form.

Let $M$ be a $g$ -module and $V = C^{N}$ the natural module (Representations of Lie algebras). On the tensor product $M \otimes V^{\otimes d}$ (The tensor product $M \otimes_{R} N$ from the additive group underlying the free $Z$ -module on $M \times N$ , elementary tensors, and finite tensor sums, Associativity of tensor products for compatible bimodules) label the factors $0, 1, \dots, d$ , with factor $0$ equal to $M$ . For $0 \leq a \neq b \leq d$ let $Ω_{a b}$ denote the operator obtained by applying $Ω$ in the two labelled factors, and for distinct $1 \leq a, b \leq d$ let $P_{a b}$ denote the operator that interchanges the two labelled factors (the flip of $V \otimes V$ on two $V$ -factors). Define, for $1 \leq j \leq d$ and $1 \leq i \leq d - 1$ , $x_{j} : = Ω_{0 j} + \sum_{k < j} P_{k j}, s_{i} : = P_{i, i + 1} .$ Finally put $Ω^{s l} : = Ω - \frac{1}{N} z \otimes z$ and define the Arakawa-Suzuki normalized operators $y_{j} : = x_{j} - \frac{1}{N} z_{M} + \frac{N - 1}{2},$ where $z_{M}$ denotes the endomorphism of $M \otimes V^{\otimes d}$ given by the action of $z$ on the factor $M$ . This is a central endomorphism, and is a scalar only when $z$ acts scalarly on $M$ .

The verification below records the properties of these operators used on this page: $Ω$ acts on $V \otimes V$ as the flip; the operators $x_{j}$ and $s_{i}$ are natural in $M$ and commute with the $g$ -action; the recursion $x_{j + 1} = s_{j} x_{j} s_{j} + s_{j}$ holds; $Ω^{s l}$ is the Casimir of ${s l}_{N}$ relative to the restricted trace form; and the operators $y_{j}$ are exactly the source operators of [AS, (2.2.2)]. For $d = 0$ the family $(x_{j})$ is empty; for the trivial module $M = C$ one has $Ω_{0 j} = 0$ , so $x_{j} = \sum_{k < j} P_{k j}$ and $x_{1} = 0$ .

Facts & Assumptions

Given: $N \geq 1$ , $g = {g l}_{N} (C)$ with matrix units $E_{i j}$ and trace form $B$ , the central element $z = \sum_{i} E_{i i}$ , a $g$ -module $M$ , the natural module $V = C^{N}$ , and $d \geq 0$ .

[F1]

Matrix units multiply by $E_{i j} E_{k l} = δ_{j k} E_{i l}$ , $B (E_{i j}, E_{k l}) = δ_{i l} δ_{j k}$ , and $B$ is symmetric and invariant; $z$ is central, so its action $z_{M}$ on any module is an endomorphism commuting with the action (Classical complex matrix Lie algebras, Trace forms are symmetric and invariant).

[F2]

The Casimir tensor $Ω = \sum_{i j} E_{i j} \otimes E_{j i} \in U (g) \otimes U (g)$ is independent of dual bases and invariant under the diagonal adjoint action: $[X \otimes 1 + 1 \otimes X, Ω] = 0$ for every $X \in g$ . Indeed, for $X = E_{a b}$ the matrix-unit relations give the cancelling sum $\sum_{j} E_{a j} \otimes E_{j b} - \sum_{i} E_{i b} \otimes E_{a i} + \sum_{i} E_{i b} \otimes E_{a i} - \sum_{j} E_{a j} \otimes E_{j b} = 0$ , and linearity gives every $X$ . Therefore its action on a tensor product of $g$ -modules is a $g$ -intertwiner. Module actions extend uniquely to $U (g)$ (Casimir operator relative to an invariant form, The Casimir operator is basis-independent and intertwining, The quadratic Casimir element is independent of the choice of dual bases, Universal enveloping algebra).

[F3]

Tensor products of modules are modules with the factorwise action, associative up to canonical isomorphism, and pure-tensor bases multiply to tensor bases (Representations of Lie algebras, The tensor product $M \otimes_{R} N$ from the additive group underlying the free $Z$ -module on $M \times N$ , elementary tensors, and finite tensor sums, Associativity of tensor products for compatible bimodules, The elementary tensors of two bases form the product basis of the tensor product).

[F4]

The degenerate affine Hecke algebra $H_{d}$ is presented by $x_{1}, \dots, x_{d}, s_{1}, \dots, s_{d - 1}$ with the Coxeter relations and the cross relations $s_{i} x_{i + 1} = x_{i} s_{i} + 1$ , $x_{i + 1} s_{i} = s_{i} x_{i} + 1$ ; the recursion $x_{j + 1} = s_{j} x_{j} s_{j} + s_{j}$ holds in $H_{d}$ (The degenerate affine Hecke algebra).

[F5]

The source operators: [AS, (2.2.1)-(2.2.2)] define, on $X \otimes V^{\otimes ℓ}$ for the ${s l}_{N}$ Casimir, the transposition operators $s_{i j} = Ω_{i j} + 1 / N$ and the generators $y_{i} = Ω_{0 i} + \sum_{1 \leq j < i} (Ω_{j i} + 1 / N) + (N - 1) / 2$ .

Proof

technique · direct

Given: The data of [F1]-[F5] and the operators of the Definition.

1.1F1F3algebra

$Ω$ is the flip on $V \otimes V$ . On the tensor basis, $Ω (v_{a} \otimes v_{b}) = \sum_{i, j} (E_{i j} v_{a}) \otimes (E_{j i} v_{b}) = \sum_{i, j} δ_{j a} v_{i} \otimes δ_{i b} v_{j} = v_{b} \otimes v_{a} = P_{01} (v_{a} \otimes v_{b})$ by [F1]. Consequently $Ω_{a b} = P_{a b}$ whenever the two labelled factors $a, b$ are both natural factors; in particular $Ω_{j, j + 1} = s_{j}$ on the $V$ -factors.

1.2F2F3algebra

Naturality and commutation with $g$ . For a $g$ -module map $f ⁣ : M \to M^{'}$ , the map $f \otimes 1^{\otimes d}$ intertwines $x_{j}$ and $s_{i}$ : it plainly intertwines the place permutations $P_{a b}$ , and it intertwines $Ω_{0 j}$ because the Casimir operator on a tensor product of modules is an intertwiner by [F2]. For the same reason each $x_{j}$ and $s_{i}$ commutes with the $g$ -action on $M \otimes V^{\otimes d}$ , the action on the $V$ -factors being the natural one.

1.3F1F2algebra

The ${s l}_{N}$ Casimir. The trace form restricts to a nondegenerate form on ${s l}_{N}$ : ${s l}_{N} = {X \in g : B (X, z) = 0}$ because $B (X, z) = tr (X)$ and $B (z, z) = N \neq 0$ . Hence $Ω^{s l} = Ω - \frac{1}{N} z \otimes z$ is the inverse tensor of $B ∣_{{s l}_{N}}$ : for $X \in {s l}_{N}$ , contracting the second factor of $Ω$ with $X$ gives $\sum_{i j} B (X, E_{j i}) E_{i j} = X$ , while contracting the second factor of $z \otimes z$ with $X$ gives $B (X, z) z = tr (X) z = 0$ ; so contraction with $Ω^{s l}$ also returns $X$ , and by [F2] the dual tensor is unique. Thus $Ω^{s l}$ is the ${s l}_{N}$ -Casimir for the restricted trace form.

2.1F4step 1.1algebra

The recursion. Since $s_{j} = P_{j, j + 1}$ and $P_{a b}$ are involutions with $P_{a b} Ω_{a b} P_{a b} = Ω_{a b}$ and conjugation by $P_{j, j + 1}$ replaces the label $j$ by $j + 1$ in every operator supported on labels ${0, 1, \dots, j}$ , one has $s_{j} Ω_{0 j} s_{j} = Ω_{0, j + 1}$ , $s_{j} P_{k j} s_{j} = P_{k, j + 1}$ for $k < j$ , and $s_{j} s_{j} = 1$ . Hence $s_{j} x_{j} s_{j} + s_{j} = Ω_{0, j + 1} + \sum_{k < j} P_{k, j + 1} + P_{j, j + 1} = x_{j + 1}$ , using $x_{j + 1} = Ω_{0, j + 1} + \sum_{k < j + 1} P_{k, j + 1}$ and $P_{j, j + 1} = s_{j}$ . Thus the operators satisfy the recursion of [F4]. The remaining Hecke relations require the independent commutator check in The Casimir and place-permutation operators satisfy the degenerate affine Hecke relations ↗.

2.2F5step 1.1step 1.3algebra

The source dictionary. Let $Ω$ now denote the ${s l}_{N}$ -Casimir in the formula (2.2.1) of the source [F5], so that on two natural factors $Ω_{i j} = P_{i j} - 1 / N$ by steps 1.1 and 1.3, i.e. $P_{i j} = Ω_{i j} + 1 / N$ , which is exactly [AS, (2.2.1)]. On $M \otimes V^{\otimes d}$ the library ${g l}_{N}$ -Casimir satisfies $Ω_{0 j}^{{g l}_{N}} = Ω_{0 j}^{s l} + z_{M} / N$ (the extra term $\frac{1}{N} z \otimes z$ acts on factor $0$ as $z_{M} / N$ and on factor $j$ as the identity on $V$ ), while $Ω_{k j}^{{g l}_{N}} = Ω_{k j}^{s l} + 1 / N$ for two natural factors; therefore $y_{j} = x_{j} - \frac{1}{N} z_{M} + \frac{N - 1}{2} = Ω_{0 j}^{s l} + \sum_{k < j} (Ω_{k j}^{s l} + \frac{1}{N}) + \frac{N - 1}{2}$ , which is precisely the source formula (2.2.2) in [F5].

3.1F1F3given∎

Special cases. For $d = 0$ there are no labels $j \geq 1$ , so the family $(x_{j})$ and $(s_{i})$ is empty and the Definition records only the module $M$ ; the recursion and the dictionary are vacuous. For the trivial module $M = C$ the action of $g$ is zero, so $Ω_{0 j} = 0$ and $z_{M} = 0$ ; hence $x_{j} = \sum_{k < j} P_{k j}$ , with the empty sum $x_{1} = 0$ .

Remarks

The operators $x_{j}$ are the universal (trace-form) Casimir operators; the operators $y_{j}$ differ from them by the central endomorphism $z_{M} / N - (N - 1) / 2$ recorded in the Definition, and this dictionary is the one used whenever the source normalization [AS] is compared with the tensor-space computations of this page.
The sign of the shift in the recursion is fixed by the cross relation $s_{i} x_{i + 1} = x_{i} s_{i} + 1$ of The degenerate affine Hecke algebra: the computation of step 2.1 gives $x_{j + 1} = s_{j} x_{j} s_{j} + s_{j}$ , whose right-hand side contains $+ s_{j}$ , not $- s_{j}$ .

Depends on

Used by

Dependency tree · two levels

51 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Tomoyuki Arakawa and Takeshi Suzuki, Lie algebras and degenerate affine Hecke algebras of type A, J. Algebra 209 (1998) 288-304 (complete 16-page arXiv preprint of the origin paper) (standard reference, not scraped)
Jonathan Brundan and Alexander Kleshchev, Schur-Weyl duality for higher levels, Selecta Math. (N.S.) 14 (2008) 1-57 (complete 50-page arXiv article) (standard reference, not scraped)