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The balanced triple and global basis for the two-dimensional $U_{q} ({s l}_{2})$ -module

Example

In the two-dimensional module $V (ω) = L_{1} (1)$ let $v$ be highest and $w = F v$ , with $E v = 0$ , $F v = w$ , $E w = v$ , $F w = 0$ , $K v = q v$ , $K w = q^{- 1} w$ . Then $V_{A} = A v \oplus A w$ , $L_{0} = A_{0} v \oplus A_{0} w$ , $L_{\infty} = A_{\infty} v \oplus A_{\infty} w$ and $E_{Z} = Z v \oplus Z w$ . The canonical reduction maps $E_{Z}$ isomorphically onto the $Z$ -span of its two crystal classes; its inverse is the integral globalization map, and rational extension identifies the full $Q$ residue with $Q \otimes_{Z} E_{Z}$ . The lower global basis is $v, w$ . The field string image $F V$ is $Q (q) w$ , its integral image ideal inside $V_{A}$ is $A w$ , and its balanced part is $Z w$ .

On $V \otimes V$ the exact quasi- $R$ is $Θ = 1 + (q - q^{- 1}) E \otimes F$ , so $Ψ (v v) = v v, Ψ (v w) = v w, Ψ (w v) = w v + (q - q^{- 1}) v w, Ψ (w w) = w w,$ with concatenation for tensor notation. This preserves total weight and the $A$ -tensor lattice and is a semilinear involution. The actual tensor global basis is $v v, v w, w v + q v w, w w$ ; the raw vector $w v$ is not bar fixed. Its zero crystal lattice is the $A_{0}$ tensor lattice, and its balanced $Z$ -part is the $Z$ -span of the corrected vectors.

For the mixed free lattice $L^{'} = A_{0} (v + w) \oplus q A_{0} v$ with actual infinity chart $ψ (L^{'}) = A_{\infty} (v + w) \oplus q^{- 1} A_{\infty} v$ , the integral balanced part is only $Z (v + w)$ ; it misses the valid residue class of $q v$ . Thus $L^{'}$ is genuinely nonbalanced even though $v + w$ is bar fixed, and $L^{'}$ is not claimed to be a crystal lattice.

Facts & Assumptions

Given: The rank-one module and string formulas of Divided-power commutation and the simple $U_{q_{i}} ({s l}_{2})$ string modules and the $i$ -string decomposition Uniqueness of the $i$ -string decomposition of an integrable module; the balanced triple definitions Balanced triples over the two charts of $P^{1}$ and the globalization map, The balanced-triple linear algebra over $P^{1}$ ; the integral globalization theorem and lower global basis The highest weight and negative half integral crystal triples are balanced, Bar invariance, congruence and uniqueness of the lower global basis, The lower global basis of $U_{q}^{-}$ and of a highest weight module; and the tensor bar results Existence and uniqueness of the weightwise quasi- $R$ intertwiner, Tensor bars on finite products of based integrable modules.

[L1]

Module structure: $V = L_{1} (1)$ has weights $1$ and $- 1$ with one-dimensional weight spaces, $F v = w$ , $E w = v$ , $E v = 0 = F w$ , $K v = q v$ , $K w = q^{- 1} w$ (Divided-power commutation and the simple $U_{q_{i}} ({s l}_{2})$ string modules).

[L2]

Divided powers: in this module $F^{(n)} = 0$ for $n \geq 2$ , $F v = w$ , so the integral lattice generated by divided powers is exactly $V_{A} = A v \oplus A w$ (Uniqueness of the $i$ -string decomposition of an integrable module).

[L3]

Charts and balanced intersection: the crystal lattice is $L_{0} = A_{0} v \oplus A_{0} w$ , its bar image is $L_{\infty} = A_{\infty} v \oplus A_{\infty} w$ , and $A \cap A_{0} = Z [q]$ , $A \cap A_{\infty} = Z [q^{- 1}]$ , $A \cap A_{0} \cap A_{\infty} = Z$ , with $A / q A = 0$ since $q$ is a unit; the chart residues are computed through $Z [q] / q Z [q] = Z$ (Balanced triples over the two charts of $P^{1}$ and the globalization map, The balanced-triple linear algebra over $P^{1}$ , The local ring $A_{0}$ of rational functions regular at $q = 0$ ).

[L4]

Quasi- $R$ : $Θ_{α} = (q - q^{- 1}) E \otimes F$ in degree $α$ , and the tensor bar is $Ψ = Θ (ψ \otimes ψ)$ with $ψ$ fixing $v, w$ and inverting $q$ (Existence and uniqueness of the weightwise quasi- $R$ intertwiner, Tensor bars on finite products of based integrable modules).

Proof

technique · direct

1.1L2L3

The module triple. By [L1] and [L3], $V_{A} \cap L_{0} = Z [q] v \oplus Z [q] w$ , whose quotient by $q (V_{A} \cap L_{0})$ has $Z$ -basis the classes of $v$ and $w$ ; the same two vectors are exactly $E_{Z} = Z v \oplus Z w$ , so the canonical reduction map is an isomorphism and the triple is balanced; its inverse, the integral globalization map, sends the two crystal classes to $v$ and $w$ , respectively. The rational extension has basis the two classes over $Q$ . The lower global basis is $v, w$ , congruent to the crystal classes.

1.2L1L3

The string image. The field string image is $F V = Q (q) w$ ; the integral image ideal is $F V \cap V_{A} = A w$ , whose balanced part is $A w \cap L_{0} \cap L_{\infty} = Z w$ . The three spaces are genuinely different, as displayed.

1.3L4

The tensor bar. Applying [L4]: $E \otimes F$ kills $v v, v w, w w$ since $E v = 0$ and $F w = 0$ , and sends $w v$ to $E w \otimes F v = v \otimes w = v w$ ; hence the four displayed formulas. $Ψ$ preserves total weight because $E \otimes F$ does, preserves the $A$ -tensor lattice because $Θ = 1 + (q - q^{- 1}) E \otimes F$ has $A$ -coefficients, and is involutive because $(q - q^{- 1}) + \overline{(q - q^{- 1})} = 0$ and $(E \otimes F)^{2} = 0$ on this module.

2.1L4step 1.3

The corrected tensor global basis. The bar-fixed combinations are $v v, v w, w w$ and $w v + c v w$ with $c \in q Z [q]$ satisfying $c - \overset{ˉ}{c} = q - q^{- 1}$ . Comparing positive Laurent coefficients gives uniquely $c = q$ , and directly $Ψ (w v + q v w) = w v + (q - q^{- 1}) v w + q^{- 1} v w = w v + q v w$ ; the transition matrix from the raw basis is unitriangular with off-diagonal $q$ , so the corrected family $v v, v w, w v + q v w, w w$ is an $A$ -basis congruent to the raw tensor basis modulo $q$ , with zero crystal lattice the $A_{0}$ -tensor lattice and balanced $Z$ -part its $Z$ -span; the raw $w v$ is not bar fixed.

2.2L3step 1.2

The nonbalanced mixed lattice. For $L^{'} = A_{0} (v + w) \oplus q A_{0} v$ , an element $α (v + w) + β v$ with $α \in A_{0}$ , $β \in q A_{0}$ lies in $V_{A}$ exactly when $α \in Z [q]$ and $β \in q Z [q]$ ; lying in $ψ (L^{'}) = A_{\infty} (v + w) \oplus q^{- 1} A_{\infty} v$ forces in addition $α \in Z [q^{- 1}]$ and $β \in q^{- 1} Z [q^{- 1}]$ with Laurent degree bounds, so $α \in Z$ and $β = 0$ : the balanced part is exactly $Z (v + w)$ . The integral zero residue contains the independent class of $q v$ , which is not in the image of $Z (v + w)$ , so $L^{'}$ is not balanced; the bar-fixed vector $v + w$ alone does not repair this, and since $\tilde{e} (v + w) = v \notin L^{'}$ the lattice is not crystal stable either.

3.1step 1.2step 2.1step 2.2∎

Conclusion. Steps 1.1–1.2 compute the rank-one balanced triple and its string-image strata; steps 1.3–2.1 compute the exact quasi- $R$ action, the corrected tensor global basis and its lattice; step 2.2 exhibits the nonbalanced mixed lattice while distinguishing $v + w$ from the undefined class of $v$ . All computations are finite, and the zero quotient $A / q A = 0$ is never used for residues.

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Sources

Masaki Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Mathematical Journal 63 (1991), 465–516 (standard reference, not scraped)
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)

The balanced triple and global basis for the two-dimensional Uq(sl2)Uq​(sl2​)-module