A (1) A Note on U q -Demazure Crystals n 1 Maggie Rahmoeller - - PowerPoint PPT Presentation

Preliminaries Crystals Path Realization

A Note on Uq

• A(1)

n−1

• Demazure Crystals

Maggie Rahmoeller

Roanoke College

June 4, 2018 QAA Conference

A Note on Uq

• A(1)

n−1

• Demazure Crystals

Maggie Rahmoeller

Preliminaries Crystals Path Realization

Motivation

1990’s: Kashiwara and Lusztig developed crystal base theory This theory provides a combinatorial tool to study Lie algebra representation theory Applications arise in statistical physics, conformal field theory, differential equations, number theory, combinatorics, and algebraic geometry

A Note on Uq

• A(1)

n−1

• Demazure Crystals

Maggie Rahmoeller

Preliminaries Crystals Path Realization

Affine Special Linear Lie Algebras

We focus on the affine special linear Lie algebra: A(1)

n−1 = ˆ

sl(n, C) = An−1 ⊗ C[t, t−1] ⊕ Cc ⊕ Cd, which has the following bracket structure: [x ⊗ ti, y ⊗ tj] = [x, y] ⊗ ti+j + tr(x, y)iδi+j,0c, [d, x ⊗ ti] = i(x ⊗ ti), [d, c] = 0.

A Note on Uq

• A(1)

n−1

• Demazure Crystals

Maggie Rahmoeller

Preliminaries Crystals Path Realization

Quantum Affine Algebras

Definition The quantum affine algebra Uq(g) is the associative alge- bra over C(q) with unity generated by the elements ei, fi, qh

• h ∈ ˇ

P

• with the following relations:

1 q0 = 1, qhqh′ = qh+h′ for h, h′ ∈ ˇ

P,

2 qheiq−h = qαi(h)ei for h ∈ ˇ

P,

3 qhfiq−h = q−αi(h)fi for h ∈ ˇ

P,

4 eifj − fjei = δij qsi hi −q−si hi qsi −q−si , 5 1−aij

• k=0

(−1)k 1 − aij k

• qsi

e1−aij−k

i

ejek

i = 0, for i = j, 6 1−aij

• k=0

(−1)k 1 − aij k

• qsi

f 1−aij−k

i

fjf k

i = 0, for i = j.

A Note on Uq

• A(1)

n−1

• Demazure Crystals

Maggie Rahmoeller

Preliminaries Crystals Path Realization

Crystal Lattices

Definition Define A0 = g(q) h(q)

• g(q), h(q) ∈ C[q], h(0) = 0
• to be the

principal ideal domain with C(q) as its field of quotients. Definition A free A0-submodule L of integrable g-module V q is a crystal lattice if

1 C(q) ⊗A0 L ∼

= V q,

2 L = λ∈P Lλ, Lλ = L ∩ V q λ , 3 ˜

ei(L) ⊆ L, ˜ fi(L) ⊆ L for all i = 0, 1, . . . , n.

A Note on Uq

• A(1)

n−1

• Demazure Crystals

Maggie Rahmoeller

Preliminaries Crystals Path Realization

Crystal Bases

Definition A crystal base for V q is a pair (L, B) such that

1 L is a crystal lattice, 2 B is a C-basis of L/qL, 3 B = ∪λ∈PBλ, Bλ = B ∩ (Lλ/qLλ), 4 ˜

ei(B) ⊆ B ∪ {0}, ˜ fi(B) ⊆ B ∪ {0},

5 For b, b′ ∈ B, ˜

fib = b′ if and only if ˜ eib′ = b.

A Note on Uq

• A(1)

n−1

• Demazure Crystals

Maggie Rahmoeller

Preliminaries Crystals Path Realization

Crystal Graphs

Definition Given a crystal base (L, B) for V q, we can define a crystal graph of V q by letting the elements of B be the set of vertices and by joining b ∈ B to b′ ∈ B with an i-colored arrow b

i

→ b′ if and only if ˜ fib = b′.

A Note on Uq

• A(1)

n−1

• Demazure Crystals

Maggie Rahmoeller

Preliminaries Crystals Path Realization

Perfect Crystals

Although the crystal B(λ) for the irreducible highest weight Uq(A(1)

n−1)-module V q(λ) is infinite, it can be realized by a finite

crystal called a perfect crystal. Suppose λ(c) = ℓ ≥ 1. By [KKMMNN], there exists a finite crystal Bℓ called a perfect crystal of level ℓ such that B(λ) ∼ = · · · ⊗ Bℓ ⊗ Bℓ ⊗ Bℓ.

A Note on Uq

• A(1)

n−1

• Demazure Crystals

Maggie Rahmoeller

Preliminaries Crystals Path Realization

Demazure Modules

For a dominant integral weight λ, consider the unique irreducible integrable highest weight Uq(A(1)

n−1)-module V q(λ).

Definition For any w ∈ W, the extremal weight space V q(λ)wλ is one- dimensional with basis vector uwλ, which is called the ex- tremal vector. Definition For w ∈ W and weight space V q(λ)wλ = C(q)uwλ, the Demazure module is Vw(λ) = Uq

• A(1)

n−1

+ uwλ, where Uq(A(1)

n−1)+ is the subalgebra generated by the ei’s.

A Note on Uq

• A(1)

n−1

• Demazure Crystals

Maggie Rahmoeller

Preliminaries Crystals Path Realization

Demazure Crystals

Definition For each w ∈ W, the Demazure module Vw(λ) has a crystal Bw(λ), which we call the Demazure crystal. Kashiwara, in 1993, proved that the Demazure crystal is a subset

• f the crystal for the associated integrable highest weight module

[5]. He also showed that the Demazure crystal has the following recursive property: w ≺ riw ⇒ Briw(λ) =

• m≥0

˜ fi

mBw(λ) \ {0}.

A Note on Uq

• A(1)

n−1

• Demazure Crystals

Maggie Rahmoeller

Preliminaries Crystals Path Realization

Perfect Crystals

For A(1)

n−1 and ℓ ≥ 1, consider the perfect crystal:

Bℓ =

• m1, m2, . . . , mn−1, m0) ∈ Zn

≥0

• n−1
• i=0

mi = ℓ

• .

The Kashiwara operators act on b ∈ Bℓ by the following actions: ˜ f0(b) = (m1 + 1, m2, . . . , mn−1, m0 − 1) , ˜ fi(b) = (m1, . . . , mi − 1, mi+1 + 1, . . . , mn−1, m0) . We also define the following: ϕi(b) = mi, ϕ0(b) = m0, ϕ(b) =

n−1

• i=0

ϕi(b)Λi εi(b) = mi+1, εn(b) = m0, ε(b) =

n−1

• i=0

εi(b)Λi.

A Note on Uq

• A(1)

n−1

• Demazure Crystals

Maggie Rahmoeller

Preliminaries Crystals Path Realization

Perfect Crystals

Perfect crystal for Uq

• A(1)

2

• f level 2: λ = 2Λ0

(2,0,0) (1,0,1) (0,0,2) (1,1,0) (0,1,1) (0,2,0) 2 2 2 1 1 1

A Note on Uq

• A(1)

n−1

• Demazure Crystals

Maggie Rahmoeller

Preliminaries Crystals Path Realization

Path Realizations

Definition For fixed λ, let bλ be the unique element of B such that ϕ (bλ) = λ. Then set λ1 = λ, λk+1 = ε (bλk) b1 = bλ, bk+1 = bλk+1 The sequence

pλ = (· · · ⊗ bk+1 ⊗ · · · ⊗ b2 ⊗ b1)

is called the ground-state path. λ = 2Λ0

(2,0,0) (1,0,1) (0,0,2) (1,1,0) (0,1,1) (0,2,0) 2 2 2 1 1 1

Example:

A Note on Uq

• A(1)

n−1

• Demazure Crystals

Maggie Rahmoeller

Preliminaries Crystals Path Realization

Path Realizations

Definition For fixed λ, let bλ be the unique element of B such that ϕ (bλ) = λ. Then set λ1 = λ, λk+1 = ε (bλk) b1 = bλ, bk+1 = bλk+1 The sequence

pλ = (· · · ⊗ bk+1 ⊗ · · · ⊗ b2 ⊗ b1)

is called the ground-state path. λ = 2Λ0

(2,0,0) (1,0,1) (0,0,2) (1,1,0) (0,1,1) (0,2,0) 2 2 2 1 1 1

Example: (0, 0, 2)

A Note on Uq

• A(1)

n−1

• Demazure Crystals

Maggie Rahmoeller

Preliminaries Crystals Path Realization

Path Realizations

Definition For fixed λ, let bλ be the unique element of B such that ϕ (bλ) = λ. Then set λ1 = λ, λk+1 = ε (bλk) b1 = bλ, bk+1 = bλk+1 The sequence

pλ = (· · · ⊗ bk+1 ⊗ · · · ⊗ b2 ⊗ b1)

is called the ground-state path. λ = 2Λ0

(2,0,0) (1,0,1) (0,0,2) (1,1,0) (0,1,1) (0,2,0) 2 2 2 1 1 1

Example: (0, 2, 0) ⊗ (0, 0, 2)

A Note on Uq

• A(1)

n−1

• Demazure Crystals

Maggie Rahmoeller

Preliminaries Crystals Path Realization

Path Realizations

Definition For fixed λ, let bλ be the unique element of B such that ϕ (bλ) = λ. Then set λ1 = λ, λk+1 = ε (bλk) b1 = bλ, bk+1 = bλk+1 The sequence

pλ = (· · · ⊗ bk+1 ⊗ · · · ⊗ b2 ⊗ b1)

is called the ground-state path. λ = 2Λ0

(2,0,0) (1,0,1) (0,0,2) (1,1,0) (0,1,1) (0,2,0) 2 2 2 1 1 1

Example: (2, 0, 0) ⊗ (0, 2, 0) ⊗ (0, 0, 2)

A Note on Uq

• A(1)

n−1

• Demazure Crystals

Maggie Rahmoeller

Preliminaries Crystals Path Realization

Path Realizations

Definition For fixed λ, let bλ be the unique element of B such that ϕ (bλ) = λ. Then set λ1 = λ, λk+1 = ε (bλk) b1 = bλ, bk+1 = bλk+1 The sequence

pλ = (· · · ⊗ bk+1 ⊗ · · · ⊗ b2 ⊗ b1)

is called the ground-state path. λ = 2Λ0

(2,0,0) (1,0,1) (0,0,2) (1,1,0) (0,1,1) (0,2,0) 2 2 2 1 1 1

Example: · · · ⊗ (0, 0, 2) ⊗ (2, 0, 0) ⊗ (0, 2, 0) ⊗ (0, 0, 2)

A Note on Uq

• A(1)

n−1

• Demazure Crystals

Maggie Rahmoeller

Preliminaries Crystals Path Realization

Path Realizations

Definition A sequence p = (· · · ⊗ p(k + 1) ⊗ p(k) ⊗ · · · ⊗ p(2) ⊗ p(1)) is called a λ-path if p(k) = bk for k ≫ 1. Example: λ-path: · · · ⊗ (0, 0, 2) ⊗ (2, 0, 0) ⊗ (1, 1, 0) ⊗ (0, 1, 1) We can use λ-paths to find a realization of the affine crystal graph

• f V (λ) and hence for the Demazure crystal.

A Note on Uq

• A(1)

n−1

• Demazure Crystals

Maggie Rahmoeller

Preliminaries Crystals Path Realization

Main Theorem [Misra, R.]

Theorem Choose dominant integral weight λ = m0Λ0 + m1Λ1 + · · · + mn−1Λn−1. Then

t≥0

Bw(L,t)(λ) = · · · ⊗ bL ⊗ B⊗(L−1)

• t≥0

Bw(L,t)(λ) = · · · ⊗ bL+1 ⊗ B⊗(L).

A Note on Uq

• A(1)

n−1

• Demazure Crystals

Maggie Rahmoeller

Preliminaries Crystals Path Realization

References [1] J. Hong, S.-J. Kang, Introduction to Quantum Groups and Crystal Bases, Volume 42 of Graduate Studies in Mathematics. American Mathematical Society, 2002. [2] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Volume 9 of Graduate Texts in Mathematics. Springer-Verlang, 1972. [3] V. G. Kac, Infinite-dimensional Lie algebras, Third edition, Cambridge, 1990. [4] S-J. Kang, M. Kashiwara, K.C. Misra, T. Miwa, T. Nakashima, and

• A. Nakayashiki, “Affine crystals and vertex models”, Int. J. Mod.
• Phys. 7 (suppl. 1A) (1992), 449-484.

[5] M. Kashiwara, ”The crystal base and Littelmann’s refined Demazure character formula”. Duke Math. J. 71 (1993), 839-858. [6] A. Kuniba, K. C. Misra, M. Okado and J. Uchiyama, “Demazure modules and perfect crystals”, Commun. Math Phys. 192 (1998), 555-567. [7] A. Kuniba, K. C. Misra, M. Okado, T. Takagi and J. Uchiyama, “Crystals for Demazure Modules of Classical Affine Lie Algebras”, J. Algebra 208 (1998), 185-215.

A Note on Uq

• A(1)

n−1

• Demazure Crystals

Maggie Rahmoeller