Alphabeta MathQuantum Affine Type A and Level-One Fock Space
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1 · Prerequisites
Abstract Algebra
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2 · Logical flowchart
3 · Items
Definition
The cyclic affine type A Cartan datum and its residues
Definition
The level-zero vector representation, its affinization and the level-one crystal energy
Definition
The affine Hecke algebra and its commuting action on tensor powers of the vector representation
Lemma
Integrable rank-one actions force the quantum Serre sums
Lemma
The q-antisymmetrizer splits the tensor power and its kernel is the sum of the Hecke kernels
Lemma
The vector representation has a perfect crystal of level one and a lower global basis
Definition
The finite q-wedge space and the normal-ordering relations
Lemma
The q-wedge relations coincide with the type-A energy braiding kernel
Theorem
Normally ordered q-wedges form a basis for every finite wedge length
Definition
The semi-infinite q-wedge Fock space and its q-adic topology
Lemma
Tail estimates: q-adic vanishing against the vacuum and finiteness of the weight strings
Lemma
Semi-infinite wedges are indexed by partitions and the action gives the node formulas
Definition
The level-one partition Fock space and node residues
Lemma
The addable-minus-removable node count is the level-one Cartan pairing
Definition
The node order and the signed exponents Nᵢˡ and Nᵢʳ
Lemma
Fixed-residue node cubes realize the actual Fock strings
Definition
Good and cogood nodes by residue-signature reduction
Theorem
The Fock space node formulas satisfy the quantum affine relations
Proposition
The empty partition generates an integrable highest weight submodule of highest weight Λ₀
Theorem
The crystal of the level-one Fock space is the good-node partition graph
Theorem
The semi-infinite action on the Fock space is well defined and the Fock space is an integrable U_q(slₑ)-module
Theorem
The empty component is the highest weight crystal B(Λ₀)
Definition
The regular-partition label convention and the Mullineux crystal involution
Theorem
The basic representation global basis is unitriangular in the partition basis
Sources
- Masaki Kashiwara, Tetsuji Miwa and Eugene Stern, Decomposition of q-deformed Fock spaces, arXiv:q-alg/9508006
- Masaki Kashiwara, Tetsuji Miwa, Jens-Ulrik H. Petersen and Chong Ming Yung, Perfect crystals and q-deformed Fock spaces, arXiv:q-alg/9603025
- Bernard Leclerc, Fock space representations (author-hosted complete 38-page treatment)
- Matthew Fayers, Partition models for the crystal of the basic $U_q(\mathfrak{sl}_n)$-module, arXiv:0906.4129
- Eugene Stern, Semi-infinite wedges and vertex operators, arXiv:q-alg/9504013
- Kashiwara-Miwa-Petersen-Yung, Perfect crystals and q-deformed Fock spaces
- Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390
- Kailash C. Misra and Tetsuji Miwa, Crystal base for the basic representation of $U_q(\mathfrak{sl}(n))$, Comm. Math. Phys. 134 (1990), 79-88
- Bernard Leclerc and Jean-Yves Thibon, Canonical bases of q-deformed Fock spaces, arXiv:q-alg/9602025
- Masaki Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras