Abstract
Let V denote a finite dimensional vector space over an algebraically closed field. Let U
0, U
1,…, U
d
denote a sequence of nonzero subspaces whose direct sum is V. Let R:V → V and L:V → V denote linear transformations with the following properties: for 0 ≤ i ≤ d, R U
i
⊆ U
i+1 and L U
i
⊆ U
i−1 where U
−1 = 0, U
d+1 = 0; for 0 ≤ i ≤ d/2, the restrictions R
d−2i
|
U
i
: U
i
→ U
d−i
and L
d−2i
|
U
d−i
: U
d−i
→ U
i
are bijections; the maps R and L satisfy the cubic q-Serre relations where q is nonzero and not a root of unity. Let K:V → V denote the linear transformation such that (K − q
2i−d
I)U
i
= 0 for 0 ≤ i ≤ d. We show that there exists a unique -module structure on V such that each of
,
, K − K
0, and K
−1 − K
1 vanish on V, where
are Chevalley generators for
. We determine which
-modules arise from our construction.
2000 Mathematics Subject Classification:
ACKNOWLEDGMENT
I would like to express my gratitude to my thesis advisor Paul Terwilliger for introducing me to this subject and for his many useful suggestions.
Notes
Communicated by K. C. Misra.