TY - JOUR
T1 - Finite-dimensional modules of the universal Racah algebra and the universal additive DAHA of type (C1∨,C1)
AU - Huang, Hau Wen
N1 - Publisher Copyright:
© 2020 Elsevier B.V.
PY - 2021/8
Y1 - 2021/8
N2 - Assume that F is an algebraically closed field with characteristic zero. The universal Racah algebra ℜ is a unital associative F-algebra defined by generators and relations. The generators are A,B,C,D and the relations state that [A,B]=[B,C]=[C,A]=2D and each of [A,D]+AC−BA,[B,D]+BA−CB,[C,D]+CB−AC is central in ℜ. The universal additive DAHA (double affine Hecke algebra) H of type (C1∨,C1) is a unital associative F-algebra generated by t0,t1,t0∨,t1∨ and the relations state that t0+t1+t0∨+t1∨=−1 and each of t02,t12,t0∨2,t1∨2 is central in H. Each H-module is an ℜ-module by pulling back via the algebra homomorphism ℜ→H given by [Formula presented] Let V denote any finite-dimensional irreducible H-module. The set of ℜ-submodules of V forms a lattice under the inclusion partial order. We classify the lattices that arise by this construction. As a consequence, the ℜ-module V is completely reducible if and only if t0 is diagonalizable on V.
AB - Assume that F is an algebraically closed field with characteristic zero. The universal Racah algebra ℜ is a unital associative F-algebra defined by generators and relations. The generators are A,B,C,D and the relations state that [A,B]=[B,C]=[C,A]=2D and each of [A,D]+AC−BA,[B,D]+BA−CB,[C,D]+CB−AC is central in ℜ. The universal additive DAHA (double affine Hecke algebra) H of type (C1∨,C1) is a unital associative F-algebra generated by t0,t1,t0∨,t1∨ and the relations state that t0+t1+t0∨+t1∨=−1 and each of t02,t12,t0∨2,t1∨2 is central in H. Each H-module is an ℜ-module by pulling back via the algebra homomorphism ℜ→H given by [Formula presented] Let V denote any finite-dimensional irreducible H-module. The set of ℜ-submodules of V forms a lattice under the inclusion partial order. We classify the lattices that arise by this construction. As a consequence, the ℜ-module V is completely reducible if and only if t0 is diagonalizable on V.
KW - Additive DAHA
KW - Irreducible modules
KW - Racah algebras
UR - http://www.scopus.com/inward/record.url?scp=85099514388&partnerID=8YFLogxK
U2 - 10.1016/j.jpaa.2020.106653
DO - 10.1016/j.jpaa.2020.106653
M3 - 期刊論文
AN - SCOPUS:85099514388
SN - 0022-4049
VL - 225
JO - Journal of Pure and Applied Algebra
JF - Journal of Pure and Applied Algebra
IS - 8
M1 - 106653
ER -