Abstract
This paper presents a $q$-analogue of an extension of the tensor algebra given by the same author. This new algebra naturally contains the ordinary tensor algebra and the Iwahori-Hecke algebra type $A$ of infinite degree. Namely this algebra can be regarded as a natural mix of these two algebras. Moreover, we can consider natural "derivations" on this algebra. Using these derivations, we can easily prove the $q$-Schur-Weyl duality (the duality between the quantum enveloping algebra of the general linear Lie algebra and the Iwahori-Hecke algebra of type $A$).
Highlights
This paper presents a q-analogue of an extension of the tensor algebra given in [4]
We denote by π the natural representation of the quantum enveloping algebra Uq (gl(V )) on V ⊗p
This fact is proved quickly as follows
Summary
This paper presents a q-analogue of an extension of the tensor algebra given in [4] Using this algebra, we can prove the q-Schur–Weyl duality (the duality between the quantum enveloping algebra Uq (gln) and the Iwahori–Hecke algebra of type A). Let us recall the algebra T (V ) given in [4] This algebra T (V ) naturally contains the ordinary tensor algebra T (V ) and the infinite symmetric group S∞. We can consider natural “derivations” on this algebra, which satisfy an analogue of canonical commutation relations. This algebra and these derivations are useful to study representations on the tensor algebra.
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