What are the relations between the notation in Lusztig's book introduction to quantum groups and the usual notation about quantum groups. For example, v in Lusztig's book corresponds to the usual q. Are E,F,K in Lusztig's book the same as the usual ones? I think ′U and U are quantum groups. What are the algebras ′f and f used for? What are the relations between f and U? Thank you.
1 Answer
Yes, the E, F, and K are standard generators of Uν(sl2). Sometimes q=ν−1.
I don't know what is standard. More generally, the standard (Chevalley) generators for Uν are Ei,Fi,Ki (i∈I).
The algebra f (generated by θi,i∈I, say) is isomorphic (as an algebra) to the algebra U− generated by the Fi. However, U−ν is not a co-subalgebra of U with respect to the coproduct Δ(Ki)=Ki⊗Ki, Δ(Ei)=Ki⊗Ei+Ei⊗1, Δ(Fi)=K−1i⊗Fi+Fi⊗1.
The algebra f is a co-algebra with respect to the comultiplication δ(θi)=1⊗θi+θi⊗1. However, it is not a bialgebra (that is, comultiplication δ:f→f⊗f is not an algebra homomorphism) unless we equip f⊗f with a twisted multiplication: (x1⊗x2)(y1⊗y2)=ν−(|x2|,|y1|)x1y1⊗x2y2.
To explain the notation above, associated to Uν is a Cartan matrix A=(aij)i,j∈I, a root system Φ with simple roots Π={αi|i∈I}, and bilinear form on Q=∑i∈IZαi normalized so that aij=2(αi,αj)(αi,αi). Now, let Q+=∑i∈IZ≥0αi. Then f is Q+-graded by assigning the degree αi to θi (written |θi|=αi). In the formula above, x2 and y1 are homogeneous with respect to the Q+-grading and the formula extends linearly.
Strictly speaking, it is the canonical basis of f which admits a geometric realization in terms of simple perverse sheaves (see chapter 13 in Lusztig's book). The algebra U−ν is then related to f via a process called "bosonisation" described by Majid in Double-Bosonisation and the Construction of {Uq(g)}.
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3Notation in the subject is definitely a headache, but probably can't be sorted out completely in this forum. Anyway, the symbol
$v$
(not greek$\nu$
) is often used by Lusztig in contexts going back to his 1979 paper with Kazhdan on Hecke algebras. It may denote a square root of$q$
. Even though results for Hecke algebras or quantum groups might end up with$q$
-formulations, there is sometimes a subtle need to work for a while with square roots (as in the development of Kazhdan-Lusztig polynomials). Commented Apr 19, 2011 at 18:12