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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.

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up vote 5 down vote accepted

Yes, the $E$, $F$, and $K$ are standard generators of $U_\nu(\mathfrak{sl}_2)$. Sometimes $$q=\nu^{-1}.$$

I don't know what is standard. More generally, the standard (Chevalley) generators for $U_\nu$ are $E_i,F_i,K_i$ ($i\in I$).

The algebra $\mathbf{f}$ (generated by $\theta_i,i\in I$, say) is isomorphic (as an algebra) to the algebra $U^-$ generated by the $F_i$. However, $U_\nu^-$ is not a co-subalgebra of $U$ with respect to the coproduct $\Delta(K_i)=K_i\otimes K_i$, $\Delta(E_i)=K_i\otimes E_i+E_i\otimes 1$, $\Delta(F_i)=K_i^{-1}\otimes F_i+F_i\otimes 1$.

The algebra $\mathbf{f}$ is a co-algebra with respect to the comultiplication $\delta(\theta_i)=1\otimes\theta_i+\theta_i\otimes1$. However, it is not a bialgebra (that is, comultiplication $\delta:\mathbf{f}\to\mathbf{f}\otimes\mathbf{f}$ is not an algebra homomorphism) unless we equip $\mathbf{f}\otimes\mathbf{f}$ with a twisted multiplication: $$(x_1\otimes x_2)(y_1\otimes y_2)=\nu^{-(|x_2|,|y_1|)}x_1y_1\otimes x_2y_2.$$

To explain the notation above, associated to $U_\nu$ is a Cartan matrix $A=(a_{ij})_{i,j\in I}$, a root system $\Phi$ with simple roots $\Pi=\lbrace\alpha_i|i\in I\rbrace$, and bilinear form on $Q=\sum_{i\in I}\mathbb{Z}\alpha_i$ normalized so that $$a_{ij}=\frac{2(\alpha_i,\alpha_j)}{(\alpha_i,\alpha_i)}.$$ Now, let $Q^+=\sum_{i\in I}\mathbb{Z}_{\geq0}\alpha_i$. Then $\mathbf{f}$ is $Q^+$-graded by assigning the degree $\alpha_i$ to $\theta_i$ (written $|\theta_i|=\alpha_i$). In the formula above, $x_2$ and $y_1$ are homogeneous with respect to the $Q^+$-grading and the formula extends linearly.

Strictly speaking, it is the canonical basis of $\mathbf{f}$ which admits a geometric realization in terms of simple perverse sheaves (see chapter 13 in Lusztig's book). The algebra $U_\nu^-$ is then related to $\mathbf{f}$ via a process called "bosonisation" described by Majid.

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Notation 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). – Jim Humphreys Apr 19 '11 at 18:12

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