What are the relations between the notation in Lusztig's book [introduction to quantum groups](http://books.google.com/books?id=HKPjCUiOUQ0C&printsec=frontcover&dq=introduction+to+quantum+groups+lusztig&hl=en&ei=UrWtTeeNKIfZgQewic3sCw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCwQ6AEwAA#v=onepage&q&f=false) 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 ${}^{'}{\bf f}$ and ${\bf f}$ used for? What are the relations between $f$ and $U$? Thank you. 



notation in Lusztig's book: introduction to quantum groups
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 in *[Double-Bosonisation and the Construction of $\{U\_q(g)\}$](https://arxiv.org/abs/q-alg/9511001)*.


