MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.