User brandyn - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T11:29:58Z http://mathoverflow.net/feeds/user/12709 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83747/sign-conventions-for-a-chevalley-basis-of-a-simple-complex-lie-algebra Sign conventions for a Chevalley basis of a simple complex Lie algebra brandyn 2011-12-17T22:57:24Z 2011-12-18T11:54:10Z <p>Let $R$ be the root system of a simple complex Lie algebra $g$ with respect to some Cartan subalgebra $h$. Chevalley showed there is a basis of $g$ given by the simple coroots {$H_{\alpha_i}=\alpha_i^\vee\in h$} and root vectors $X_\alpha\in g_\alpha$ for each $\alpha\in R$. This basis has the following properties:</p> <p>$[H_{\alpha_i},H_{\alpha_j}]=0$</p> <p>$[H_{\alpha_i},X_\beta]=\beta(H_{\alpha_i})X_\beta$</p> <p>$[X_{\alpha},X_{-\alpha}]=H_\alpha=\alpha^\vee\in h$</p> <p>($\ast$) $[X_\alpha,X_\beta]=\pm(p+1)X_{\alpha+\beta}$, when $\alpha+\beta\in R$ and $p$ is the greatest positive integer such that $\beta-p\alpha\in R$. Otherwise, if $\alpha+\beta$ is not a root, then the bracket is zero.</p> <p>References for this can be found in Serre's book on semisimple complex Lie algebras or Humphrey's book or Wikipedia.</p> <blockquote> <p>Does anybody know a simple way to determine the sign $\pm$ in the fourth property ($\ast$)?</p> </blockquote> <p>I cannot find a reference and my French is not good, so reading the original works by Chevalley and Tits isn't a viable option. In particular, I need to find a sign convention that will work for $g$ of type $F_4$.</p> <p>Thanks so much.</p> http://mathoverflow.net/questions/25592/what-role-does-the-dual-coxeter-number-play-in-lie-theory-and-should-it-be-cal/83767#83767 Answer by brandyn for What role does the "dual Coxeter number" play in Lie theory (and should it be called the "Kac number")? brandyn 2011-12-18T04:27:27Z 2011-12-18T04:27:27Z <p>The dual Coxeter number $h$ comes up in a conjecture of Cachazo-Douglas-Seilberg-Witten which was motivated by supersymmetric gauge theory. Let $R:=\bigwedge( g\oplus g) = \bigwedge g\otimes\bigwedge g$, where $g$ is a finite dimensional simple complex Lie algebra. Let {$e_i$} be some basis of $g$ and let {$f_i$} denote the dual basis with respect to the normalized Killing form. Consider three different embeddings of $g$ into the 2-graded part of $R$ (which are independent of our chosen basis): </p> <p>$C_1=${$\sum_i [x,e_i]\wedge f_i \otimes 1: x\in g$}$\subset \bigwedge^2 g\otimes \bigwedge^0 g$</p> <p>$C_2=${$1\otimes \sum_i [x,e_i]\wedge f_i : x\in g$}$\subset \bigwedge^0 g\otimes \bigwedge^2 g$</p> <p>$C_3=${$\sum_i [x,e_i]\otimes f_i : x\in g$}$\subset \bigwedge^1 g\otimes \bigwedge^1 g$</p> <p>Let $J$ be the ideal of $R$ generated by $C_1,C_2,C_3$ and let $A$ denote the $g$-algebra $A:=R/J$. Lastly, let $$S=\sum_i e_i\otimes f_i\in \bigwedge^1g\otimes\bigwedge^1g,$$ which also does not depend on the choice of basis.</p> <p>The CDSW conjecture is:</p> <blockquote> <p>The subalgebra $A^g$ of $g$-invariants in $A$ is generated as an algebra by the element $S$. Furthermore, $S^h=0$ and $S^{h-1}\neq 0$. Thus, $$A^g\simeq \mathbb{C}[S]/\langle S^h\rangle.$$</p> </blockquote> <p>I know this isn't an answer to your question, but it is another interesting example of where the dual Coxeter number makes the numerology work. The conjecture is open for type $F_4$ and $E_6,E_7,E_8$, but settled in the other cases. Also, I recently asked a question on MathOverFlow related to this topic and Jim helped me out on it considerably. </p> <p>Lastly, for a reference, see <em>On the Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras</em> paper by Shrawan Kumar, [J. Amer. Math. Soc. 21 (2008), no. 3, 797--808; MR2393427 (2009e:17013)]. </p> http://mathoverflow.net/questions/54929/multiplication-tables-for-hg-p Multiplication tables for H*(G/P)? brandyn 2011-02-09T20:59:08Z 2011-02-11T12:34:50Z <p>Hi everyone. My recent work has me developing software to compute in $H^\ast(G/P)$, where $G$ is a complex connected semisimple algebraic group and $P$ is a standard parabolic subgroup (usually, $B$ or a maximal $P$). While my programs are built on sound theory, one can never be too sure. It's always good practice to check your work.</p> <p>I'm looking for references of multiplication tables of these cohomology rings. </p> <p>In particular, I'm interested in cases where $G$ is NOT simply-laced (that is, Lie type $B_n$, $C_n$, $F_4$, $G_2$), though simply-laced tables would be nice too. Any tables would depend on a choice of additive basis for $H^\ast(G/P)$. I typically use cohomology classes either Hom-dual or Poincare dual to the usual Schubert varieties living in $G/P$, and I like to parameterize my Schubert varieties with $W^P$, the minimal length coset representatives of $W/W_P$ where $W$ is the Weyl group and $W_P$ is the Weyl group of the Levi associated to $P$. Tables using this convention would be great. Of course, tables in any basis would be fine. :D</p> <p>Thanks so much. </p> http://mathoverflow.net/questions/83747/sign-conventions-for-a-chevalley-basis-of-a-simple-complex-lie-algebra/83748#83748 Comment by brandyn brandyn 2011-12-18T02:42:50Z 2011-12-18T02:42:50Z Thank you very much for this reference. In particular, because I'm already working with the $F_4\rightarrow E_6$ embedding in a related project and I'm very familiar with Mathematica. THANKS! :D http://mathoverflow.net/questions/83747/sign-conventions-for-a-chevalley-basis-of-a-simple-complex-lie-algebra Comment by brandyn brandyn 2011-12-17T23:33:28Z 2011-12-17T23:33:28Z I'm working on a calculation in the exterior algebra $\bigwedge(g\oplus g)$ where $g$ is type $F_4$. I was hoping to do a small part of the calculation via a computer program, but getting a working model of $g$ is the first step. http://mathoverflow.net/questions/54929/multiplication-tables-for-hg-p/55122#55122 Comment by brandyn brandyn 2011-02-11T15:47:20Z 2011-02-11T15:47:20Z Actually, I was using Haibao's result's for my program (which I've written in Mathematica).