8
votes
What is Pressley and Segal's "basic inner product" for compact simple Lie algebras of types B and C?
Section 4 of
Gawȩdzki, Krzysztof; Reis, Nuno, Basic gerbe over non-simply connected compact groups, J. Geom. Phys. 50, No. 1-4, 28-55 (2004). ZBL1067.22009.
lists, in an absolutely concrete way, the ...
- 4,519
7
votes
Why should affine lie algebras and quantum groups have equivalent representation theories?
(Written on my phone - apologies for any typos.)
A few comments:
a) First, as to the source of the braided monoidal structure on the Kazhdan-Lusztig category. The category of integrable affine Lie ...
- 5,958
6
votes
Central extensions of loop groups
Fix a cocycle $\omega\in C^3(G, \mathbb R)$ such that $\omega(H_3(G, \mathbb Z)) = \mathbb Z$. (For $G = SU(2)$, we can take $\omega$ to be the standard volume form.) Fix a loop $L$ in $G$, and let $...
- 12.8k
6
votes
Accepted
Holomorphic line bundles on $\mathbb{P}^1$ from gluing data
I am just posting my comment above as an answer. Let $U$ and $V$ be subsets of $\mathbb{CP}^1$, or any other projective algebraic curve $C$, whose complement sets are finite, and such that $\{U,V\}$ ...
5
votes
Accepted
Sheafification of loop scheme/group
Despite Will Sawin's answer in comment, it was not so obvious for me at first glance, so I give a detailed (at least for $X$ affine) proof here.
Lemma. Let $A$ be a $K$-algebra and let $A\to B$ be ...
- 1,121
4
votes
Sheafification of loop scheme/group
Let me give an answer beynd the case when $X$ is affine.
Upshot: Minimally restricting the setup, the answer is "LX is a sheaf, so you don't need to sheafify" for all quasi-projective ...
- 755
4
votes
Reconciling the affine grassmannian and the based loop group
There are many spaces of maps $S^1\to K$ (hence many version of the affine Grassmannian) one might want to consider.
Let's list them:
• Algebraic maps (i.e. algebraic maps from $\mathbb C^\times$ to $...
- 43.2k
3
votes
Accepted
Difference between two definitions of affine Lie algebras
In my experience, the Laurent polynomial construction is more suited to the "algebraic" aspects of the theory--in particular, if the power of $t$ corresponds to the coefficient of $\delta$ ...
- 550
2
votes
Cartan decomposition of loop group
Apparently, this is based on a result by Iwahori and Matsumoto (Corollary 2.17 of [IM]). A modern proof of this result can be found on [DHLH]. This result is one of the main elements of the proof of ...
- 553
1
vote
Accepted
Non-invariant forms on loop Lie algebra of semisimple Lie group
Thanks to Yves Cornulier, for suggesting to look at the paper of Neeb and Wagemann. After reading Example 6.2 of that paper (arxiv version), I think the answer to my question is that in fact all 2-...
- 9,853
1
vote
Accepted
What is Pressley and Segal's "basic inner product" for compact simple Lie algebras of types B and C?
I found a reference that gave the correct inner products explicitly, without requiring the reader to assemble the relevant facts: Chapter II, section 1.2 (bottom of page 583) of:
McKenzie Y. Wang, ...
- 35.5k
Only top scored, non community-wiki answers of a minimum length are eligible