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4 - Fusion Ring and Explicit Verlinde Formula
Shrawan Kumar, University of North Carolina, Chapel Hill
Book:

Conformal Blocks, Generalized Theta Functions and the Verlinde Formula

Published online:

19 November 2021

Print publication:

25 November 2021, pp 125-162
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Summary

As mentioned in Chapter 3, to determine the dimension of the space of vacua on a genus-g curve, it suffices to determine it on the projective line with three marked points. To achieve this, a general algebraic framework in the form of a fusion ring of the simple Lie algebra g at level c is introduced in this chapter. It is a finite rank-reduced algebra. We determine its set of characters explicitly by using the combinatorics of the affine Weyl group and the affine analogue of the Borel--Weil--Bott theorem, as well as a Lie algebra cohomology vanishing result of Teleman. Once we have explicitly determined the characters of the fusion ring (as we have), one of the most important results of the book -- the Verlinde dimension formula -- follows easily by using simple representation theory for finite groups.

A p-adic analogue of the Borel regulator and the Bloch–Kato exponential map
Part of
- $K$-theory in number theory
- Algebraic number theory: local and $p$-adic fields
Annette Huber, Guido Kings
Journal:

Journal of the Institute of Mathematics of Jussieu / Volume 10 / Issue 1 / January 2011

Published online by Cambridge University Press:

05 August 2010, pp. 149-190

Print publication:

January 2011
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In this paper we define a p-adic analogue of the Borel regulator for the K-theory of p-adic fields. The van Est isomorphism in the construction of the classical Borel regulator is replaced by the Lazard isomorphism. The main result relates this p-adic regulator to the Bloch–Kato exponential and the Soulé regulator. On the way we give a new description of the Lazard isomorphism for certain formal groups. We also show that the Soulé regulator is induced by continuous and even analytic classes.

The Second Cohomology of Current Algebras of General Lie Algebras
Karl-Hermann Neeb, Friedrich Wagemann
Journal:

Canadian Journal of Mathematics / Volume 60 / Issue 4 / 01 August 2008

Published online by Cambridge University Press:

20 November 2018, pp. 892-922

Print publication:

01 August 2008
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Let $A$ be a unital commutative associative algebra over a field of characteristic zero, $\mathfrak{k}$ a Lie algebra, and $\mathfrak{z}$ a vector space, considered as a trivial module of the Lie algebra $\mathfrak{g}:=A\otimes \mathfrak{k}$. In this paper, we give a description of the cohomology space ${{H}^{2}}(\mathfrak{g},\mathfrak{z})$ in terms of easily accessible data associated with $A$ and $\mathfrak{k}$. We also discuss the topological situation, where $A$ and $\mathfrak{k}$ are locally convex algebras.