Affine Krichever-Novikov algebras, their representations and applications

Module structure of the center of the universal central extension of a genus zero Krichever-Novikov algebra

We describe how the center of the universal central extension of the genus zero Krichever-Novikov current algebra decomposes as a direct sum of irreducible modules for automorphism group of the coordinate ring of this algebra.

The Fermion Model of Representations of Affine Krichever--Novikov Algebras

To a generic holomorphic vector bundle on an algebraic curve and an irreducible finite-dimensional representation of a semisimple Lie algebra, we assign a representation of the corresponding affine Krichever--Novikov algebra in the space of semi-infinite exterior forms. It is shown that equivalent pairs of data give rise to equivalent representations and vice versa.

Second order casimirs for the affine Krichever--Novikov algebras $\hat{\mathfrak{gl}}_{g,2}$ and $\hat{\mathfrak{sl}}_{g,2}$

The second order casimirs for the affine Krichever--Novikov algebras $\hat{\mathfrak{gl}}_{g,2}$ and $\hat{\mathfrak{sl}}_{g,2}$ are described. More general operators which we call semi-casimirs are introduced. It is proven that the semi-casimirs induce well-defined operators on conformal blocks and, for a certain moduli space of Riemann surfaces with two marked points and fixed jets of local coordinates, there is a natural projection of its tangent space onto the space of these operators. It is (non-formally) explained how semi-casimirs appear in course of operator quantization of the second …

On the module structure of the center of hyperelliptic Krichever-Novikov algebras

We consider the coordinate ring of a hyperelliptic curve and let $\mathfrak{g}\otimes R$ be the corresponding current Lie algebra where $\mathfrak g$ is a finite dimensional simple Lie algebra defined over $\mathbb C$. We give a generator and relations description of the universal central extension of $\mathfrak{g}\otimes R$ in terms of certain families of polynomials $P_{k,i}$ and $Q_{k,i}$ and describe how the center $Ω_R/dR$ decomposes into a direct sum of irreducible representations when the automorphism group is $C_{2k}$ or $D_{2k}$.

正交多项式与超椭圆Krichever--Novikov代数的通用中心扩张的中心

Orthogonal polynomials and the centres of the universal central extensions of superelliptic Krichever--Novikov algebras

该研究确定了超椭圆Krichever--Novikov代数通用中心扩张的中心结构，对任意覆盖次数完成了导子侧的类约化，明确了正交多项式链的测度承载情况，并闭式计算了二次多项式下的混合上闭链。

On the module structure of the center of hyperelliptic Krichever-Novikov algebras II

Let $R := R_{2}(p)=\mathbb{C}[t^{\pm 1}, u : u^2 = t(t-α_1)\cdots (t-α_{2n})] $ be the coordinate ring of a nonsingular hyperelliptic curve and let $\mathfrak{g}\otimes R$ be the corresponding current Lie algebra. \color{black} Here $\mathfrak g$ is a finite dimensional simple Lie algebra defined over $\mathbb C$ and \begin{equation*} p(t)= t(t-α_1)\cdots (t-α_{2n})=\sum_{k=1}^{2n+1}a_kt^k. \end{equation*} In earlier work, Cox and Im gave a generator and relations description of the universal central extension of $\mathfrak{g}\otimes R$ in terms of certain families of polynomials $P_{k,i}$ and…