A note on affine representable algebras

Affine structures on filiform Lie algebras

In this note we prove that every non characteristically filiform Lie algebra is endowed with an affine structure.

关于从与 $\widehat{\mathfrak{sl}}(n)$ 相伴的 affine Yangian 到与 $\widehat{\mathfrak{sl}}(n+1)$ 相伴的 affine Yangian 的同态的注记

Notes on a homomorphism from the affine Yangian associated with $\hat{\mathfrak{sl}}(n)$ to the affine Yangian associated with $\widehat{\mathfrak{sl}}(n+1)$

本文利用先前构造的affine Yangian同态，构造了从affine Yangian到A型非矩形W-代数泛包络代数的同态，探讨了affine shifted Yangian的新定义，并证明了其与量子仿射代数同态的关联。

Finite representations for two small relation algebras

In this note, we give two different proofs that relation algebra $52_{65}$ is representable over a finite set. The first is probabilistic, and uses Johnson schemes. The second is an explicit group representation over $ (\mathbb{Z}/2\mathbb{Z})^{10}$. We also give a finite representation of $59_{65}$ over $\mathbb{Z}/113\mathbb{Z}$ using a technique due to Comer.

仿射代数幺半群作为有限维代数的自同态幺半群

Affine algebraic monoids as endomorphisms' monoids of finite-dimensional algebras

本文证明了任意仿射代数幺半群均可实现为某个有限维非结合代数的自同态幺半群。

Semisimplicity of affine cellular algebras

In this note, we prove that an affine cellular algebra $A$ is semisimple if and only if the scheme associated to $A$ is reduced and 0-dimensional, and the bilinear forms with respect to all layers of $A$ are isomorphisms. Moreover, if the ground ring is a perfect field, then $A$ is semisimple if and only if it is separable. We also give a sufficient condition for an affine cellular algebra being Jacobson semisimple.

关于秩2自由代数的模空间的一个注记

A note on the moduli spaces of free algebras of rank 2

本文用函子化方法刻画一般基环上秩2代数的模问题，并将其模空间构造为仿射概形在群概形作用下的预层商。

Unitary highest weight modules of locally affine Lie algebras

Locally affine Lie algebras are generalizations of affine Kac--Moody algebras with Cartan subalgebras of infinite rank whose root system is locally affine. In this note we study a class of representations of locally affine algebras generalizing integrable highest weight modules. In particular, we construct such an integrable representation for each integral weight not vanishing on the center and show that, over the complex numbers, we thus obtain unitary representations w.r.t. a unitary real form. We also use Yoshii's recent classification of locally affine root systems to derive a classificat…

Lectures on extended affine Lie algebras

We give an introduction to the structure theory of extended affine Lie algebras, which provide a common framework for finite-dimensional semisimple, affine and toroidal Lie algebras. The notes are based on a lecture series given during the Fields Institute summer school at the University of Ottawa in June 2009.