Moduli Algebra Research Articles

On the Generalized Cartan Matrices Arising from k-th Yau Algebras of Isolated Hypersurface Singularities

Let (V,0) be an isolated hypersurface singularity defined by the holomorphic function $f: (\mathbb {C}^{n}, 0)\rightarrow (\mathbb {C}, 0)$ . The k-th Yau algebra Lk(V ) is defined to be the Lie algebra of derivations of the k-th moduli algebra $A^{k}(V) := \mathcal {O}_{n}/(f, m^{k}J(f))$ , where k ≥ 0, m is the maximal ideal of $\mathcal {O}_{n}$ . I.e., Lk(V ) := Der(Ak(V ),Ak(V )). These new series of derivation Lie algebras are quite subtle invariants since they capture enough information about the complexity of singularities. In this paper we formulate a conjecture for the complete characterization of ADE singularities by using generalized Cartan matrix Ck(V ) associated to k-th Yau algebras Lk(V ), k ≥ 1. In this paper, we provide evidence for the conjecture and give a new complete characterization for ADE singularities. Furthermore, we compute their other various invariants that arising from the 1-st Yau algebra L1(V ).

On two inequality conjectures for the k-th Yau numbers of isolated hypersurface singularities

Let (V, 0) be an isolated hypersurface singularity defined by the holomorphic function $$f: ({\mathbb {C}}^n, 0)\rightarrow ({\mathbb {C}}, 0)$$ . In our previous work, we introduced a series of novel Lie algebras associated to (V, 0), i.e., k-th Yau algebra $$L^k(V), k\ge 0$$ . It was defined to be the Lie algebra of derivations of the k-th moduli algebras $$A^k(V)= {\mathcal {O}}_n/(f, m^k J(f)), k\ge 0$$ , where m is the maximal ideal of $${\mathcal {O}}_n$$ . I.e., $$L^k(V):=\text {Der}(A^k(V), A^k(V))$$ . The dimension of $$L^k(V)$$ was denoted by $$\lambda ^k(V)$$ . The number $$\lambda ^k(V)$$ , which was called k-th Yau number, is a subtle numerical analytic invariant of (V, 0). Furthermore, we formulated two conjectures for these k-th Yau number invariants: a sharp upper estimate conjecture of $$\lambda ^k(V)$$ for weighted homogeneous isolated hypersurface singularities (see Conjecture 1.2) and an inequality conjecture $$\lambda ^{(k+1)}(V) > \lambda ^k(V), k\ge 0$$ (see Conjecture 1.1). In this article, we verify these two conjectures when k is small for large class of singularities.

Integrating quantum groups over surfaces

We apply the mechanism of factorization homology to construct and compute category-valued two-dimensional topological field theories associated to braided tensor categories, generalizing the (0,1,2)-dimensional part of Crane–Yetter–Kauffman four-dimensional TFTs associated to modular categories. Starting from modules for the Drinfeld–Jimbo quantum group U q ( g ) we obtain in this way an aspect of topologically twisted four-dimensional N = 4 super Yang–Mills theory, the setting introduced by Kapustin–Witten for the geometric Langlands program. For punctured surfaces, in particular, we produce explicit categories which quantize character varieties (moduli of G-local systems) on the surface; these give uniform constructions of a variety of well-known algebras in quantum group theory. From the annulus, we recover the reflection equation algebra associated to U q ( g ) , and from the punctured torus we recover the algebra of quantum differential operators associated to U q ( g ) . From an arbitrary surface we recover Alekseev's moduli algebras. Our construction gives an intrinsically topological explanation for well-known mapping class group symmetries and braid group actions associated to these algebras, in particular the elliptic modular symmetry (difference Fourier transform) of quantum D-modules.

Application of classical invariant theory to biholomorphic classification of plane curve singularities, and associated binary forms

We use classical invariant theory to solve the biholomorphic equivalence problem for two families of plane curve singularities previously considered in the literature. Our calculations motivate an intriguing conjecture that proposes a method for extracting a complete set of invariants of homogeneous plane curve singularities from their moduli algebras.

Extracting invariants of isolated hypersurface singularities from their moduli algebras

We use classical invariant theory to construct invariants of complex graded Gorenstein algebras of finite vector space dimension. As a consequence, we obtain a way of extracting certain numerical invariants of quasi-homogeneous isolated hypersurface singularities from their moduli algebras, which extends an earlier result due to the first author. Furthermore, we conjecture that the invariants so constructed solve the biholomorphic equivalence problem in the homogeneous case. The conjecture is easily verified for binary quartics and ternary cubics. We show that it also holds for binary quintics and sextics. In the latter cases the proofs are much more involved. In particular, we provide a complete list of canonical forms of binary sextics, which is a result of independent interest.

Unimodularity of Poincaré Polynomials of Lie Algebras for Semisimple Singularities

We single out a large class of semisimple singularities with the property that all roots of the Poincare polynomial of the Lie algebra of derivations of the corresponding suitably (not necessarily quasihomogeneously) graded moduli algebra lie on the unit circle; for a still larger class there might occur exactly four roots outside the unit circle. This is a corrected version of Theorem 4.5 from [3]. 1. Formulations 1.1. Tensor product of graded algebras and their gradings. An important invariant of an isolated singularity S given by a holomorphic germ f : (C, 0) → (C, 0) is the algebra C[[x]] /(

The Centre of the Graph¶and Moduli Algebras at Roots of 1

The structure of the centres ${\cal Z}(\Lg)$ and ${\cal Z}(\Mg)$ of the graph algebra ${\cal L}_g(sl_2)$ and the moduli algebra ${\cal M}_g(sl_2)$ is studied at roots of 1. It it shown that ${\cal Z}(\Lg)$ can be endowed with the structure of the Poisson graph algebra. The elements of $Spec({\cal Z}(\Mg))$ are shown to satisfy the defining relation for the holonomies of a flat connection along the cycles of a Riemann surface. The irreducible representations of the graph algebra are constructed.

Deformation of isolated hypersurface singularities and their moduli algebras

By the using of determinantal varieties from moduli algebras of hypersurface singularites the relation of the deformation of hypersurface singularities and the deformation of their moduli algebras is studied. For a type of hypersurface singularities a weak Torelli type result is proved. This weak Torelli type result showes that for families of hypersurface singularities the moduli algebras can be used to distinguish the complex structures of singularities at least in some weak sence.

Micro-local characterization of quasi-homogeneous singularities

A moduli algebra A ( V ) of hypersurface singularity ( V , 0) is a finite dimensional C -algebra. In 1982, Mather and Yau proved that two germs of complex analytic hypersurfaces of the same dimension with isolated singularities are biholomorphically equivalent if and only if their moduli algebra are isomorphic. It is a natural question to ask for a necessary and sufficient condition for a complex analytic isolated hypersurface singularity to be quasi-homogeneous in terms of its moduli algebra. In this paper we prove that ( V , 0) admits a quasi-homogeneous structure if and only if its moduli algebra is isomorphic to a finite dimensional nonnegatively graded algebra. In 1983, Yau introduced a finite dimensional Lie algebra L ( V ) to an isolated hypersurface singularity ( V , 0). L ( V ) is defined to be the algebra of derivations of the moduli algebra A ( V ) and is finite dimensional. We prove that ( V , 0) is quasi-homogeneous singularity if (1) L ( V ) is isomorphic to a nonnegatively graded Lie algebra without center, (2) There exists E in L ( V ) of degree zero such that [ E, D i ] = i D i for any D i in L ( V ) of degree i , and (3) For any element a ∈ m - m 2 where m is the maximal ideal of A ( V ), aE is not in degree zero part of L ( V ).

Classification of isolated hypersurface singularities by their moduli algebras

Classification of isolated hypersurface singularities by their moduli algebras

Criterion for biholomorphic equivalence of isolated hypersurface singularities

Two germs of complex analytic hypersurfaces with isolated singularities are biholomorphically equivalent if and only if they have the same dimension and their moduli algebras are isomorphic.

Moduli of Isolated Hypersurface Singularities

It was shown by J.N. Mather and S.S.-T. Yau that an isolated complex hypersurface singularity is completely determined by its moduli algebra. In this article it is shown, for the simple elliptic singularities, how to construct continuous invariants from the moduli algebras and, hence, associate invariants to the singularities themselves.