Related papers: Vogel's universality and the classification proble…
Kleinian singularities, i.e., the varieties corresponding to the algebras of invariants of Kleinian groups are of fundamental importance for Algebraic geometry, Representation theory and Singularity theory. The filtered deformations of…
The integrality of the Kontsevich integral and perturbative invariants is discussed. We show that the denominator of the degree $n$ part of the Kontsevich integral of any knot or link is a divisor of $(2!3!... n!)^4(n+1)!$. We also show…
This is the second part of the revised versions of the notes of three consecutive expository lectures given by Chongying Dong, Haisheng Li and Yi-Zhi Huang in the conference on Monster and vertex operator algebras at the Research Institute…
In a previous paper by the author a universal ring of invariants for algebraic structures of a given type was constructed. This ring is a polynomial algebra that is generated by certain trace diagrams. It was shown that this ring admits the…
We present the universal, in Vogel's sense, expression for the quantum dimension of Cartan product of an arbitrary number of adjoint and $X_2$ representations of simple Lie algebras. The same formula mysteriously gives quantum dimensions of…
Multi-indexed Jacobi polynomials are defined by the Wronskian of four types of eigenfunctions of a deformed P\"oschl-Teller Hamiltonian. We give a correspondence between multi-indexed Jacobi polynomials and pairs of Maya diagrams, and we…
For a quiver with potential, we can associate a vanishing cycle to each representation space. If there is a nice torus action on the potential, the vanishing cycles can be expressed in terms of truncated Jacobian algebras. We study how…
We establish a connection between knot theory and cluster algebras via representation theory. To every knot diagram (or link diagram), we associate a cluster algebra by constructing a quiver with potential. The rank of the cluster algebra…
We study the arithmetic Fourier transforms of trace functions on general connected commutative algebraic groups. To do so, we first prove a generic vanishing theorem for twists of perverse sheaves by characters, and using this tool, we…
The classical involutive division theory by Janet decomposes in the same way both the ideal and the escalier. The aim of this paper, following Janet's approach, is to discuss the combinatorial properties of involutive divisions, when…
Using an appropriate notion of locally convex Kasparov modules, we show how to induce isomorphisms under a large class of functors on the category of locally convex algebras; examples are obtained from spectral triples. Our considerations…
By definition the identities $[x_1, x_2] + [x_2, x_1] = 0$ and $[x_1, x_2, x_3] + [x_2, x_3, x_1] + [x_3, x_1, x_2] = 0$ hold in any Lie algebra. It is easy to check that the identity $[x_1, x_2, x_3, x_4] + [x_2, x_1, x_4, x_3] + [x_3,…
We find a single two-parameter skein relation on trivalent graphs, the quantum exceptional relation, that specializes to a skein relation associated to each exceptional Lie algebra (in the adjoint representation). If a slight strengthening…
It has been folklore for several years in the knot theory community that certain integrals on configuration space, originally motivated by perturbation theory for the Chern-Simons field theory, converge and yield knot invariants. This was…
We consider polynomial maps, which we call degree $d$-linear maps, that satisfy the Jacobian condition. We prove that certain infinite families of elements, which appear in the coefficients of the formal inverse of such maps, are in the…
Original proofs of the AGT relations with the help of the Hubbard-Stratanovich duality of the modified Dotsenko-Fateev matrix model did not work for beta different from one, because Nekrasov functions were not properly reproduced by…
Weight systems are functions on chord diagrams satisfying Vassiliev's $4$-term relations. They originate in the theory of finite type knot invariants. Recent developments in understanding weight systems arising from Lie algebras are based…
We outline the proof of a conjecture of Kontsevich on the isomorphism between the group of polynomial symplectomorphisms in $2n$ variables and the group of automorphisms of the $n$-th Weyl algebra over complex numbers. Our proof uses…
We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi-Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use…
We introduce a new method for studying universality of random matrices. Let T_n be the Jacobi matrix associated to the Dyson beta ensemble with uniformly convex polynomial potential. We show that after scaling, T_n converges to the…