Related papers: Codimension sequence, grade, and generating degree…
We introduce a generalization of the notion of operad that we call a contractad, whose set of operations is indexed by connected graphs and whose composition rules are numbered by contractions of connected subgraphs. We show that many…
We study the structure of graded Lie superalgebras with arbitrary dimension and over an arbitrary field ${\mathbb K}$. We show that any of such algebras ${\mathfrak L}$ with a symmetric $G$-support is of the form ${\mathfrak L} = U +…
We define and study the space of $q$-opers associated with Bethe equations for integrable models of XXZ type with quantum toroidal algebra symmetry. Our construction is suggested by the study of the enumerative geometry of cyclic quiver…
We give a definition of weak n-categories based on the theory of operads. We work with operads having an arbitrary set S of types, or `S-operads', and given such an operad O, we denote its set of operations by elt(O). Then for any S-operad…
Associated to a presentable $\infty$-category $\mathcal{C}$ and an object $X \in \mathcal{C}$ is the tangent $\infty$-category $\mathcal{T}_X\mathcal{C}$, consisting of parameterized spectrum objects over $X$. This gives rise to a…
Let A be a finite dimensional algebra over a field of characteristic zero graded by a finite abelian group G. Here we study a growth function related to the graded polynomial identities satisfied by A by computing the exponential rate of…
We study the quantization of the moduli space of multiplicative Higgs bundles through the lens of five-dimensional $\mathcal{N}=1$ supersymmetric gauge theories in $\Omega$-background. We extend the 4d $\mathcal{N}=2$ gauge theoretical…
Graded contractions of certain non-toral $\mathbb{Z}_2^3$-gradings on the simple Lie algebras $\mathfrak{so}(7,\mathbb C)$ and $\f{so}(8,\mathbb C)$ are classified up to two notions of equivalence. In particular, there arise two large…
Let Uq(g) be the quantum affine superalgebra associated with an affine Kac-Moody superalgebra g which belongs to the three series osp(1|2n)^(1),sl(1|2n)^(2) and osp(2|2n)^(2). We develop vertex operator constructions for the level 1…
In this paper we study the (Cohen-Macaulay) type of orders over Dedekind domains in \'etale algebras. We provide a bound for the type, and give formulas to compute it. We relate the type of the overorders of a given order to the size of…
We introduce a construction that associates, to each finite dimensional k-vector space V, a family of projective k-varieties that comes equipped with the structure of a operad in the category of k-schemes. When dim V = 1, this operad…
We describe a Grothendieck construction for non-symmetric operads with values in categories, and hence in groupoids and posets. The construction produces a 2-category which is operadically fibered over the category D of finite non-empty…
We characterize vertex algebras (in a suitable sense) as algebras over a certain graded co-operad. We also discuss some examples and categorical implications of this characterization.
For a PI-algebra R over a field of characteristic 0 let T(R) be the T-ideal of the polynomial identities of R and let c(R,t) be the codimension series of R (i.e., the generating function of the codimension sequence of R). Let A, B and R be…
The notion of 2--monoidal category used here was introduced by B.~Vallette in 2007 for applications in the operadic context. The starting point for this article was a remark by Yu. Manin that in the category of quadratic algebras (that is,…
The purpose of this dissertation is to set up a theory of generalized operads and multicategories, and to use it as a language in which to propose a definition of weak n-category. Included is a full explanation of why the proposed…
We show that if $T$ is any of four semigroups of two elements that are not groups, there exists a finite dimensional associative $T$-graded algebra over a field of characteristic $0$ such that the codimensions of its graded polynomial…
Starting from a Hopf algebra endowed with an action of a group G by Hopf automorphisms, we construct (by a twisted double method) a quasitriangular Hopf G-coalgebra. This method allows us to obtain non-trivial examples of quasitriangular…
Let $A$ be an associative algebra with a superinvolution $*$ over a field of characteristic zero, and let $c_n^*(A)$, $n = 1, 2, \ldots$, denote its sequence of $*$-codimensions. It is well known that this sequence is either polynomially…
In this paper we describe the right coideal subalgebras containing all group-like elements of the two-parameter quantum groups Uq(g) and uq(g), where g is a simple Lie algebra of type G2. As a consequence, we determine that there are…