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A supersymmetric gauge invariant action is constructed over any 4-dimensional Riemannian manifold describing Witten's theory of 4-monopoles. The topological supersymmetric algebra closes off-shell. The multiplets include the auxiliary…

We investigate a spectrum of positive self-adjoint operator $\Gk$ (Laplace operator) acting in the external complexes of some interesting subalgebras $\Lk$ of Lie algebra $A_1^{(1)}$. We obtain an explicit formula for the action of $\G_k$.…

Let $G$ be a connected reductive algebraic group defined over an algebraically closed field $\mathbbm k$ of characteristic zero. We consider the commuting variety $\mathcal C(\mathfrak u)$ of the nilradical $\mathfrak u$ of the Lie algebra…

A Lie version of Turaev's $\overline{G}$-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a \textit{$\frak{g}$-quasi-Frobenius Lie algebra} for…

The Quillen-Barr-Beck cohomology of augmented algebras with divided powers is defined as the derived functor of Beck derivations. The main theorem of this paper states that the K\"ahler differentials of an augmented algebra with divided…

Let $\mathcal{L}$ be a finite-dimensional semisimple Lie algebra of rank $N$ over an algebraically closed field of characteristic $0$. Associated to $\mathcal{L}$ is a family of polynomial folding maps…

We calculate the dg algebra of global functions on commuting stacks of complex reductive groups using tools from Betti Geometric Langlands. In particular, we prove that the ring of invariant functions on the commuting scheme is reduced. Our…

The curious connection between the spacings of the eigenvalues of random matrices and the corresponding spacings of the non trivial zeros of the Riemann zeta function is analyzed on the basis of the geometric dynamical global program of…

We show that if $S,T$ are two commuting automorphisms of standard Borel space such that they generate a free Borel $\Z^2$-action then $S$ and $T$ do not have same sets of real valued bounded coboundaries. We also prove a weaker form of…

We examine the commuting elements $\theta_i=\sum_{j\neq i} \frac{s_{ij}}{z_i-z_j}$, $z_i\neq z_j$, $s_{ij}$ the transposition swapping $i$ and $j$, and we study their actions on irreducible $S_n$ representations. By applying Schur-Weyl…

Let $G$ be a linear Lie group acting properly and isometrically on a $G$-spin$^c$ manifold $M$ with compact quotient. We show that Poincar\'e duality holds between $G$-equivariant $K$-theory of $M$, defined using finite-dimensional…

In this work, we construct an alternative formulation to the traditional Algebraic Bethe ansatz for quantum integrable models derived from a generalised rational Gaudin algebra realised in terms of a collection of spins 1/2 coupled to a…

We study the Dehn function of connected Lie groups. We show that this function is always exponential or polynomially bounded, according to the geometry of weights and of the 2-cohomology of their Lie algebras. Our work, which also addresses…

The space of Lie algebra cohomology is usually described by the dimensions of components of certain degree even for the adjoint module as coefficients when the spaces of cochains and cohomology can be endowed with a Lie superalgebra…

We consider non-selfadjoint operator algebras $\mathfrak{L}(G,\lambda)$ generated by weighted creation operators on the Fock-Hilbert spaces of countable directed graphs $G$. These algebras may be viewed as noncommutative generalizations of…

Let k be an algebraically closed field of characteristic p>0 and let G be a connected reductive group over k. Let B be a Borel subgroup of G and let g and b be the Lie algebras of G and B. Denote the first Frobenius kernels of G and B by…

The phenomenon of BCJ duality implies that gauge theories possess an abstract kinematic algebra, mirroring the non-abelian Lie algebra underlying the colour information. Although the nature of the kinematic algebra is known in certain…

This paper presents an $\mathcal{E}$-derivation analogue of a result on derivations due to van den Essen, Wright and Zhao. We prove that the image of a locally finite $K$-$\mathcal{E}$-derivation of polynomial algebras in two variables over…

When $A = \mathbb{k}[x_1, \ldots, x_n]$ and $G$ is a small subgroup of $\operatorname{GL}_n(\mathbb{k})$, Auslander's Theorem says that the skew group algebra $A \# G$ is isomorphic to $\operatorname{End}_{A^G}(A)$ as graded algebras. We…

We here present rudiments of an approach to geometric actions in noncommutative algebraic geometry, based on geometrically admissible actions of monoidal categories. This generalizes the usual (co)module algebras over Hopf algebras which…