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A loop-algebraic presentation is given for toroidal Lie superalgebras of classical types. Based on the loop superalgebra presentation free field realizations of toroidal Lie superalgebras are constructed for types $A(m,n)$, $B(m,n)$, C(n)…
We introduce a diagram category, study its structure, and investigate some of its applications to the representation theory of Lie algebras and Lie superalgebras. The morphisms of the category, which contains a subcategory isomorphic to the…
We construct the space of vector fields on a generic quantum group. Its elements are products of elements of the quantum group itself with left invariant vector fields. We study the duality between vector fields and 1-forms and generalize…
Let $(\mathrm{M}, \omega_{0})$ be a connected paracompact smooth oriented manifold. We establish a necessary and sufficient conditions on Lie subalgebra $\mathfrak{a}$ of $\mathrm{T M}$ such that its orbits becomes diffeomorphic to an open…
For T $\Omega$ a Hermitian symmetric tube-type domain, a family ($\pi$ $\mu$) $\mu$$\in$C of holomorphic vector-valued representations is studied. The corresponding Wallach set is determined. The main tool is a realization of the…
We show that singular Riemannian foliations, or, more generally, manifold submetries, defined on a compact normal homogeneous space, have algebraic nature. Moreover, in this case there exists a one-to-one correspondence between algebras of…
In quantum groups coproducts of Lie-algebras are twisted in terms of generators of the corresponding universal enveloping algebra. If representations are considered, twists also serve as starproducts that accordingly quantize representation…
The vector space of holomorphic polyvector fields on any complex manifold has a natural Gerstenhaber algebra structure. In this paper, we study BV operators of the Gerstenhaber algebras of holomorphic polyvector fields on smooth compact…
Superderivations for the eight families of finite or infinite dimensional graded Lie superalgebras of Cartan-type over a field of characteristic $p>3$ are completely determined by a uniform approach: The infinite dimensional case is reduced…
We study the geometry of hyperconvex representations of surface groups in ${\rm PSL}(d,\mathbb{C})$ and their deformation spaces: We produce a natural holomorphic extension of the classical Ahlfors--Bers map to a product of Teichm\"uller…
Let $m\in N$, $P(t)\in C[t]$. Then we have the Riemann surfaces (commutative algebras) $R_m(P)=C[t^{\pm1},u | u^m=P(t)]$ and $S_m(P)=C[t , u| u^m=P(t)].$ The Lie algebras $\mathcal{R}_m(P)=Der(R_m(P))$ and $\mathcal{S}_m(P)=Der(S_m(P))$ are…
Important subalgebras of a Lie algebra of an algebraic group are its toral subalgebras, or equivalently (over fields of characteristic 0) its Cartan subalgebras. Of great importance among these are ones that are split: their action on the…
We give a full classification of 6-dimensional nilpotent Lie algebras over an arbitrary field, including fields that are not algebraically closed and fields of characteristic~2. To achieve the classification we use the action of the…
Dynkin's classification of maximal subalgebras of simple finite dimensional complex Lie algebras is generalized to linear Lie superalgebras. Namely, the maximal non-simple irreducible subalgebras of $\mathfrak{gl}(p|q), \mathfrak{q}(n),…
We define a natural basis for the algebra of $\frak{gosp}(1|2n)$-invariant differential operators on the affine superspace $\mathbb{C}^{1|2n}$. We prove that these operators lie in the image of the centre of the enveloping algebra of…
We introduce the notion of N=1 supergeometric vertex operator superalgebra motivated by the worldsheet geometry underlying genus-zero, two-dimensional, holomorphic N=1 superconformal field theory. We then show, assuming the convergence of…
The "superconformal index" is a character-valued invariant attached by theoretical physics to unitary representations of Lie superalgebras, such as $\mathfrak{su}(2,2\vert n)$, that govern certain quantum field theories. The index can be…
We define Lie algebroids over infinite jet spaces and establish their equivalent representation through homological evolutionary vector fields.
A construction of supersymmetric field-theoretical models in non-commutative geometry is reviewed. The underlying superstructure of the models is encoded in $osp(2,2)$ superalgebra.
A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface M…