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Throughout this paper, we show on one hand, that there are nilpotent and solvable Lie superalgebras with infinitely many local superderivations which are not standard superderivations. On the other hand, we show that every local superderivation is a superderivation on the maximal-dimensional solvable Lie superalgebras with model filiform or model nilpotent nilradical. Moreover, we extend the latter result for Leibniz superalgebras by showing that every local superderivation is a superderivation on the maximal-dimensional solvable Leibniz superalgebras with model filiform or model nilpotent non-Lie nilradical.

The aim of this research project is to seek new developments in areas where integrable systems and quantum groups intersect such as quantum (super)group, quantum symmetric pair, crystal base, and symmetric function.

Throughout this paper, we show on one hand, that there are nilpotent and solvable Lie superalgebras with infinitely many local superderivations which are not standard superderivations. On the other hand, we show that every local superderivation is a superderivation on the maximal-dimensional solvable Lie superalgebras with model filiform or model nilpotent nilradical. Moreover, we extend the latter result for Leibniz superalgebras by showing that every local superderivation is a superderivation on the maximal-dimensional solvable Leibniz superalgebras with model filiform or model nilpotent non-Lie nilradical.

The aim of this research project is to seek new developments in areas where integrable systems and quantum groups intersect such as quantum (super)group, quantum symmetric pair, crystal base, and symmetric function.

The aim of this research project is to seek new developments in areas where integrable systems and quantum groups intersect such as quantum (super)group, quantum symmetric pair, crystal base, and symmetric function.

For classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule Delta(lambda) to be such that every nonzero homomorphism from another Verma supermodule to Delta(lambda) is injective. This is applied to describe the socle of the cokernel of an inclusion of Verma supermodules over the periplectic Lie superalgebras pe(n) and, furthermore, to reduce the problem of description of Ext(O)(1)(L(mu), Delta(lambda)) for pe(n) to the similar problem for the Lie algebra gl(n). Additionally, we study the projective and injective dimensions of structural supermodules in parabolic category O-p for classical Lie superalgebras. In particular, we completely determine these dimensions for structural supermodules over the periplectic Lie superalgebra pe(n) and the orthosymplectic Lie superalgebra osp(2 vertical bar 2n).

The Bol operators are unary differential operators between spaces of weighted densities on the 1-dimensional manifold invariant under projective transformations of the manifold. On the 1 vertical bar n-dimensional supermanifold (superstring) M, we classify analogs of Bol operators invariant under the simple maximal subalgebra h of the same rank as its simple ambient superalgebra g of vector fields on M and containing all elements of negative degree of g in a Z-grading. We also consider the Lie superalgebras of vector fields g preserving a contact structure on the superstring M. We have discovered many new operators.

We prove that the tensor product of a simple and a finite dimensional sln-module has finite type socle. This is applied to reduce classification of simple q(n)-supermodules to that of simple sln-modules. Rough structure of simple q(n)-supermodules, considered as sln-modules, is described in terms of the combinatorics of category O.

We prove that the tensor product of a simple and a finite dimensional sln-module has finite type socle. This is applied to reduce classification of simple q(n)-supermodules to that of simple sln-modules. Rough structure of simple q(n)-supermodules, considered as sln-modules, is described in terms of the combinatorics of category O.

We prove that the tensor product of a simple and a finite dimensional sln-module has finite type socle. This is applied to reduce classification of simple q(n)-supermodules to that of simple sln-modules. Rough structure of simple q(n)-supermodules, considered as sln-modules, is described in terms of the combinatorics of category O.

We prove that the tensor product of a simple and a finite dimensional sln-module has finite type socle. This is applied to reduce classification of simple q(n)-supermodules to that of simple sln-modules. Rough structure of simple q(n)-supermodules, considered as sln-modules, is described in terms of the combinatorics of category O.

We prove that the tensor product of a simple and a finite dimensional sln-module has finite type socle. This is applied to reduce classification of simple q(n)-supermodules to that of simple sln-modules. Rough structure of simple q(n)-supermodules, considered as sln-modules, is described in terms of the combinatorics of category O.

The inverses of indecomposable Cartan matrices are computed for finite-dimensional Lie algebras and Lie superalgebras over fields of any characteristic, and for hyperbolic (almost affine) complex Lie (super)algebras. This enables one to express the fundamental weights in terms of simple roots corresponding to the Cartan matrix. We discovered three yet inexplicable new phenomena, of which (a) and (b) concern hyperbolic (almost affine) complex Lie (super)algebras: (a) several of the inverses of Cartan matrices have all their elements negative (not just non-positive, as they should be according to an a priori characterization due to Zhang Hechun); (b) the Os only occur on the main diagonals of the inverses; (c) the determinants of inequivalent Cartan matrices of the simple Lie (super)algebra of any dimension may differ (in any characteristic). We interpret most of the results of Wei and Zou (2017) [31] as inverses of the Gram matrices of non-degenerate invariant symmetric bilinear forms on the (super)algebras considered, not of Cartan matrices, and give more adequate references. In particular, the inverses of Cartan matrices of simple Lie algebras were already published, starting with Dynkin's paper in 1952, see also Table 2 in Springer's book by Onishchik and Vinberg (1990).

We review the list of non-degenerate invariant (super)symmetric bilinear forms (briefly: NIS) on the following simple (relatives of) Lie (super)algebras: (a) with symmetrizable Cartan matrix of any growth, (b) with non-symmetrizable Cartan matrix of polynomial growth, (c) Lie (super)algebras of vector fields with polynomial coefficients, (d) stringy a.k.a. superconformal superalgebras, (e) queerifications of simple restricted Lie algebras. Over algebraically closed fields of positive characteristic, we establish when the deform (i.e., the result of deformation) of the known finite-dimensional simple Lie (super)algebra has a NIS. Amazingly, in most of the cases considered, if the Lie (super)algebra has a NIS, its deform has a NIS with the same Gram matrix after an identification of bases of the initial and deformed algebras. We do not consider odd parameters of deformations. Closely related with simple Lie (super)algebras with NIS is the notion of doubly extended Lie (super)algebras of which affine Kac-Moody (super)algebras are the most known examples.

We give explicit formulas proving that the following Lie (super)algebras are restricted: known exceptional simple vectorial Lie (super)algebras in characteristic 3, deformed Lie (super)algebras with indecomposable Cartan matrix, simple subquotients over an algebraically closed field of characteristic 3 of these (super)algebras (under certain conditions), and deformed divergence-free Lie superalgebras of a certain type with any finite number of indeterminates in any characteristic.

As a step towards the structure theory of Lie algebras in symmetric monoidal categories we establish results involving the Killing form. The proper categorical setting for discussing these issues are symmetric ribbon categories.

We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category O associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to classical Lie superalgebras in many cases. Along the way we prove that category O and its parabolic generalizations for classical Lie superalgebras are categories with full projective functors. As an application we prove that in many cases the endomorphism algebra of the basic projective-injective module in (parabolic) category O for classical Lie superalgebras is symmetric. As a special case we obtain that in these cases the algebras describing blocks of the category of finite dimensional modules are symmetric. We also compute the latter algebras for the superalgebra q(2).

Over algebraically closed fields of characteristic p > 2, -prolongations of simple finite dimensional Lie algebras and Lie superalgebras with Cartan matrix are studied for certain simplest gradings of these algebras. We discover several new simple Lie superalgebras, serial and exceptional, including super versions of Brown and Melikyan algebras, and thus corroborate the super analog of the Kostrikin-Shafarevich conjecture. Simple Lie superalgebras with 2 x 2 Cartan matrices are classified., authorCount :3

For the Kac-Moody superalgebra associated with the loop superalgebra with values in a finite-dimensional Lie superalgebra g, we show what its quadratic Casimir element is equal to if the Casimir element for g is known (if g has an even invariant supersymmetric bilinear form). The main tool is the Wick normal form of the even quadratic Casimir operator for the Kac-Moody superalgebra associated with g; this Wick normal form is independently interesting. If g has an odd invariant supersymmetric bilinear form, then we compute the cubic Casimir element. In addition to the simple Lie superalgebras g = g(A) with a Cartan matrix A for which the Shapovalov determinant was known, we consider the Poisson Lie superalgebra poi(0 vertical bar n) and the related Kac-Moody superalgebra., authorCount :2