Narrow Results

In this paper, we study moduli spaces of low-dimensional complex Lie superalgebras. We discover a similar pattern for the structure of these moduli spaces as we observed for ordinary Lie algebras, namely, that there is a stratification of the moduli space by projective orbifolds. The moduli spaces consist of some families as well as some singleton elements. The different strata are linked by jump deformations, which gives a unique manner of decomposing the moduli space which is consistent with deformation theory.

The cohomology of Lie (super)algebras has many important applications in mathematics and physics. It carries most fundamental ("topological") information about algebra under consideration. At present, because of the need for very tedious algebraic computation, the explicitly computed cohomology for different classes of Lie (super)algebras is known only in a few cases. That is why application of computer algebra methods is important for this problem. We describe here an algorithm and its C implementation for computing the cohomology of Lie algebras and superalgebras. The program can proceed finite-dimensional algebras and infinite-dimensional graded algebras with finite-dimensional homogeneous components. Among the last algebras, Lie algebras and superalgebras of formal vector fields are most important. We present some results of computation of cohomology for Lie superalgebras of Buttin vector fields and related algebras. These algebras being super-analogs of Poisson and Hamiltonian algebras have found many applications to modern supersymmetric models of theoretical and mathematical physics.

In this paper, we derive a general formulation for thermal Lie superalgebras motivated by thermofield dynamics formalism (TFD). Particularly, we construct the thermal Poincaré superalgebras. The operators in TFD are defined through the doubling of the degrees of freedom of the system and it can be related to Hopf algebras. In this way, we explore the notion of quantum group associated with these superalgebras and we show the noncommutativity in this thermal scenario. Furthermore, the thermal M-superalgebra is also derived from TFD prescription.

We give a list of all possible schemes for performing amino acid and codon assignments in algebraic models for the genetic code, which are consistent with a few simple symmetry principles, in accordance with the spirit of the algebraic approach to the evolution of the genetic code proposed by Hornos and Hornos. Our results are complete in the sense of covering all the algebraic models that arise within this approach, whether based on Lie groups/Lie algebras, on Lie superalgebras or on finite groups.

The super extensions of g-cKdV and mKdV integrable systems are proposed. Two hierarchies of super-integrable nonlinear evolution equations are found. In addition, making use of the super-trace identity, we construct the super-Hamiltonian structures of zero-curvature equations associated with Lie superalgebras.

We prove a refinement of Vogel's statement that the Vassiliev invariants of knots coming from semisimple Lie algebras do not generate all Vassiliev invariants. This refinement takes into account the second grading on the Vassiliev invariants induced by cabling of knots. As an application we get an amelioration of the actually known lower bounds for the dimensions of the space of Vassiliev invariants.

In 1998 Victor Kac classified infinite-dimensional, transitive, irreducible ℤ-graded Lie superalgebras of finite depth. Here we classify bitransitive, irreducible ℤ-graded Lie superalgebras of infinite depth and finite growth which are not contragredient. In particular we show that the growth of every such superalgebra is equal to one.

Based on Kostant's cohomological interpretation of the Amitsur–Levitzki theorem, we prove a super version of this theorem for the Lie superalgebras 𝔬𝔰𝔭(1, 2n). We conjecture that no other classical Lie superalgebra can satisfy an Amitsur–Levitzki type super identity. We show several (super) identities for the standard super polynomials. Finally, a combinatorial conjecture on the standard skew supersymmetric polynomials is stated.

In this work we investigate the complex Leibniz superalgebras with characteristic sequence (n - 1, 1|m₁, …, m_k) and with nilindex equal to n + m. We prove that such superalgebras with the condition m₂ ≠ 0 have nilindex less than n + m. Therefore the complete classification of Leibniz algebras with characteristic sequence (n - 1, 1|m₁, …, m_k) and with nilindex equal to n + m is reduced to the classification of filiform Leibniz superalgebras of nilindex equal to n + m, which was provided in [3, 7].

The maximal graded subalgebras for four families of Lie superalgebras of Cartan type over a field of prime characteristic are studied. All maximal graded subalgebras are described completely by a constructive method and their isomorphism classes, dimension formulas are found except for maximal irreducible graded subalgebras. The classification of maximal irreducible graded subalgebras is reduced to the classification of the maximal irreducible subalgebras for the classical Lie superalgebras 𝔤𝔩(m, n), 𝔰𝔩(m, n) and 𝔬𝔰𝔭(m, n).

We study bosonic–fermionic constructions of the root spaces of the orthosymplectic Lie superalgebras. We also present explicitly bosonic–fermionic realizations of the supertrace forms for the orthosymplectic Lie superalgebras. As by-products, we obtain bosonic–fermionic realizations of both the general linear Lie superalgebras and vertex representations of the affine orthosymplectic Lie superalgebras.

We introduce the definition of the typical irreducible modules of the generalized quantum groups, and prove the Weyl–Kac-type formulas of their characters. As a by-product, we obtain the Weyl–Kac-type character formulas of the typical irreducible modules of the quantum superalgebras associated with the basic classical Lie superalgebras, which is explained in Introduction.

The set of standard Capelli bitableaux and the set of standard Young–Capelli bitableaux are bases of $U(\mathrm{\text{gl}}(n))$, whose action on the Gordan–Capelli basis of polynomial algebra $ℂ[M_{n,n}]$ have remarkable properties (see, e.g. [A. Brini, A. Palareti and A. Teolis, Gordan–Capelli series in superalgebras, Proc. Natl. Acad. Sci. USA 85 (1988) 1330–1333; A. Brini and A. Teolis, Young–Capelli symmetrizers in superalgebras, Proc. Natl. Acad. Sci. USA 86 (1989) 775–778; A. Brini and A. Teolis, Capelli bitableaux and $\mathbb{Z}$-forms of general linear Lie superalgebras, Proc. Natl. Acad. Sci. USA 87 (1990) 56–60; A. Brini and A. Teolis, Capelli’s theory, Koszul maps, and superalgebras, Proc. Natl. Acad. Sci. USA 90 (1993) 10245–10249.]). We introduce a new class of elements of $U(\mathrm{\text{gl}}(n))$, called the Capelli immanants, that can be efficiently computed and provide a system of linear generators of $U(\mathrm{\text{gl}}(n))$. The Okounkov quantum immanants [A. Okounkov, Quantum immanants and higher Capelli identities, Transform Groups 1 (1996) 99–126; A. Okounkov, Young basis, Wick formula, and higher Capelli identities, Int. Math. Res. Not. 1996(17) (1996) 817–839.] — quantum immanants, for short — are proved to be simple linear combinations of diagonal Capelli immanants, with explicit coefficients. Quantum immanants can also be expressed as sums of double Young–Capelli bitableaux. Since double Young–Capelli bitableaux uniquely expands into linear combinations of standard Young–Capelli bitableaux, this leads to canonical presentations of quantum immanants, and, furthermore, it does not involve the computation of the irreducible characters of symmetric groups.

Let $𝔤={𝔤}_{\bar{0}}\oplus {𝔤}_{\bar{1}}$ be a finite-dimensional simple Lie superalgebra of type $D(2,1;\alpha )$, $G(3)$ or $F(4)$ over $ℂ$. Let $G$ be the simply connected semisimple algebraic group over $ℂ$ such that $\mathrm{\text{Lie}}(G)={𝔤}_{\bar{0}}$. Suppose $e\in {𝔤}_{\bar{0}}$ is nilpotent. We describe the centralizer ${𝔤}^e$ of $e$ in $𝔤$ and its center $𝔷({𝔤}^e)$ especially. We also determine the labeled Dynkin diagram for $e$. We prove theorems relating the dimension of ${\left(𝔷({𝔤}^e)\right)}^{G^e}$ and the labeled Dynkin diagram.

In this paper, we study holomorphic $\mathbb{Z}$-graded vertex superalgebras. We prove that all such vertex superalgebras of central charge $8$ and $16$ are purely even. For the case of central charge $24$ we prove that the weight-one Lie superalgebra is either zero, of superdimension $24$, or else is one of an explicit list of 1332 semisimple Lie superalgebras.

This paper is primarily concerned with (α, β, γ)-derivations of finite-dimensional Lie superalgebras over the field of complex numbers. Some properties of (α, β, γ)-derivations of the Lie superalgebras are obtained. In particular, two examples for (α, β, γ)-derivations of low-dimensional non-simple Lie superalgebras are presented and the super-spaces of (α, β, γ)-derivations for simple Lie superalgebras are determined. Using certain complex parameters we generalize the concept of cohomology cocycles of Lie superalgebras. A special case for the generalization of 1-cocycles with respect to the adjoint representation is exactly (α, β, γ)-derivations. Furthermore, two-dimensional twisted cocycles of the adjoint representation are investigated in detail.

We formalize higher-dimensional and higher gauge WZW-type sigma-model local prequantum field theory, and discuss its rationalized/perturbative description in (super-)Lie n-algebra homotopy theory (the true home of the "FDA"-language used in the supergravity literature). We show generally how the intersection laws for such higher WZW-type σ-model branes (open brane ending on background brane) are encoded precisely in (super-)L_∞-extension theory and how the resulting "extended (super-)space-times" formalize spacetimes containing σ-model brane condensates. As an application we prove in Lie n-algebra homotopy theory that the complete super-p-brane spectrum of superstring/M-theory is realized this way, including the pure σ-model branes (the "old brane scan") but also the branes with tensor multiplet worldvolume fields, notably the D-branes and the M5-brane. For instance the degree-0 piece of the higher symmetry algebra of 11-dimensional (11D) spacetime with an M2-brane condensate turns out to be the "M-theory super-Lie algebra". We also observe that in this formulation there is a simple formal proof of the fact that type IIA spacetime with a D0-brane condensate is the 11D sugra/M-theory spacetime, and of (prequantum) S-duality for type IIB string theory. Finally we give the non-perturbative description of all this by higher WZW-type σ-models on higher super-orbispaces with higher WZW terms in stacky differential cohomology.

Let $W(\Gamma )$ be a class of not-finitely graded Lie algebras related to generalized Virasoro algebras with basis $\{L_{\alpha ,i},C | \alpha \in \Gamma , i\in {\mathbb{Z}}_{+}\}$, which satisfies relations $[L_{\alpha ,i},L_{\beta ,j}]=(\beta -\alpha )L_{\alpha +\beta ,i+j}+(j-i)L_{\alpha +\beta ,i+j+1}+{\delta }_{\alpha +\beta ,0}{\delta }_{i+j,0}\frac{{\alpha }^3-\alpha }{12}C$ and $[C,L_{\alpha ,i}]=0$. In this paper, $W(\Gamma )$-modules of the intermediate series satisfying some conditions are constructed and classified. We also obtain modules of the intermediate series over the related Lie superalgebra.

We define perfect ideals, near perfect ideals and upper bounded ideals of a finite-dimensional Lie superalgebra, and study the properties of these three kinds of ideals through their relevant sequences. We prove that a Lie superalgebra is solvable if and only if its maximal perfect ideal is zero, or its quotient superalgebra by the maximal perfect ideal is solvable. We also show that a Lie superalgebra is nilpotent if and only if its maximal near perfect ideal is zero. Moreover, we prove that a nilpotent Lie superalgebra has only one upper bounded ideal, which is the nilpotent Lie superalgebra itself.

Let $L$ be a simple modular Lie superealgebra of Witt type or special type over an algebraically closed field of characteristic $p>2$. In this paper, all the superbiderivations of $L$ are studied. By means of weight space decompositions with respect to a suitable torus and the standard $\mathbb{Z}$-grading structures of $L$, we show that the supersymmetric superbiderivations of $L$ are zero. Generalizing a result on the skewsymmetric biderivation of Lie algebras to the super case, we find that all the superbiderivations of $L$ are inner. As applications, the linear supercommuting maps and the supercommutative post-Lie superalgebra structures on $L$ are described.