This note is devoted to a more detailed description of one of the five simple exceptional Lie superalgebras of vector fields, cvect(0|3)_*, a subalgebra of vect(4|3). We derive differential equations for its elements, and solve these equations. Hence we get an exact form for the elements of cvect(0|3)_*. Moreover we realize cvect(0|3)_* by "glued" pairs of generating functions on a (3|3)-dimensional periplectic (odd symplectic) supermanifold and describe the bracket explicitly.
We examine the structure of gauge transformations in extended geometry, the framework unifying double geometry, exceptional geometry, etc. This is done by giving the variations of the ghosts in a Batalin-Vilkovisky framework, or equivalently, an $L_\infty$ algebra. The $L_\infty$ brackets are given as derived brackets constructed using an underlying Borcherds superalgebra ${\scr B}({\mathfrak g}_{r+1})$, which is a double extension of the structure algebra ${\mathfrak g}_r$. The construction includes a set of "ancillary" ghosts. All brackets involving the infinite sequence of ghosts are given …
We consider Lie superalgebras under constraints of Hamiltonian reduction, yielding finite $W$-superalgebras which provide candidates for quadratic spacetime superalgebras. These have an undeformed bosonic symmetry algebra (even generators) graded by a fermionic sector (supersymmetry generators) with anticommutator brackets which are quadratic in the even generators. We analyze the reduction of several Lie superalgebras of type $gl(M|N)$ or $osp(M|2N)$ at the classical (Poisson bracket) level, and also establish their quantum (Lie bracket) equivalents. Purely bosonic extensions are also conside…
Dual Pfaff equations (of the form \tilde D^a = 0, \tilde D^a some vector fields of degree -1) preserved by the exceptional infinite-dimensional simple Lie superalgebras ksle(5|10), vle(3|6) and mb(3|8) are constructed, yielding an intrinsic geometric definition of these algebras. This leads to conditions on the vector fields, which are solved explicitly. Expressions for preserved differential form equations (Pfaff equations), brackets (similar to contact brackets) and tensor modules are written down. The analogous construction for the contact superalgebra k(1|m) (a.k.a. the centerless N=m supe…
We introduce and study a construction of higher derived brackets generated by a (not necessarily inner) derivation of a Lie superalgebra. Higher derived brackets generated by an element of a Lie superalgebra were introduced in our earlier work. Examples of higher derived brackets naturally appear in geometry and mathematical physics. From a totally different viewpoint, we show that higher derived brackets arise when one wants to turn the inclusion map of a subalgebra of a differential Lie superalgebra, with a given complementary subalgebra, into a fibration. (For a non-Abelian complementary su…
A linear odd Poisson bracket realized solely in terms of Grassmann variables is suggested. It is revealed that with the bracket, corresponding to a semi-simple Lie group, both a Grassmann-odd Casimir function and invariant (with respect to this group) nilpotent differential operators of the first, second and third orders are naturally related and enter into a finite-dimensional Lie superalgebra. A connection of the quantities, forming this Lie superalgebra, with the BRST charge, $Δ$-operator and ghost number operator is indicated.
The aim of this paper is to generalise the construction of $n$-ary Hom-Lie bracket by means of an $(n-2)$-cochain of given Hom-Lie algebra to super case inducing a $n$-Hom-Lie superalgebras. We study the notion of generalized derivation and Rota-Baxter operators of $n$-ary Hom-Nambu and $n$-Hom-Lie superalgebras and their relation with generalized derivation and Rota-Baxter operators of Hom-Lie superalgebras. We also introduce the notion of $3$-Hom-pre-Lie superalgebras which is the generalization of $3$-Hom-pre-Lie algebras.
The purpose of this article is to investigate relations between W-superalgebras and integrable super-Hamiltonian systems. To this end, we introduce the generalized Drinfel'd-Sokolov (D-S) reduction associated to a Lie superalgebra $g$ and its even nilpotent element $f$, and we find a new definition of the classical affine W-superalgebra $W(g,f,k)$ via the D-S reduction. This new construction allows us to find free generators of $W(g,f,k)$, as a differential superalgebra, and two independent Lie brackets on $W(g,f,k)/\partial W(g,f,k).$ Moreover, we describe super-Hamiltonian systems with the P…
A linear odd Poisson bracket (antibracket) realized solely in terms of Grassmann variables is suggested. It is revealed that the bracket, which corresponds to a semi-simple Lie group, has at once three Grassmann-odd nilpotent $Δ$-like differential operators of the first, the second and the third orders with respect to Grassmann derivatives, in contrast with the canonical odd Poisson bracket having the only Grassmann-odd nilpotent differential $Δ$-operator of the second order. It is shown that these $Δ$-like operators together with a Grassmann-odd nilpotent Casimir function of this bracket form…
We prove that the hierarchy of higher antibrackets (aka higher Koszul brackets, aka Koszul braces) of a linear operator $Δ$ on a commutative superalgebra can be defined by some universal formulas involving iterated Nijenhuis-Richardson brackets having as arguments $Δ$ and the multiplication operators. As a byproduct, we can immediately extend higher antibrackets to noncommutative algebras in a way preserving the validity of generalized Jacobi identities.
The uniqueness of (the class of) deformation of Poisson Lie algebra has long been a completely accepted folklore. Actually, it is wrong as stated, because its validity depends on the class of functions that generate Poisson Lie algebra, Po(2n): it is true for polynomials but false for Laurent polynomials. We show that unlike the Lie superalgebra Po(2n|m), its quotient modulo center, the Lie superalgebra H(2n|m) of Hamiltonian vector fields with polynomial coefficients, has exceptional extra deformations for (2n|m)=(2|2) and only for this superdimension. We relate this result to the complete de…
We obtain an infinite sequence of bosonic non-local conserved quantities for the N=1 supersymmetric Korteweg-de Vries equation. It is generated from a bosonic non-local conserved quantity of Super Gardner equation. In distinction to the already known one with odd parity and dimension 1/2, it has even parity and dimension 1. It fits exactly in the supersymmetric cohomology in the space of conserved quantities that we also introduce here. Using results from this cohomology we obtain the Poisson bracket of several non-local conserved quantities, including the already known odd ones and the new ev…
From the four normed division algebras--the real numbers, complex numbers, quaternions and octonions, of dimension k=1, 2, 4 and 8, respectively--a systematic procedure gives a 3-cocycle on the Poincare superalgebra in dimensions k+2=3, 4, 6 and 10, and a 4-cocycle on the Poincare superalgebra in dimensions k+3=4, 5, 7 and 11. The existence of these cocycles follow from spinor identities that hold only in these dimensions, and which are closely related to the existence of the superstring in dimensions k+2, and the super-2-brane in dimensions k+3. In general, an (n+1)-cocycle on a Lie superalge…
