Exceptional Superalgebra Research Articles

The superconformal gaugings in three dimensions

We show how three-dimensional superconformal theories for any number N <= 8 of supersymmetries can be obtained by taking a conformal limit of the corresponding three-dimensional gauged supergravity models. The superconformal theories are characterized by an embedding tensor that satisfies a linear and quadratic constraint. We analyze these constraints and give the general solutions for all cases. We find new N = 4,5 superconformal theories based on the exceptional Lie superalgebras F(4), G(3) and D(2|1;\alpha). Using the supergravity connection we discuss which massive deformations to expect. As an example we work out the details for the case of N = 6 supersymmetry.

$S_4$-symmetry on the Tits construction of exceptional Lie algebras and superalgebras

The classical Tits construction provides models of the exceptional simple Lie algebras in terms of a unital composition algebra and a degree three simple Jordan algebra. A couple of actions of the symmetric group $S_4$ on this construction are given. By means of these actions, the models provided by the Tits construction are related to models of the exceptional Lie algebras obtained from two different types of structurable algebras. Some models of exceptional Lie superalgebras are discussed too.

A SECRET SYMMETRY OF AdS/CFT S-MATRIX

We find a new quantum Yangian symmetry of AdS/CFT S-matrix, which does not have the original Lie algebra 𝔰𝔲(2|2) analogue, and persue the origin of this symmetry in exceptional superalgebra 𝔡(2, 1; ε) , which recovers 𝔰𝔲(2|2) when the paramater ε taken to zero.

An exceptional algebraic origin of the AdS/CFT Yangian symmetry

In the su(2|2) spin chain motivated by the AdS/CFT correspondence, a novel symmetry extending the superalgebra su(2|2) into u(2|2) was found. We pursue the origin of this symmetry in the exceptional superalgebra d(2,1;epsilon), which recovers su(2|2) when the parameter epsilon is taken to zero. Especially, we rederive the Yangian symmetries of the AdS/CFT spin chain using the exceptional superalgebra and find that the epsilon-correction corresponds to the novel symmetry. Also, we reproduce the non-canonical classical r-matrix of the AdS/CFT spin chain expressed with this symmetry from the canonical one of the exceptional algebra.

Embedding of the Lie superalgebra D(2,1;α) into the Lie superalgebra of pseudodifferential symbols on S1∣2

We obtain an embedding of the one-parameter family of exceptional simple Lie superalgebras D(2,1;α) into the Lie superalgebra of pseudodifferential symbols on the supercircle S1∣2. Correspondingly, there is an embedding of D(2,1;α) into a nontrivial central extension of the derived contact superconformal algebra K′(4) realized in terms of 4×4 matrices over a Weyl algebra.

Classification of generalized quantum statistics associated with the exceptional Lie (super)algebras

Generalized quantum statistics (GQS) associated with a Lie algebra or Lie superalgebra extends the notion of para-Bose or para-Fermi statistics. Such GQS have been classified for all classical simple Lie algebras and basic classical Lie superalgebras. In the current paper we finalize this classification for all exceptional Lie algebras and superalgebras. Since the definition of GQS is closely related to a certain Z grading of the Lie (super)algebra G, our classification reproduces some known Z gradings of exceptional Lie algebras. For exceptional Lie superalgebras such a classification of Z gradings has not been given before.

Simple Lie superalgebras and nonintegrable distributions in characteristic p

Recently, Grozman and Leites returned to the original Cartan’s description of Lie algebras to interpret the Melikyan algebras (for p ≤ 5) and several other little-known simple Lie algebras over algebraically closed fields for p = 3 as subalgebras of Lie algebras of vector fields preserving nonintegrable distributions analogous to (or identical with) those preserved by G(2), O(7), Sp(4), and Sp(10). The description was performed in terms of Cartan-Tanaka-Shchepochkina prolongs using Shchepochkina’s algorithm and with the help of SuperLie package. Grozman and Leites also found two new series of simple Lie algebras. Here we apply the same method to distributions preserved by one of the two exceptional simple finite-dimensional Lie superalgebras overℂ; for p = 3, we obtain a series of new simple Lie superalgebras and an exceptional one. Bibliography: 30 titles.

REPRESENTATIONS OF THE EXCEPTIONAL LIE SUPERALGEBRA E(3,6) III: CLASSIFICATION OF SINGULAR VECTORS

We continue the study of irreducible representations of the exceptional Lie superalgebra E(3,6). This is one of the two simple infinite-dimensional Lie superalgebras of vector fields which have a Lie algebra sℓ(3) × sℓ(2) × gℓ(1) as the zero degree component of its consistent ℤ-grading. We provide the classification of the singular vectors in the degenerate Verma modules over E(3,6), completing thereby the classification and construction of all irreducible E(3,6)-modules that are L0-locally finite.

Link Invariants and Lie Superalgebras

Berger and Stassen reviewed skein relations for link invariants coming from the simple Lie algebras . They related the invariants with decomposition of the tensor square of the -module V of minimal dimension into irreducible components. (If V ≄ V *, one should also consider the decompositions of V ⊗ V * and V * ⊗ V *.) Here we consider decompositions into irreducible components for -modules V of minimal dimension over some simple and close to simple Lie superalgebras . For the classical series , as well as for the Poisson and Hamiltonian algebras — “quasi-classical” analogs of and — the answer is rather complicated due to the lack of complete reduciblity. Contrariwise, the case of exceptional Lie superalgebras and turned out to be similar to that of Lie algebras: The -module (here the representation of minimal dimension is the adjoint one) is completely reducible and, remarkably, the spectra of highest weights for are almost identical (in certain coordinates) to that for ! We also consider for α ≠0, 1.

Grassmann speciality of Jordan supersystems

The Grassmann envelope Γ(S)=S 0 ̄ ⊗ Γ 0 ̄ +S 1 ̄ ⊗ Γ 1 ̄ is a useful tool for deciding varietal questions about supersystems S=S 0 ̄ +S 1 ̄ . It is not useful in deciding simplicity, since the envelope is always fraught with trivial ideals coming from trivial superscalars. At first glance it would not seem useful in deciding superspeciality either, but we will show that it is a more sensitive tool than it seems. We say a Jordan supersystem J s (algebra, triple, or pair) is Grassmann special if its Grassmann envelope is special as an ordinary Jordan system over Γ 0 ̄ . Certainly superspeciality J s ⊂ A s + over Φ implies Grassmann speciality Γ( J s )⊂Γ( A s ) + over Γ 0 ̄ , but it is not obvious how Grassmann speciality influences superspeciality of J s . Nevertheless, we show how to transform obstacles to speciality of J s over Φ (anti-superspecial elements) into obstacles to speciality of Γ( J s ) over Γ 0 ̄ (anti-special elements) using Grassmann boosters, so that non-superspeciality J s over Φ implies non-speciality of Γ( J s ) over Γ 0 ̄ . Thus Grassmann speciality is the same as superspeciality: a supersystem J is a superspecial Jordan supersystem over Φ iff Γ( J s ) is a special Jordan system over Γ 0 ̄ . (It is not clear this is the same as speciality of Γ( J s ) over Φ, since in general speciality of quadratic Jordan systems depends on the scalars: we give an example of a Jordan Ω -algebra which is not special, but is special over a scalar subring Φ.) As a corollary, any Jordan superalgebra with zero linear extreme radical which is viably evenly 4-interconnected is superspecial; thus exceptional superalgebras must be built over even parts of degree ⩽3.

Quaternions, octonions and the forms of the exceptional simple classical Lie superalgebras

The forms of the exceptional simple classical Lie superalgebras are determined over arbitrary fields of characteristic $\ne 2,3$.

The Tits Construction and the Exceptional Simple Classical Lie Superalgebras

We construct the exceptional simple classical Lie superalgebras D(2, 1; α) (α ≠ 0, -1), G(3), and F(4) by means of the Tits construction.

A Kronecker factorization theorem for the exceptional Jordan superalgebra

We prove that a Jordan superalgebra J containing the 10-dimensional exceptional Kac superalgebra K 10 is isomorphic to ( K 10⊗ F S)⊕ J′, where S is an associative commutative algebra.

Representations of the exceptional Lie superalgebra E(3,6) I: Degeneracy conditions

Recently one of the authors obtained a classification of simple infinite dimensional Lie superalgebras of vector fields which extends the well known classification of E. Cartan in the Lie algebra case. The list consists of many series defined by simple equations, and of several exceptional superalgebras, among themE(3, 6).In the article we study irreducible representations of the exceptional Lie superalgebraE(3, 6). This superalgebra hassℓ(3)×sℓ(2)×gℓ(1) as the zero degree component of the consistent ℤ-grading which leads us to believe that its representation theory has potential for physical applications.

Representations of the exceptional lie superalgebraE(3, 6) I: Degeneracy conditions

Recently one of the authors obtained a classification of simple infinite dimensional Lie superalgebras of vector fields which extends the well known classification of E. Cartan in the Lie algebra case. The list consists of many series defined by simple equations, and of several exceptional superalgebras, among themE(3, 6).

Complexes of modules over exceptional Lie superalgebras E(3,8) and E(5,10)

In [3, 4, 5], we constructed all degenerate irreducible modules over the exceptional Lie superalgebra E(3, 6). In the present paper, we apply the same method to the exceptional Lie superalgebras E(3, 8) and E(5, 10). The Lie superalgebra E(3, 8) is strikingly similar to E(3, 6). In particular, as in the case ofE(3, 6), themaximal compact subgroup of the group of automorphisms ofE(3, 8) is isomorphic to the group of symmetries of the StandardModel. However, as the computer calculations by Joris van Jeugt show, the fundamental particle contents in the E(3, 8) case is completely different from that in the E(3, 6) case [3]. All the nice features of the latter case, like the CPT symmetry, completely disappear in the former case. We believe that themain reasonbehind this is that, unlikeE(3, 6),E(3, 8) cannot be embedded in E(5, 10), which, we believe, is the algebra of symmetries of the SU5 Grand Unified Model (the maximal compact subgroup of the automorphism group of E(5, 10) is SU5). However, similarity with E(3, 6) allows us to apply to E(3, 8) all the arguments from [3] almost verbatim, and Figure 4.1 of the present paper, that depicts all degenerate E(3, 8)-modules, is almost the same as [3, Figure 3] for E(3, 6). The picture in the E(5, 10) case is quite different (see Figure 5.1). We believe that it depicts all degenerate irreducible E(5, 10)-modules, but we still do not have a proof.

The [formula omitted] coset theory as a Hamiltonian reduction of [formula omitted

We show that the coset sℓ ̂ (2) k 1 ⊕ sℓ ̂ (2) k 2 / sℓ ̂ (2) k 1+k 2 is a quantum Hamiltonian reduction of the exceptional affine Lie superalgebra D ̂ (2|1;α) and that the corresponding W algebra is the commutant of the U qD(2|1;α) quantum group.

Generalized scalar particle quantization in 1+1 dimensions andD(2,1;α)

The exceptional superalgebra D(2,1;?) has been classified as a candidate conformal supersymmetry algebra in two dimensions. We propose an alternative interpretation of it as an extended BFV-BRST quantization superalgebra in 2D (D(2,1;1)osp(2,2|2)). A superfield realization is presented wherein the standard extended phase space coordinates can be identified. The physical states are studied via the cohomology of the BRST operator. Finally we reverse engineer a classical action corresponding to the algebraic model we have constructed, and identify the Lagrangian equations of motion.

Representations of the Exceptional Lie Superalgebra E(3,6) II: Four Series of Degenerate Modules

Four $\ZZ_+$-bigraded complexes with the action of the exceptional infinite-dimensional Lie superalgebra E(3,6) are constructed. We show that all the images and cokernels and all but three kernels of the differentials are irreducible E(3,6)-modules. This is based on the list of singular vectors and the calculation of homology of these complexes. As a result, we obtain an explicit construction of all degenerate irreducible E(3,6)-modules and compute their characters and sizes. Since the group of symmetries of the Standard Model $SU(3) \times SU(2) \times U(1)$ (divided by a central subgroup of order six) is a maximal compact subgroup of the group of automorphisms of E(3,6), our results may have applications to particle physics.

And as Vertex Operator Extensions¶of Dual Affine Algebras

We discover a realisation of the affine Lie superalgebra sl(2|1) and of the exceptional affine superalgebra D(2|1;alpha) as vertex operator extensions of two affine sl(2) algebras with dual levels (and an auxiliary level 1 sl(2) algebra). The duality relation between the levels is (k+1)(k'+1)=1. We construct the representation of sl(2|1) at level k' on a sum of tensor products of sl(2) at level k, sl(2) at level k' and sl(2) at level 1 modules and decompose it into a direct sum over the sl(2|1) spectral flow orbit. This decomposition gives rise to character identities, which we also derive. The extension of the construction to the affine D(2|1;k') at level k is traced to properties of sl(2)+sl(2)+sl(2) embeddings into D(2|1;alpha) and their relation with the dual sl(2) pairs. Conversely, we show how the level k' sl(2) representations are constructed from level k sl(2|1) representations.