Exceptional Algebras Research Articles

On Holomorphic Realizations of Nilpotent Lie Algebras

Realizations of five-dimensional Lie algebras as algebras of holomorphic vector fields on homogeneous real hypersurfaces of a three-dimensional complex space are studied. In view of already known results, in the problem of describing such varieties only Levy nondegenerate hypersurfaces with exactly five-dimensional Lie algebras are of interest. It is shown that only two of the nine existing distinct nilpotent Lie algebras admit realizations associated with such varieties, and the varieties corresponding to these exceptional algebras are standard quadrics.

The Magic of Being Exceptional

Starting from the Jordan algebraic interpretation of the “Magic Star” embedding within the exceptional sequence of simple Lie algebras, we exploit the so-called spin factor embedding of rank-3 Jordan algebras and its consequences on the Jordan algebraic Lie symmetries, in order to provide another perspective on the origin of the Exceptional Periodicity (EP) and its “Magic Star” structure. We also highlight some properties of the special class of Vinberg’s rank-3 (dubbed exceptional) T-algebras, appearing on the tips of the “Magic Star” projection of EP(-generalized, finite-dimensional, exceptional) algebras.

Symmetry breaking in quantum curves and super Chern-Simons matrix models

It was known that quantum curves and super Chern-Simons matrix models correspond to each other. From the viewpoint of symmetry, the algebraic curve of genus one, called the del Pezzo curve, enjoys symmetry of the exceptional algebra, while the super Chern-Simons matrix model is described by the free energy of topological strings on the del Pezzo background with the symmetry broken. We study the symmetry breaking of the quantum cousin of the algebraic curve and reproduce the results in the super Chern-Simons matrix model.

Hochschild cohomology for periodic algebras of polynomial growth

We describe the dimensions of low Hochschild cohomology spaces of exceptional periodic representation-infinite algebras of polynomial growth. As an application we obtain that an indecomposable non-standard periodic representation-infinite algebra of polynomial growth is not derived equivalent to a standard self-injective algebra.

Quiver theories and formulae for nilpotent orbits of Exceptional algebras

We treat the topic of the closures of the nilpotent orbits of the Lie algebras of Exceptional groups through their descriptions as moduli spaces, in terms of Hilbert series and the highest weight generating functions for their representation content. We extend the set of known Coulomb branch quiver theory constructions for Exceptional group minimal nilpotent orbits, or reduced single instanton moduli spaces, to include all orbits of Characteristic Height 2, drawing on extended Dynkin diagrams and the unitary monopole formula. We also present a representation theoretic formula, based on localisation methods, for the normal nilpotent orbits of the Lie algebras of any Classical or Exceptional group. We analyse lower dimensioned Exceptional group nilpotent orbits in terms of Hilbert series and the Highest Weight Generating functions for their decompositions into characters of irreducible representations and/or Hall Littlewood polynomials. We investigate the relationships between the moduli spaces describing different nilpotent orbits and propose candidates for the constructions of some non-normal nilpotent orbits of Exceptional algebras.

Quaternion frames and fractal surface as tools to control orientation of a spacecraft

Reorientation of an object's (spacecraft) problem is formulated in details in the SO(3,R) and SU(2) groups matrix terms using the most optional math tool of exceptional algebra of quaternion numbers. A thorough analysis of the two approaches is made resulting in original formulas linking parameters of the assigned object's consequent 3D rotations with a single rotation about a unit vector pointing the instant rotation axis, respective operational technology described with relevant examples. It is also demonstrated that an axial quaternion frame admits fractalization so that the reorientation problem is reduced to deformations of the sub-geometric fractal surface.

Exceptional quantum geometry and particle physics

Based on an interpretation of the quark–lepton symmetry in terms of the unimodularity of the color group SU(3) and on the existence of 3 generations, we develop an argumentation suggesting that the “finite quantum space” corresponding to the exceptional real Jordan algebra of dimension 27 (the Euclidean Albert algebra) is relevant for the description of internal spaces in the theory of particles. In particular, the triality which corresponds to the 3 off-diagonal octonionic elements of the exceptional algebra is associated to the 3 generations of the Standard Model while the representation of the octonions as a complex 4-dimensional space C⊕C3 is associated to the quark–lepton symmetry (one complex for the lepton and 3 for the corresponding quark). More generally it is suggested that the replacement of the algebra of real functions on spacetime by the algebra of functions on spacetime with values in a finite-dimensional Euclidean Jordan algebra which plays the role of “the algebra of real functions” on the corresponding almost classical quantum spacetime is relevant in particle physics. This leads us to study the theory of Jordan modules and to develop the differential calculus over Jordan algebras (i.e. to introduce the appropriate notion of differential forms). We formulate the corresponding definition of connections on Jordan modules.

Exceptional versus superPoincaré algebra as the defining symmetry of maximal supergravity

We describe how one may use either the superPoincar\'e algebra or the exceptional algebra to construct maximal supergravity theories in the light-cone formalism. The d=4 construction shows both symmetries albeit in a non-linearly realized manner. In d=11, we find that we have to choose which of these two symmetries to use, in constructing the theory. In order to understand the other "unused" symmetry, one has to perform a highly non-trivial field redefinition. We argue that this shows that one cannot trust counterterm arguments that do not take the full symmetry of the theory into account. Finally we discuss possible consequences for Superstring theory and M-theory.

On the N = 1 ∗ $$ \mathcal{N}={1}^{\ast } $$ gauge theory on a circle and elliptic integrable systems

We continue our study of the N=1* supersymmetric gauge theory and its relation to elliptic integrable systems. Upon compactification on a circle, we show that the semi-classical analysis of the massless and massive vacua depends on the classification of nilpotent orbits, as well as on the conjugacy classes of the component group of their centralizer. We demonstrate that semi-classically massless vacua can be lifted by Wilson lines in unbroken discrete gauge groups. The pseudo-Levi subalgebras that play a classifying role in the nilpotent orbit theory are also key in defining generalized Inozemtsev limits of (twisted) elliptic integrable systems. We illustrate our analysis in the N=1* theories with gauge algebras su(3), su(4), so(5) and for the exceptional gauge algebra G2. We map out modular duality diagrams of the massive and massless vacua. Moreover, we provide an analytic description of the branches of massless vacua in the case of the su(3) and the so(5) theory. The description of these branches in terms of the complexified Wilson lines on the circle invokes the Eichler-Zagier technique for inverting the elliptic Weierstrass function. After fine-tuning the coupling to elliptic points of order three, we identify the Argyres-Douglas singularities of the su(3) theory.

Just infinite alternative algebras

Alternative just infinite-dimensional algebras are studied, i.e., infinite-dimensional algebras in which every nonzero ideal has finite codimension. It is proved that these algebras are prime. In the nonassociative case, the Noetherian property with respect to one-sided ideals is proved, and the cases of Cayley–Dickson rings and exceptional algebras are investigated.

Periodicity of self-injective algebras of polynomial growth

Let A be an indecomposable representation-infinite tame finite-dimensional algebra of polynomial growth over an algebraically closed field. We prove that A is a periodic algebra with respect to action of the bimodule syzygy operator if and only if A is Morita equivalent to a socle deformation of an orbit algebra Bˆ/G where Bˆ is the repetitive category of a tubular algebra B and G is an admissible infinite cyclic automorphism group of Bˆ. The main contribution in the paper is to prove the sufficiency part of this equivalence. It is known that every orbit algebra Bˆ/G of a tubular algebra B admits a presentation as an orbit algebra T(B)(r)/H of an r-fold trivial extension algebra T(B)(r) of B with respect to free action of a finite cyclic automorphism group H of T(B)(r). A significant part of the paper is devoted to explicit descriptions of the minimal projective bimodule resolutions of properly chosen ten exceptional self-injective algebras of polynomial growth and showing that all of them are periodic algebras. Then we deduce the periodicity of all algebras socle equivalent to the orbit algebras Bˆ/G=T(B)(r)/H of tubular algebras B from the periodicity of these ten exceptional algebras, using invariance of periodicity for finite Galois coverings and derived equivalences of algebras.

Wei–Norman equations for classical groups

We show that the nonlinear autonomous Wei–Norman equations, expressing the solution of a linear system of non-autonomous equations on a Lie algebra, can be reduced to the hierarchy of matrix Riccati equations in the case of all classical simple Lie algebras. The result generalizes our previous one concerning the complex Lie algebra of the special linear group. We show that it cannot be extended to all simple Lie algebras, in particular to the exceptional G2 algebra.

“General Theory of Particle Mechanics” arising from a fractal surface

The logical line is traced of formulation of theory of mechanics founded on the basic correlations of mathematics of hypercomplex numbers and associated geometric images. Namely, it is shown that the physical equations of quantum, classical and relativistic mechanics can be regarded as mathematical consequences of a single condition of stability of exceptional algebras of real, complex and quaternion numbers under transformations of primitive constituents of their units and elements. In the course of the study a notion of basic fractal surface underlying the physical three-dimensional space is introduces, and an original geometric treatment (admitting visualization) of some formerly considered abstract functions (mechanical action, space-time interval) are suggested.

Physical theories in hypercomplex geometric description

Compact description is given of algebras of poly-numbers: quaternions, bi-quaternions, double (split-complex) and dual numbers. All units of these (and exceptional) algebras are shown to be represented by direct products of 2D vectors of a local basis defined on a fundamental surface. In this math medium a series of equalities identical or similar to known formulas of physical laws is discovered. In particular, a condition of the algebras' stability with respect to transformations of the 2D-basis turns out equivalent to the spinor (Schrödinger–Pauli and Hamilton–Jacobi) equations of mechanics. It is also demonstrated that isomorphism of SO(3, 1) and SO(3, ℂ) groups leads to formulation of a quaternion relativity theory predicting all effects of special relativity but simplifying solutions of relativistic problems in non-inertial frames. Finely it is shown that the Cauchy–Riemann type equations written for functions of quaternion variable repeat vacuum Maxwell equations of electrodynamics, while a quaternion space with non-metricity comprises main relations of Yang–Mills field theory.

Classification of Invariant Differential Operators for Non-Compact Lie Algebras via Parabolic Relations

In the present paper we review the progress of the project of classification and construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we called earlier 'conformal Lie algebras' (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduced recently the new notion of parabolic relation between two non-compact semisimple Lie algebras G and G' that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Thus, we consider the exceptional algebra E7(7) which is parabolically related to the CLA E7(−25). Other interesting examples are the orthogonal algebras so(p, q) all of which are parabolically related to the conformal algebra so(n, 2) with p + q = n + 2, the parabolic subalgebras including the Lorentz subalgebra so(n − 1,1) and its analogs so(p − 1, q − 1). Further we consider the algebras sl(2n, ℝ) and for n = 2k the algebras su* (4k) which are parabolically related to the CLA su(n,n). Further we consider the algebras sp(r,r) which are parabolically related to the CLA sp(2r, ℝ). We consider also E6(6) and E6(2) which are parabolically related to the hermitian symmetric case E6(-14),

Growth rate of cluster algebras

We complete the computation of growth rate of cluster algebras. In particular, we show that growth of all exceptional non-affine mutation-finite cluster algebras is exponential.

Degenerate alternative algebras

We prove that each degenerate alternative algebra of characteristic ≠ 2 contains a nonzero ideal with an additive basis consisting of the absolute zero divisors of an arbitrary large order. As a corollary we establish the existence of infinitely many nonisomorphic commutative prime alternative algebras and existence of infinite series of strict and nonstrict exceptional alternative algebras with different sets of proper identities.

Exceptional Moufang quadrangles and structurable algebras

In 2000, J. Tits and R. Weiss classified all Moufang spherical buildings of rank two, also known as Moufang polygons. The hardest case in the classification consists of the Moufang quadrangles. They fall into different families, each of which can be described by an appropriate algebraic structure. For the exceptional quadrangles, this description is intricate and involves many different maps that are defined ad hoc and lack a proper explanation. In this paper, we relate these algebraic structures to two other classes of algebraic structures that had already been studied before, namely to Freudenthal triple systems and to structurable algebras. We show that these structures give new insight in the understanding of the corresponding Moufang quadrangles.

Invariant differential operators for non-compact Lie algebras parabolically related to conformal Lie algebras

AbstractIn the present paper we continue the project of systematic construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we call ’conformal Lie algebras’ (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduce the new notion ofparabolic relationbetween two non-compact semisimple Lie algebrasGandG′that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Thus, we consider the exceptional algebraE7(7)which is parabolically related to the CLAE7(−25), the parabolic subalgebras includingE6(6)andE6(−26). Other interesting examples are the orthogonal algebras so(p, q) all of which are parabolically related to the conformal algebraso(n, 2) withp+q=n+ 2, the parabolic subalgebras including the Lorentz subalgebra so(n− 1, 1) and its analogs so(p− 1,q− 1). We consider also E6(6)and E6(2)which are parabolically related to the hermitian symmetric case E6(−14), the parabolic subalgebras including real forms of sl(6).We also give a formula for the number of representations in the main multiplets valid for CLAs and all algebras that are parabolically related to them. In all considered cases we give the main multiplets of indecomposable elementary representations including the necessary data for all relevant invariant differential operators. In the case ofso(p,q) we give also the reduced multiplets. We should stress that the multiplets are given in the most economic way in pairs ofshadow fields. Furthermore we should stress that the classification of all invariant differential operators includes as special cases all possibleconservation lawsandconserved currents, unitary or not.

A quantum affine algebra for the deformed Hubbard chain

The integrable structure of the one-dimensional Hubbard model is based on Shastry's R-matrix and the Yangian of a centrally extended superalgebra. Alcaraz and Bariev have shown that the model admits an integrable deformation whose R-matrix has recently been found. This R-matrix is of trigonometric type and here we derive its underlying exceptional quantum affine algebra. We also show how the algebra reduces to the above-mentioned Yangian and to the conventional quantum affine algebra in two special limits.