Construction of Exceptional Lie Algebra G2 and Non-associative Algebras Using Clifford Algebra

The recent development of superstring theory (1) has drawn attention to the potential importance of the exceptional Lie algebras (particularly E/sub 6/ and E/sub 8/) in a future unified theory. The paper consists of three parts. In the first, the authors consider normed algebras and the properties of their transformations. In the second they give the construction of the magic square and the corresponding commutation relations. The construction is anticipated by two symmetric constructions, the symbiosis of which it is. In the final part, the exceptional algebras of the series E are decomposed. The proofs of the Jacobi identities and the identification of the algebras in the magic square are omitted since the calculations are lengthy.

In the study of d-dimensional quantum channels (d⩾2), an assumption which includes many interesting examples, and which has a natural physical interpretation, is that the corresponding Kraus operators form a representation of a Lie algebra. Physically, this is a symmetry algebra for the interaction Hamiltonian. This paper begins a systematic study of channels defined by representations; the famous Werner-Holevo channel is one element of this infinite class. We show that the channel derived from the defining representation of su(n) is a depolarizing channel for all n, but for most other representations this is not the case. Since the standard Bloch sphere only exists for the qubit representation of su(2), we develop a consistent generalization of Bloch’s technique. By representing the density matrix as a polynomial in Lie algebra generators, we determine a class of positive semidefinite matrices which represent quantum states for various channels defined by finite-dimensional representations of semisimple Lie algebras. We also give a general method for finding positive semidefinite matrices using Lie algebraic trace identities. This includes an analysis of channels based on the exceptional Lie algebra g2 and the Clifford algebra.

In this paper various representations of the exceptional Lie algebra G2 are investigated in a purely algebraic manner, and multi-boson/multi-fermion realizations are obtained. Matrix elements of the master representation, which is defined on the space of the universal enveloping algebra of G2, are explicitly determined. From this master representation, different indecomposable representations defined on invariant subspaces or quotient spaces with respect to these invariant subspaces are discussed. Especially, the Verma module are investigated in detail to construct the Dyson-Maleev-like realization of G2. After obtaining explicit forms of all twelve extremal vectors of the Verma module with the dominant highest weight $\Lambda $ , representations with their respective highest weights related to $\Lambda$ are systematically discussed. Moreover, this method can be used to construct fermion realizations from the irreducible representations, without referring to the Lie algebra chains, and a three-fermion realization is constructed as an example.

In two 1966 papers, J. Tits gave a construction of exceptional Lie algebras (hence implicitly exceptional algebraic groups) and a classification of possible indexes of simple algebraic groups. For the special case of his construction that gives groups of type E6, we connect the two papers by answering the question: Given an Albert algebra A and a separable quadratic field extension K, what is the index of the resulting algebraic group?

The simple root system of each exceptional, simple Lie algebra is explicitly constructed in a variety of forms. Each construction is based on the natural embedding in the exceptional algebra of a classical, semi-simple algebra of the same rank. The procedure adopted leads to the discovery of a number of new chains of subalgebras illustrated by means of supplemented Dynkin diagrams. The various sets of simple roots are then used to determine natural labels for the irreducible representations of each of the exceptional algebras. For each such labelling scheme the modification rules for dealing with non-standard representation labels are tabulated. The connection between the natural labels and Dynkin labels is given in detail and a comparison is made with the labels of B.G. Wybourne and M.J. Bowick (1977).

We consider exceptional vertex operator algebras for which particular Casimir vectors constructed from the primary vectors of lowest conformal weight are Virasoro descendants of the vacuum. We discuss constraints on these theories that follow from an analysis of appropriate genus zero and genus one two point correlation functions. We find explicit differential equations for the partition function in the cases where the lowest weight primary vectors form a Lie algebra or a Griess algebra. Examples include the Wess-Zumino-Witten model for Deligne's exceptional Lie algebras and the Moonshine Module. We partially verify the irreducible decomposition of the tensor product of Deligne's exceptional Lie algebras and consider the possibility of similar decompositions for tensor products of the Griess algebra. We briefly discuss some conjectured extremal vertex operator algebras arising in Witten's recent work on three dimensional black holes.

The exceptional simple graded Lie algebras whose existence is suggested by the results of the preceding paper are explicitly constructed. In this way the classification of all simple graded Lie algebras whose Lie algebra is reductive is completed.

In the present paper, we continue the project of systematic construction of invariant differential operators on the example of the non-compact exceptional algebra E7(−25). Our choice of this particular algebra is motivated by the fact that it belongs to a narrow class of algebras, which we call ‘conformal Lie algebras’, which have very similar properties to the conformal algebras of n-dimensional Minkowski spacetime. This class of algebras is identified and summarized in a table. Another motivation is related to the AdS/CFT correspondence. We give the multiplets of indecomposable elementary representations, including the necessary data for all relevant invariant differential operators.

The construction of a family of real Hamiltonian forms (RHF) for the special class of affine 1+1-dimensional Toda field theories (ATFT) is reported. Thus the method, proposed in (1) for systems with finite number of degrees of freedom is generalized to infinite- dimensional Hamiltonian systems. The construction method is illustrated on the explicit nontrivial example of RHF of ATFT related to the exceptional algebras E6 and E7. The involutions of the local integrals of motion are proved by means of the classical R-matrix approach.

From the embedding E8⊇D4×D4, defining fermionic operators which transform, respectively, as the vectorial and spinorial representations of D4, the algebra of E8 is constructed in terms of a bilinear in the fermionic operators. Then, by embeddings, the adjoint and fundamental representations of the other exceptional algebras are explicitly written.

We show that the Casimir operators of the semidirect products of the exceptional Lie algebra G2 and a Heisenberg algebra can be constructed explicitly from the Casimir operators of G2.

Nilpotent coadjoint orbits in small characteristic

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.

Local derivations and automorphisms of Cayley algebras

In a previous paper, we studied the characters and Clebsch-Gordan series for the exceptional Lie algebra E7 by relating them to the quantum trigonometric Calogero-Sutherland Hamiltonian with the coupling constant κ = 1. We now extend that approach to the case of an arbitrary coupling constant.