A new functor from <Emphasis Type="Italic">D</Emphasis><Subscript>5</Subscript>-Mod to <Emphasis Type="Italic">E</Emphasis><Subscript>6</Subscript>-Mod

We find a new representation of the simple Lie algebra of type E ₆ on the polynomial algebra in 16 variables, which gives a fractional representation of the corresponding Lie group on 16-dimensional space. Using this representation and Shen’s idea of mixed product, we construct a new functor from D ₅-Mod to E ₆-Mod. A condition for the functor to map a finite-dimensional irreducible D ₅-module to an infinite-dimensional irreducible E ₆-module is obtained. Our results yield explicit constructions of certain infinite-dimensional irreducible weight E6-modules with finite-dimensional weight subspaces. In our approach, the idea of Kostant’s characteristic identities plays a key role.     

Algebras of Quotients of Jordan–Lie Algebras

In this article, we introduce the notion of algebra of quotients of a Jordan–Lie algebra. Properties such as semiprimeness or primeness can be lifted from a Jordan–Lie algebra to its algebras of quotients. Finally, we construct a maximal algebra of quotients for every semiprime Jordan–Lie algebra.     

Groups of outer type E<Subscript>6</Subscript> with trivial Tits algebras

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?     

Irreducible <Emphasis Type="Italic">A</Emphasis><Stack><Subscript>1</Subscript><Superscript>(1)</Superscript></Stack>-modules from modules over two-dimensional non-abelian Lie algebra

For any module V over the two-dimensional non-abelian Lie algebra b and scalar α ∈ C, we define a class of weight modules F ^α (V) with zero central charge over the affine Lie algebra A ₁ ⁽¹⁾ . These weight modules have infinitedimensional weight spaces if and only if V is infinite dimensional. In this paper, we will determine necessary and sufficient conditions for these modules F ^α(V) to be irreducible. In this way, we obtain a lot of irreducible weight A ₁ ⁽¹⁾ -modules with infinite-dimensional weight spaces.     

A Differential Calculus on the Z<Subscript>3</Subscript>-graded Quantum Group GL<Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>(2)

In this work, we introduce Z₃-graded quantum group GL\({_q(2,{\mathbb C})}\) with the help of Z₃-graded quantum plane and a Z₃-graded bicovariant differential calculus on the Z₃-graded quantum group GL _q (2). The corresponding Z₃-graded quantum Lie superalgebra is obtained.     

Explicit computations of low-lying eigenfunctions for the quantum trigonometric Calogero-Sutherland model related to the exceptional algebra <Emphasis Type="Italic">E</Emphasis><Subscript>7</Subscript>

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. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 283–293, February, 2008.     

Gelfand–Kirillov Dimension and Local Finiteness of Jordan Superpairs Covered by Grids and Their Associated Lie Superalgebras

Abstract

In this paper we show that a Lie superalgebra L graded by a 3-graded irreducible root system has Gelfand–Kirillov dimension equal to the Gelfand–Kirillov dimension of its coordinate superalgebra A, and that L is locally finite if and only A is so. Since these Lie superalgebras are coverings of Tits–Kantor–Koecher superalgebras of Jordan superpairs covered by a connected grid, we obtain our theorem by combining two other results. Firstly, we study the transfer of the Gelfand–Kirillov dimension and of local finiteness between these Lie superalgebras and their associated Jordan superpairs, and secondly, we prove the analogous result for Jordan superpairs: the Gelfand–Kirillov dimension of a Jordan superpair V covered by a connected grid coincides with the Gelfand– Kirillov dimension of its coordinate superalgebra A, and V is locally finite if and only if A is so.     

Real elements in groups of type <Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>

Let G be a group. An element x ∈ G is called real if x is conjugate to x ^?1 in G. In this paper we study the structure of real elements in the compact connected Lie group of type F ₄ and algebraic groups of type F ₄ defined over an arbitrary field.     

The Exceptional Compact Symmetric Spaces <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> and <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">SO</Emphasis>(4) as Tubes

In the present paper we discuss in detail the cohomogeneity one isometric actions of the Lie groups SU(3) × SU(3) and SU(3) on the exceptional compact symmetric spaces G₂ and G₂/SO(4), respectively. We show that the principal orbits coincide with the tubular hypersurfaces around the totally geodesic singular orbits, and the symmetric spaces G₂ and G₂/SO(4) can be thought of as compact tubes around SU(3) and P², respectively. Moreover, we determine the radii of these tubes and describe the shape operators of the principal orbits. Finally, we apply these results to compute the volumes of the two symmetric spaces.The author was partially supported by the Hungarian National Science and Research Foundation OTKA T032478.     

Prime Quotients of Jordan Systems and Lie Algebras

We show that, unlike alternative algebras, prime quotients of a nondegenerate Jordan system or a Lie algebra need not be nondegenerate, even if the original Jordan system is primitive, or the Lie algebra is strongly prime, both with nonzero simple hearts. Nevertheless, for Jordan systems and Lie algebras directly linked to associative systems, we prove that even semiprime quotients are necessarily nondegenerate.     

Representations and T*-Extensions of Hom–Jordan–Lie Algebras

The purpose of this article is to study representations and T*-extensions of δ-hom–Jordan–Lie algebras. In particular, adjoint representations, trivial representations, deformations, and many properties of T*-extensions of δ-hom–Jordan–Lie algebras are studied in detail. Derivations and central extensions of δ-hom–Jordan–Lie algebras are also discussed as an application.     

Spectra of the finite simple groups <Emphasis Type="Italic">E</Emphasis> <Subscript>7</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

We describe the spectra of simple and universal groups of Lie type E₇.     

Commuting <Emphasis Type="Italic">U</Emphasis>-operators and nondegenerate imbeddings of Jordan systems

Over an arbitrary ring of scalars, we build a Jordan algebra J having two elements x,y∈J such that x?y=0, but whose U-operators do not commute. This shows that nondegeneracy is a necessary condition in the main theorem of Commuting U-Operators in Jordan Algebras by J. A Anquela, T. Cortés and H. P. Petersson (Trans. Amer. Math. Soc. 366 (2014), 5877–5902). As a consequence, we obtain examples of Jordan systems over arbitrary rings of scalars that cannot be imbedded in nondegenerate systems.     

Dual Coalgebras of Jordan Bialgebras and Superalgebras

W. Michaelis showed for Lie bialgebras that the dual coalgebra of a Lie algebra is a Lie bialgebra. In the present article we study an analogous question in the case of Jordan bialgebras. We prove that a simple infinite-dimensional Jordan superalgebra of vector type possesses a nonzero dual coalgebra. Thereby, we demonstrate that the hypothesis formulated by W. Michaelis for Lie coalgebras fails in the case of Jordan supercoalgebras.     

On deformations of <Emphasis Type="Italic">F</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> in compact Lie groups

We study some properties of the varieties of deformations of free groups in compact Lie groups. In particular, we prove a conjecture of Margulis and Soifer about the density of non-virtually free points in such variety, and a conjecture of Goldman on the ergodicity of the action of Aut(F _n ) on such variety when n ≥ 3. The author was partially supported by NSF grant DMS-0404557, BSF grant 2004010, and the ‘Finite Structures’ Marie Curie Host Fellowship, carried out at the Alfréd Rényi Institute of Mathematics in Budapest.     

A Selberg integral for the Lie algebra A<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

A new q-binomial theorem for Macdonald polynomials is employed to prove an A _n analogue of the celebrated Selberg integral. This confirms the \mathfrakg = A_n \mathfrak{g} ={\rm{A}}_{n} case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for every simple Lie algebra \mathfrakg \mathfrak{g} .     

Characterization of Locally Finite Simple Groups of the Type <Superscript>3</Superscript><Emphasis Type="Italic">D</Emphasis><Subscript>4</Subscript> Over Fields of Odd Characteristic in the Class of Periodic Groups

We prove that a periodic group is locally finite, given that each finite subgroup of the group lies in a subgroup isomorphic to a finite simple group of Lie type ³D₄ over a field of odd characteristic.     

On generalized Jordan derivations of Lie triple systems

Under some conditions we prove that every generalized Jordan triple derivation on a Lie triple system is a generalized derivation. Specially, we conclude that every Jordan triple θ-derivation on a Lie triple system is a θ-derivation.     

Integral geometry under G<Subscript>2</Subscript> and Spin(7)

A Hadwiger-type theorem for the exceptional Lie groups G₂ and Spin(7) is proved. The algebras of G₂ or Spin(7) invariant, translation invariant continuous valuations are both of dimension 10. Geometrically meaningful bases are constructed and the algebra structures are computed. Finally, the kinematic formulas for these groups are determined.