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Jiri Patera

Publications and source records attributed to Jiri Patera.

27 records · Page 2Linked to original sources
arXiv2007

Invariants of Triangular Lie Algebras

Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three classes of Lie algebras, namely those which are either strictly or non-strictly triangular, and for so-called special upper triangular Lie algebras. Algebraic algorithm of [J. Phys. A: Math. Gen., 2006, V.39, 5749; math-ph/0602046], developed further in [J. Phys. A: Math. Theor., 2007, V.40, 113; math-ph/0606045], is used to determine the invariants. A conjecture of [J. Phys. A: Math. Gen., 2001, V.34, 9085], concerning the number of independent invariants and their form, is corroborated.

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arXiv2007

Antisymmetric Orbit Functions

In the paper, properties of antisymmetric orbit functions are reviewed and further developed. Antisymmetric orbit functions on the Euclidean space $E_n$ are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a Weyl group, corresponding to a Coxeter-Dynkin diagram. Properties of such functions are described. These functions are closely related to irreducible characters of a compact semisimple Lie group $G$ of rank $n$. Up to a sign, values of antisymmetric orbit functions are repeated on copies of the fundamental domain $F$ of the affine Weyl group (determined by the initial Weyl group) in the entire Euclidean space $E_n$. Antisymmetric orbit functions are solutions of the corresponding Laplace equation in $E_n$, vanishing on the boundary of the fundamental domain $F$. Antisymmetric orbit functions determine a so-called antisymmetrized Fourier transform which is closely related to expansions of central functions in characters of irreducible representations of the group $G$. They also determine a transform on a finite set of points of $F$ (the discrete antisymmetric orbit function transform). Symmetric and antisymmetric multivariate exponential, sine and cosine discrete transforms are given.

math-ph↗
arXiv2007

Discrete and continuous exponential transforms of simple Lie groups of rank two

We develop and describe continuous and discrete transforms of class functions on compact simple Lie group $G$ as their expansions into series of uncommon special functions, called here $\E$-functions in recognition of the fact that the functions generalize common exponential functions. The rank of $G$ is the number of variables in the $\E$-functions. A uniform discretization of the decomposition problem is described on lattices of any density and symmetry admissible for the Lie group $G$.

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arXiv2006

Invariants of Lie Algebras with Fixed Structure of Nilradicals

An algebraic algorithm is developed for computation of invariants ('generalized Casimir operators') of general Lie algebras over the real or complex number field. Its main tools are the Cartan's method of moving frames and the knowledge of the group of inner automorphisms of each Lie algebra. Unlike the first application of the algorithm in [J. Phys. A: Math. Gen., 2006, V.39, 5749; math-ph/0602046], which deals with low-dimensional Lie algebras, here the effectiveness of the algorithm is demonstrated by its application to computation of invariants of solvable Lie algebras of general dimension $n<\infty$ restricted only by a required structure of the nilradical. Specifically, invariants are calculated here for families of real/complex solvable Lie algebras. These families contain, with only a few exceptions, all the solvable Lie algebras of specific dimensions, for whom the invariants are found in the literature.

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arXiv2006

Orthogonality within the Families of C-, S-, and E-Functions of Any Compact Semisimple Lie Group

The paper is about methods of discrete Fourier analysis in the context of Weyl group symmetry. Three families of class functions are defined on the maximal torus of each compact simply connected semisimple Lie group $G$. Such functions can always be restricted without loss of information to a fundamental region $\check F$ of the affine Weyl group. The members of each family satisfy basic orthogonality relations when integrated over $\check F$ (continuous orthogonality). It is demonstrated that the functions also satisfy discrete orthogonality relations when summed up over a finite grid in $\check F$ (discrete orthogonality), arising as the set of points in $\check F$ representing the conjugacy classes of elements of a finite Abelian subgroup of the maximal torus $\mathbb T$. The characters of the centre $Z$ of the Lie group allow one to split functions $f$ on $\check F$ into a sum $f=f_1+...+f_c$, where $c$ is the order of $Z$, and where the component functions $f_k$ decompose into the series of $C$-, or $S$-, or $E$-functions from one congruence class only.

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arXiv2006

Orbit Functions

In the paper, properties of orbit functions are reviewed and further developed. Orbit functions on the Euclidean space $E_n$ are symmetrized exponential functions. The symmetrization is fulfilled by a Weyl group corresponding to a Coxeter-Dynkin diagram. Properties of such functions will be described. An orbit function is the contribution to an irreducible character of a compact semisimple Lie group $G$ of rank $n$ from one of its Weyl group orbits. It is shown that values of orbit functions are repeated on copies of the fundamental domain $F$ of the affine Weyl group (determined by the initial Weyl group) in the entire Euclidean space $E_n$. Orbit functions are solutions of the corresponding Laplace equation in $E_n$, satisfying the Neumann condition on the boundary of $F$. Orbit functions determine a symmetrized Fourier transform and a transform on a finite set of points.

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arXiv2005

Compact Simple Lie Groups and Their C-, S-, and E-Transforms

New continuous group transforms, together with their discretization over a lattice of any density and admissible symmetry, are defined for a general compact simple Lie groups of rank $2\leq n<\infty$. Rank 1 transforms are known. Rank 2 exposition of $C$- and $S$-transforms is in the literature. The $E$-transforms appear here for the first time.

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