Infinite Lie Algebras Research Articles

The derivation algebras of the classical infinite Lie algebras

The derivation algebras of the classical infinite Lie algebras

Vortices in He II, current algebras and quantum knots

A canonical quantization scheme is developed for vertices in superfluid He II, using Dirac's technique for constrained hamiltonian systems. Quantization introduces in the theory in natural way the structure of the infinite Lie algebra of incompressible flows. We argue that all the topological invariants of the vortex, considered as a knot, can be regarded as observables of the system. Finally unitary representations of measure preserving flows on R 3 and current algebra are discussed.

Axioms for finite and infinite classical Lie algebras

Axioms for finite and infinite classical Lie algebras

Some Unusual Applications of Lie Algebra Representations in Quantum Theory

Several new applications of Lie algebra representations in quantum theory are discussed. These include a method of self adjoint extension of the Dirac Hamiltonian, description of continuous spectra, dynamical groups and infinite-component wave equations, induced and indecomposable representations and infinite parameter Lie algebras.

Infinite Nodal Noncommutative Jordan Algebras; Differentiably Simple Algebras

The first result is that any differentiably simple algebra of the form $A = F1 + R$, for R a proper ideal, 1 the identity element, and F the base field, must be a subalgebra of a (commutative associative) power series algebra over F, and is truncated if the characteristic is not zero. Moreover the algebra A contains the polynomial subalgebra generated by the indeterminates and identity of the power series algebra. This is used to prove that if A is any simple flexible algebra of the form $A = F1 + R$, R an ideal of ${A^ + }$, then ${A^ + }$ is a subalgebra of a power series algebra and multiplication in A is determined by certain elements ${c_{ij}}$ in A as in \[ fg = f \cdot g + \frac {1}{2}\sum {\frac {{\partial f}}{{\partial {x_i}}} \cdot \frac {{\partial g}}{{\partial {x_j}}} \cdot {c_{ij}},} \] where ${c_{ij}} = - {c_{ji}}$ and “$\cdot$” is the multiplication in ${A^ + }$. This applies in particular to simple nodal noncommutative Jordan algebras (of characteristic not 2). These results suggest a method of constructing noncommutative Jordan algebras of the given form. We have done this with the restriction that the ${c_{ij}}$ lie in F1. The last result is that if A is a finitely generated simple noncommutative algebra of characteristic 0 of this form, then Der (A) is an infinite simple Lie algebra of a known type.

Infinite nodal noncommutative Jordan algebras; differentiably simple algebras

The first result is that any differentiably simple algebra of the form A = F 1 + R A = F1 + R , for R a proper ideal, 1 the identity element, and F the base field, must be a subalgebra of a (commutative associative) power series algebra over F, and is truncated if the characteristic is not zero. Moreover the algebra A contains the polynomial subalgebra generated by the indeterminates and identity of the power series algebra. This is used to prove that if A is any simple flexible algebra of the form A = F 1 + R A = F1 + R , R an ideal of A + {A^ + } , then A + {A^ + } is a subalgebra of a power series algebra and multiplication in A is determined by certain elements c i j {c_{ij}} in A as in \[ f g = f ⋅ g + 1 2 ∑ ∂ f ∂ x i ⋅ ∂ g ∂ x j ⋅ c i j , fg = f \cdot g + \frac {1}{2}\sum {\frac {{\partial f}}{{\partial {x_i}}} \cdot \frac {{\partial g}}{{\partial {x_j}}} \cdot {c_{ij}},} \] where c i j = − c j i {c_{ij}} = - {c_{ji}} and “ ⋅ \cdot ” is the multiplication in A + {A^ + } . This applies in particular to simple nodal noncommutative Jordan algebras (of characteristic not 2). These results suggest a method of constructing noncommutative Jordan algebras of the given form. We have done this with the restriction that the c i j {c_{ij}} lie in F1. The last result is that if A is a finitely generated simple noncommutative algebra of characteristic 0 of this form, then Der (A) is an infinite simple Lie algebra of a known type.

Embedding and Existence theorems of infinite Lie algebra

Embedding and Existence theorems of infinite Lie algebra

The classification of the real primitive infinite Lie algebras

The classification of the real primitive infinite Lie algebras

The classification of real primitive infinite Lie algebras

The classification of real primitive infinite Lie algebras

CLASSIFICATION OF THE REAL INFINITE SIMPLE AND REAL INFINITE PRIMITIVE LIE ALGEBRAS

In this paper the real infinite simple and real infinite primitive Lie algebras are classified.

SQUARES OF ADJOINT ENDOMORPHISMS IN SIMPLE LIEp-ALGEBRAS

A new characterization of simple Lie p-algebras of classical type based on the properties of nilpotent elements is given herein. By this means an analogy is also established between degenerate Lie algebras of a special class and infinite Lie algebras of zero characteristic.