We give an account of the construction of exterior differential systems based on the notion of tableaux over Lie algebras as developed in [33]. The definition of a tableau over a Lie algebra is revisited and extended in the light of the formalism of the Spencer cohomology; the question of involutiveness for the associated systems and their prolongations is addressed; examples are discussed.

Let 9 be a reductive complex Lie algebra, with adjoint group G, Cartan subalgebra ~ and Weyl group W. Then G acts naturally on the algebra of polynomial functions &'(g) and hence on the ring of differential operators with polynomial coefficients, .97(g). Similarly, W acts on ~ and hence on .97(~). In [BC2], Harish-Chandra defined an algebra homomorphism J : .97(g)G -t .97(~)w. Recently, Wallach...

We introduce 3-irreducible modules, even roots and odd roots for Leibniz algebras, produce a basis for a root space of a Leibniz algebra with a semisimple Lie factor, and classify finite dimensional simple Leibniz algebras with Lie factor sl2. In this paper, all vector spaces are vector spaces over an algebraically closed field k of characteristic 0. This paper consists of three sections. In Se...

Consider the Lie algebras Lr,t : [K1,K2] = sK3, [K3,K1] = rK1, [K3,K2] = −rK2, [K3,K4] = 0, [K4,K1] = −tK1, and [K4,K2] = tK2, subject to the physical conditions, K3 and K4 are real diagonal operators representing energy, K2 = K † 1, and the Hamiltonian H = ω1K3 + (ω1 + ω2)K4 + λ(t)(K1e−iφ +K2e) is a Hermitian operator. Matrix representations are discussed and faithful representations of least ...

In the framework of McKay correspondence we determine, for every finite subgroup Γ of SL4C, how the finite dimensional irreducible representations of SL4C decompose under the action of Γ. Let h be a Cartan subalgebra of sl4C and let ̟1, ̟2, ̟3 be the corresponding fundamental weights. For (p, q, r) ∈ N, the restriction πp,q,r|Γ of the irreducible representation πp,q,r of highest weight p̟1 + q̟2 + r̟...

Drinfeld twist is applied to the Lie algebra gl(2) so that a two-parametric deformation of it is obtained, which is identical to the Jordanian deformation of the gl(2) obtained by Aneva et al. The same twist element is applied to deform the Lie superalgebra sl(1/2), since the gl(2) is embedded into the sl(1/2). By making use of the FRT-formalism, we construct a deformation of the Lie supergroup...

Suppose F is a finite set of selfadjoint elements in a tracial von Neumann algebra M . For α > 0, F is α-bounded if Pα(F ) < ∞ where Pα is the α-packing entropy of F introduced in [7]. We say that M is strongly 1-bounded if M has a 1-bounded finite set of selfadjoint generators F such that there exists an x ∈ F with χ(x) > −∞. It is shown that if M is strongly 1-bounded, then any finite set of ...

We characterize the support of the free Lie algebra LZm(A) on a finite alphabet A over the ring Zm of integers mod m as the set of all words w such that m c(w), where c(w) is the greatest common divisor of the coefficients in the polynomial l∗(w), for the adjoint endomorphism l∗ of the left-normed Lie bracketing l of the free Lie ring. For words of length n this can be interpreted combinatorial...

The paper concerns two versions of the notion of real forms of Lie superalgebras. One is the standard approach, where a real form of a complex Lie superalgebra is a real Lie superalgebra such that its complexification is the original complex Lie superalgebra. The second is related to considering A-points of a Lie superalgebra over a commutative complex superalgebra A equipped with superconjugat...

We consider Yang-Mills theories with general gauge groups G and twists on the four torus. We find consistent boundary conditions for gauge fields in all instanton sectors. An extended Abelian projection with respect to the Polyakov loop operator is presented, where A 0 is independent of time and in the Cartan subalgebra. Fundamental domains for the gauge fixed A 0 are constructed for arbitrary ...