Representations of q-analogue of the Virasoro Algebra

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Tensor Product Weight Representations of the Neveu–Schwarz Algebra

We study the tensor product of a highest weight module with an intermediate series module over the Neveu–Schwarz algebra. If the highest weight module is nontrivial, the weight spaces of such a tensor product are infinite dimensional. We show that such a tensor product is indecomposable. Using a “shifting technique” developed by H. Chen, X. Guo, and K. Zhao for the Virasoro algebra case, we give necessary and sufficient conditions for such a tensor product to be irreducible. Furthermore, we give necessary and sufficient conditions for two such tensor products to be isomorphic.     

Nonrelativistic Contraction of a Canonically Deformed Super-Poincaré Hopf Algebra

We consider the nonrelativistic contraction of a canonically deformed N = 1{\mathcal{N}} = 1 super-Poincaré Hopf algebra. In such a way we get the new quantum supergroup – a canonically twisted super-Galilei Hopf algebra equipped with the Clifford-like spinorial sector of the corresponding super-space. Besides, we also propose the new Lie-algebraic twist deformation of a super-Galilei algebra leading to classical time and quantum space as well as to the Clifford-like fermionic part of the twisted N = 1{\mathcal{N}} = 1 superspace.     

ℤ2-Codimensions of the Grassmann Algebra

Let E be the infinite-dimensional Grassmann algebra over a field F of characteristic zero, and consider L the F-vector space spanned by all generators of E. Let ? _l be any fixed automorphism of E of order 2 such that L is an homogeneous subspace.

Our goal is to finish the computation of the sequences of ? _l -codimensions, by finding its exact value for the unique open case, that is, when the subspace of L corresponding to the eigenvalue 1 is finite-dimensional. As a consequence we get the ?-codimensions for a large amount of arbitrary automorphisms ? of E of order 2.     

Construction of the Gelfand–Tsetlin Basis for Unitary Principal Series Representations of the Algebra sln(ℂ)

Theoretical and Mathematical Physics - We consider infinite-dimensional unitary principal series representations of the algebra sln(?), implemented on the space of functions of n(n?1)/2...     

Convergence of the reweighted ℓ 1 minimization algorithm for ℓ 2–ℓ p minimization

The iteratively reweighted ? ₁ minimization algorithm (IRL1) has been widely used for variable selection, signal reconstruction and image processing. In this paper, we show that any sequence generated by the IRL1 is bounded and any accumulation point is a stationary point of the ? ₂–? _p minimization problem with 0<p<1. Moreover, the stationary point is a global minimizer and the convergence rate is approximately linear under certain conditions. We derive posteriori error bounds which can be used to construct practical stopping rules for the algorithm.     

Gelfand–Kirillov Dimensions of the Z~2-graded Oscillator Representations of sl(n)

We find an exact formula of Gelfand–Kirillov dimensions for the infinite-dimensional explicit irreducible sl(n, F)-modules that appeared in the Z2-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. Three infinite subfamilies of these modules have the minimal Gelfand–Kirillov dimension. They contain weight modules with unbounded weight multiplicities and completely pointed modules.     

Irreducible Representations for the Baby Tits–Kantor–Koecher Algebra

The baby Tits–Kantor–Koecher (TKK) algebra constructed from the smallest (nonlattice) semilattice is related to the “smallest” extended affine Lie algebras other than the finite dimensional simple Lie algebras and the affine Kac–Moody algebras. In this article, we classify the finite dimensional irreducible representations for the baby TKK algebra. It turns out that such representations can be lifted from modules of direct sums of finitely many copies of the simple Lie algebra sp ₄(?).     

Cohomology of the Schrdinger Algebra S(1)

We explicitly compute the first and second cohomology groups of the Schrdinger algebra S(1) with coefficients in the trivial module and the finite-dimensional irreducible modules.We also show that the first and second cohomology groups of S(1) with coefficients in the universal enveloping algebras U(S(1))(under the adjoint action) are infinite dimensional.     

Realization of the Universal sℓ q (2)-Symmetric R-Operator in a Function Space for General and Exceptional Values of the Deformation Parameter

Based on the realization of representations of the algebra s _q (2) in the space of polynomials for general values of the deformation parameter q and on a finite tuple of theta functions, which are a natural generalization of polynomials, we construct the eigenstates and find the related eigenvalues of the universal R-operator for cyclic representations corresponding to q ^N = ±1.     

Defining Relations for the Algebra of Invariants of 2×2 Matrices

We obtain defining relations of the algebra of invariants of the classical subgroups of GL ₂(C) acting by simultaneous conjugation on m-tuples of 2×2 complex matrices. The sets of defining relations look uniformly for all m2 and are derived by translation of classical results on invariant theory of orthogonal groups in the language of 2×2 matrix invariants, combined with arguments of representation theory of the general linear group GL _m (C) and ideas coming from the theory of algebras with polynomial identities.     

Representations of a Noncommutative Associative Algebra Related to Quantum Torus of Rank Three

In this paper, we present some modules over the rank-three quantized Weyl algebra, which are closely related to modules over some vertex algebras. The isomorphism classes among these modules are also determined.     

Polarizations on Abelian Varieties and Self-dual ℓ-adic Representations of Inertia Groups

It is well known that every finite subgroup of GL _d (Q ) is conjugate to a subgroup of GL _d (Z ). However, this does not remain true if we replace general linear groups by symplectic groups. We say that G is a group of inertia type of G is a finite group which has a normal Sylow-p-subgroup with cyclic quotient. We show that if >d+1, and G is a subgroup of Sp _2d (Q ) of inertia type, then G is conjugate in GL _2d (Q ) to a subgroup of Sp _2d (Z ). We give examples which show that the bound is sharp. We apply these results to construct, for every odd prime , isogeny classes of Abelian varieties all of whose polarizations have degree divisible by ². We prove similar results for Euler characteristic of invertible sheaves on Abelian varieties over fields of positive characteristic.     

Solvability of a Commutative Algebra which Satisfies (x 2)2 = 0

We studied the solvability of the algebra which satisfies the polynomial identity (x ²)² = 0. We believe that, if A is a finite dimensional commutative algebra over a field F of characteristic not 2 which satisfies (x ²)² = 0 for all x ∈ A, then A is solvable. In this article we proved this when dim  _F A ≤ 7.     

Applications of Clifford’s Geometric Algebra

We survey the development of Clifford’s geometric algebra and some of its engineering applications during the last 15 years. Several recently developed applications and their merits are discussed in some detail. We thus hope to clearly demonstrate the benefit of developing problem solutions in a unified framework for algebra and geometry with the widest possible scope: from quantum computing and electromagnetism to satellite navigation, from neural computing to camera geometry, image processing, robotics and beyond.