Related papers: Lowest dimensional example on non-universality of …
We prove that for each dimension not less than five there exists a contraction between solvable Lie algebras that can be realized only with matrices whose Euclidean norms necessarily approach infinity at the limit value of contraction…
We present a simple and rigorous proof of the claim by Weimar-Woods [Rev. Math. Phys. 12 (2000) 1505-1529] that any diagonal contraction (e.g., a generalized In\"on\"u-Wigner contraction) is equivalent to a generalized In\"on\"u-Wigner…
Theoretical background of continuous contractions of finite-dimensional Lie algebras is rigorously formulated and developed. In particular, known necessary criteria of contractions are collected and new criteria are proposed. A number of…
In this contributed presentation, we discuss and compare the mutually opposite procedures of deformations and contractions of Lie algebras. We suggest that with appropriate combinations of both procedures one may construct new Lie algebras.…
We consider a class of generalized Inonu-Wigner contraction for semidirect product of two particularly related semisimple Lie (super)algebras. The special class of such contractions provides D=4 Maxwell algebra and recently introduced…
After reviewing the three well-known methods to obtain Lie algebras and superalgebras from given ones, namely, contractions, deformations and extensions, we describe a fourth method recently introduced, the expansion of Lie (super)algebras.…
We introduce a new construction of bilinear invariant forms on Lie algebras, based on the method of graded contractions. The general method is described and the $\Bbb Z_2$-, $\Bbb Z_3$-, and $\Bbb Z_2\otimes\Bbb Z_2$-contractions are found.…
We introduce a duality for In\"{o}n\"{u}-Wigner contractions attached to real symmetric Lie algebras. Starting from a symmetric pair $(\mathfrak{g},\theta)$, we define a dual real form $\mathfrak{g}^{*}$ inside the complexification of…
Infinite-dimensional Galilean conformal algebras can be constructed by contracting pairs of symmetry algebras in conformal field theory, such as $W$-algebras. Known examples include contractions of pairs of the Virasoro algebra, its $N=1$…
For any Inonu-Wigner contraction of a three dimensional Lie algebra we construct the corresponding contractions of representations. Our method is quite canonical in the sense that in all cases we deal with realizations of the…
This note is devoted to the construction of two very easy examples, of respective dimensions 4 and 6, of graded Lie algebras whose grading is not given by a semigroup, the latter one being a semisimple algebra. It is shown that 4 is the…
We consider a proper parabolic subalgebra p of a simple Lie algebra g and the Inonu-Wigner contraction of p with respect to its decomposition into its standard Levi factor and its nilpotent radical : this is the Lie algebra which is…
Graded contractions of certain non-toral $\mathbb{Z}_2^3$-gradings on the simple Lie algebras $\mathfrak{so}(7,\mathbb C)$ and $\f{so}(8,\mathbb C)$ are classified up to two notions of equivalence. In particular, there arise two large…
Quadratic algebras are generalizations of Lie algebras which include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical…
In this paper we formulate a conjecture about the minimal dimensional representations of the finite $W$-superalgebra $U(\mathfrak{g}_\bbc,e)$ over the field of complex numbers and demonstrate it with examples including all the cases of type…
We call a finite-dimensional complex Lie algebra $\mathfrak{g}$ strongly rigid if its universal enveloping algebra $\Ug$ is rigid as an associative algebra, i.e. every formal associative deformation is equivalent to the trivial deformation.…
In this note we give an example of a set $\W\subset \R^4$ such that $L^2(\W)$ admits an orthonormal basis of exponentials $\{\frac{1}{|\W |^{1/2}}e^{2\pi i x, \xi}\}_{\xi\in\L}$ for some set $\L\subset\R^4$, but which does not tile $\R^4$…
We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra A=K[W] is said to be of Kostant type, if its centre Z(A) is freely generated by homogeneous polynomials…
Contractions are a procedure to construct a new Lie algebra out of a given one via a singular limit. Specifically, the \.In\"on\"u--Wigner construction starts with a Lie algebra $\mathfrak{g}$ with Lie subalgebra $\mathfrak{h} \subseteq…
We determine all the contractions within the class of finite-dimensional real Lie algebras whose coadjoint orbits have dimensions $\le2$.