Quantum scars as embeddings of weakly broken Lie algebra representations

A metric Lie algebra is a Lie algebra equipped with an invariant non-degenerate symmetric bilinear form. It is called indecomposable if it is not the direct sum of two metric Lie algebras. We are interested in describing the isomorphism classes of indecomposable metric Lie algebras. In the present paper we restrict ourselves to a certain class of solvable metric Lie algebras which includes all indecomposable metric Lie algebras with maximal isotropic centre. We will see that each metric Lie algebra belonging to this class is a twofold extension associated with an orthogonal representation of an abelian Lie algebra. We will describe equivalence classes of such extensions by a certain cohomology set. In particular we obtain a classification scheme for indecomposable metric Lie algebras with maximal isotropic centre and the classification of metric Lie algebras of index 2.

We will be interested in complex representations of both real and complex Lie algebras. There is an important distinction to be made. If g is a real Lie algebra, then a complex representation is an ℝ-linear homomorphism g → End(V), where V is a complex vector space. On the other hand, if g is a complex Lie algebra, we require that the homomorphism be (ℂ-linear. The reader should note that we ask more of a complex representation of a complex Lie algebra than we do of a complex representation of a real Lie algebra.

Motivated by an example of Inönü and Wigner, a general procedure for the contraction of irreducible unitary representations of semisimple compact Lie algebras is given. In this situation, contractions mean generalized Inönü–Wigner contractions. In particular, this means that by a simple limiting process, one obtains representations of nonsemisimple noncompact Lie algebras. As examples, a general treatment of SO(3) is given as well as some results for SU(3).

ABSTRACTAs it is well-known there exist matrix representations of any given finite-dimensional complex Lie algebra. More concretely, such representations can be obtained by means of an isomorphic matrix Lie algebra consisting of upper-triangular square matrices. However, there is no general information about the minimal order for the matrices involved in such representations. In this way, our main goal is to revisit, debug and implement an algorithm which provides the minimal order for matrix representations of any finite-dimensional nilpotent Lie algebra from its law, as well as returning a matrix representative of such an algebra by using the minimal order previously computed. In order to show the applicability of this procedure, we have computed minimal representative for each nilpotent Lie algebra of dimensions 6 and 7 and we have also obtained the representation of some families with an arbitrary dimension.

We study M-theory and D-brane quantum partition functions for microscopic black hole ensembles within the context of the AdS/CFT correspondence in terms of highest weight representations of infinite-dimensional Lie algebras, elliptic genera, and Hilbert schemes, and describe their relations to elliptic modular forms. The common feature in our examples lies in the modular properties of the characters of certain representations of the pertinent affine Lie algebras, and in the role of spectral functions of hyperbolic three-geometry associated with q-series in the calculation of elliptic genera. We present new calculations of supergravity elliptic genera on local Calabi–Yau threefolds in terms of BPS invariants and spectral functions, and also of equivariant D-brane elliptic genera on generic toric singularities. We use these examples to conjecture a link between the black hole partition functions and elliptic cohomology.

Representations of twisted multi-loop Lie algebras

Killing – An algebraic computational package for Lie algebras

Irreducible representations of nongraded Witt type Lie algebras

Lie groups are a concept that has originate from few strong intuitive ideas. We have illustrated the operations of Lie groups (LG) as well as Lie algebras (LA) on multitudinous in this paper. The matrix part and Bi-invariant systems of Lie groups, as well as representations of Lie Algebras, can be used. A Lie group which is a multitudinous on the product and inverse section operations are well-defined. It is extremely significant in the theory of fiber bundles, as well as having numerous applications in physics. Lie algebras are a product of Lie groups in the past (the tangent space introduced a Lie group). On smooth manifolds, we worked nonstop to describe the mathematical aspects of Lie groups along with Lie algebras.

abstract: We introduce the notion of almost representations of Lie algebras and quantum tori, and establish an Ulam-stability type phenomenon: every irreducible almost representation is close to a genuine irreducible representation. As an application, we prove that geometric quantizations of the two-dimensional sphere and the two-dimensional torus are conjugate in the semi-classical limit up to a small error.

Periodic driving has been established as a powerful technique for engineering novel phases of matter and intrinsically out-of-equilibrium phenomena such as time crystals. Recent work by Bluvstein et al. [Science 371, 1355 (2021)] has demonstrated that periodic driving can also lead to a significant enhancement of quantum many-body scarring, whereby certain non-integrable systems can display persistent quantum revivals from special initial states. Nevertheless, the mechanisms behind driving-induced scar enhancement remain poorly understood. Here we report a detailed study of the effect of periodic driving on the PXP model describing Rydberg atoms in the presence of a strong Rydberg blockade - the canonical static model of quantum many-body scarring. We show that periodic modulation of the chemical potential gives rise to a rich phase diagram, with at least two distinct types of scarring regimes that we distinguish by examining their Floquet spectra. We formulate a toy model, based on a sequence of square pulses, that accurately captures the details of the scarred dynamics and allows for analytical treatment in the large-amplitude and high-frequency driving regimes. Finally, we point out that driving with a spatially inhomogeneous chemical potential allows to stabilize quantum revivals from arbitrary initial states in the PXP model, via a mechanism similar to prethermalization.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

We prove most of Lusztig's conjectures on the canonical basis in homology of a Springer fiber.The conjectures predict that this basis controls numerics of representations of the Lie algebra of a semisimple algebraic group over an algebraically closed field of positive characteristic.We check this for almost all characteristics.To this end we construct a noncommutative resolution of the nilpotent cone which is derived equivalent to the Springer resolution.On the one hand, this noncommutative resolution is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and Rumynin.On the other hand, it is compatible with the t-structure arising from an equivalence with the derived category of perverse sheaves on the affine flag variety of the Langlands dual group.This equivalence established by Arkhipov and the first author fits the framework of local geometric Langlands duality.The latter compatibility allows one to apply Frobenius purity theorem to deduce the desired properties of the basis.We expect the noncommutative counterpart of the Springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric Langlands duality.

We obtain minimal dimension matrix representations for each of the Lie algebras of dimensions five, six, seven and eight obtained by Turkowski that have a non-trivial Levi decomposition. The key technique involves using the invariant subspaces associated to a particular representation of a semi-simple Lie algebra to help in the construction of the radical in the putative Levi decomposition.

Integrating Unitary Representations of Infinite-Dimensional Lie Groups