scholarly journals A Plactic Algebra Action on Bosonic Particle Configurations

2020 ◽  
Author(s):  
Joanna Meinel
Keyword(s):  
Normal Form ◽  
Lie Algebra ◽  
Symmetric Tensor ◽  
Quotient Algebra ◽  
Special Linear ◽  
Quantum Analogue ◽  
Tensor Representations ◽  
Plactic Algebra

AbstractWe study an action of the plactic algebra on bosonic particle configurations. These particle configurations together with the action of the plactic generators can be identified with crystals of the quantum analogue of the symmetric tensor representations of the special linear Lie algebra $\mathfrak {s} \mathfrak {l}_{N}$ s l N . It turns out that this action factors through a quotient algebra that we call partic algebra, whose induced action on bosonic particle configurations is faithful. We describe a basis of the partic algebra explicitly in terms of a normal form for monomials, and we compute the center of the partic algebra.

scholarly journals CONSTRUCTIVE NONCOMMMUTATIVE INVARIANT THEORY

2021 ◽  
Author(s):  
MÁTYÁS DOMOKOS ◽  
VESSELIN DRENSKY
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Invariant Theory ◽  
Reductive Group ◽  
Symmetric Tensor ◽  
Free Associative Algebra ◽  
Tensor Algebra ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Degree Bounds

AbstractThe problem of finding generators of the subalgebra of invariants under the action of a group of automorphisms of a finite-dimensional Lie algebra on its universal enveloping algebra is reduced to finding homogeneous generators of the same group acting on the symmetric tensor algebra of the Lie algebra. This process is applied to prove a constructive Hilbert–Nagata Theorem (including degree bounds) for the algebra of invariants in a Lie nilpotent relatively free associative algebra endowed with an action induced by a representation of a reductive group.

Orthogonal decompositions of Lie algebras over finite commutative rings

2021 ◽  
pp. 2350006
Author(s):  
Songpon Sriwongsa
Keyword(s):  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Rings ◽  
Orthogonal Decomposition ◽  
Necessary Condition ◽  
Special Linear ◽  
Orthogonal Lie Algebra ◽  
Orthogonal Decompositions ◽  
Finite Commutative Ring

Let [Formula: see text] be a finite commutative ring with identity. In this paper, we give a necessary condition for the existence of an orthogonal decomposition of the special linear Lie algebra over [Formula: see text]. Additionally, we study orthogonal decompositions of the symplectic Lie algebra and the special orthogonal Lie algebra over [Formula: see text].

Representations and cohomologies of regular Hom-pre-Lie algebras

2019 ◽  
Vol 19 (08) ◽  
pp. 2050149
Author(s):  
Shanshan Liu ◽  
Lina Song ◽  
Rong Tang
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Cohomology Theory ◽  
Trivial Representation ◽  
Linear Deformation ◽  
Dual Representations ◽  
Regular Representations ◽  
Equivalent Characterization ◽  
Tensor Representations

In this paper, first we study dual representations and tensor representations of Hom-pre-Lie algebras. Then we develop the cohomology theory of regular Hom-pre-Lie algebras in terms of the cohomology theory of regular Hom-Lie algebras. As applications, we study linear deformations of regular Hom-pre-Lie algebras, which are characterized by the second cohomology groups of regular Hom-pre-Lie algebras with the coefficients in the regular representations. The notion of a Nijenhuis operator on a regular Hom-pre-Lie algebra is introduced which can generate a trivial linear deformation of a regular Hom-pre-Lie algebra. Finally, we introduce the notion of a Hessian structure on a regular Hom-pre-Lie algebra, which is a symmetric nondegenerate 2-cocycle with the coefficient in the trivial representation. We also introduce the notion of an [Formula: see text]-operator on a regular Hom-pre-Lie algebra, by which we give an equivalent characterization of a Hessian structure.

scholarly journals Normal forms for elements of o(p, q) and Hamiltonians with integrals linear in momenta

1992 ◽  
Vol 33 (4) ◽  
pp. 486-507 ◽  
Author(s):  
Gerard Thompson
Keyword(s):  
Dynamical Systems ◽  
Normal Form ◽  
Lie Algebra ◽  
Normal Forms ◽  
First Integral ◽  
Inner Product ◽  
Indefinite Inner Product ◽  
Hamiltonian Dynamical Systems ◽  
Matrix Normal

AbstractWe solve the problem of finding a simultaneous matrix normal form for an element of the Lie algebra o(p, q) and the underlying indefinite inner product. The results are used to determine several classes of classical Hamiltonian dynamical systems which possess a first integral linear in the momentum variables.

scholarly journals COMMENTS ON HIGHER-SPIN SYMMETRIES

2009 ◽  
Vol 06 (02) ◽  
pp. 285-342 ◽  
Author(s):  
XAVIER BEKAERT
Keyword(s):  
Lie Algebra ◽  
Differential Operators ◽  
Symmetric Tensor ◽  
Weak Field ◽  
Gauge Fields ◽  
Gauge Parameter ◽  
Gauge Transformations ◽  
First Order ◽  
Gauge Symmetries ◽  
Higher Spin

The unconstrained frame-like formulation of an infinite tower of completely symmetric tensor gauge fields is reviewed and examined in the limit where the cosmological constant goes to zero. By partially fixing the gauge and solving the torsion constraints, the form of the gauge transformations in the unconstrained metric-like formulation are obtained till first order in a weak field expansion. The algebra of the corresponding gauge symmetries is shown to be equivalent, at this order and modulo (unphysical) gauge parameter redefinitions, to the Lie algebra of Hermitian differential operators on ℝn, the restriction of which to the spin-two sector is the Lie algebra of infinitesimal diffeomorphisms.

scholarly journals FUSION AND BRAIDING IN W-ALGEBRA EXTENDED CONFORMAL THEORIES (II): GENERALIZATION TO CHIRAL SCREENED VERTEX OPERATORS LABELLED BY ARBITRARY YOUNG TABLEAUX

1990 ◽  
Vol 05 (10) ◽  
pp. 1881-1909 ◽  
Author(s):  
ADEL BILAL
Keyword(s):  
Potts Model ◽  
Braid Group ◽  
Simple Form ◽  
Group Representation ◽  
Symmetric Tensor ◽  
Vertex Operators ◽  
Young Tableaux ◽  
Exchange Algebra ◽  
Form Closure ◽  
Tensor Representations

In a previous work, we defined the chiral screened vertex operators of W-algebra extended conformal theories by fusion of elementary ones. After reviewing how to obtain the braid group representation matrices, realizing the exchange algebra for those chiral vertex operators corresponding to the symmetric tensor representations of An, we generalize our results to chiral screened vertex operators associated with arbitrary An representations. The fused braiding matrices for antisymmetric tensor screened vertex operators are computed explicitly and shown to have a very simple form. Closure of the exchange algebra in the general case is proved using the relation with the Boltzmann weights of the An face models. Since, in the unitary case, the W-algebras are realized as cosets ĝk⊕ĝ1/ĝk+1, the present results can also be reinterpreted in terms of fusion of braiding matrices of the ĝ WZW models. As an example, the simplest W-algebra extended theory, the 3-state Potts model, is discussed in some detail.

scholarly journals Cohomology of simple modules for sl3(k) in characteristic 3

2021 ◽  
Vol 103 (3) ◽  
pp. 36-43
Author(s):  
A.A. Ibrayeva ◽  
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Quotient Algebra ◽  
Characteristic P ◽  
One Dimensional ◽  
Simple Modules ◽  
Trivial Module ◽  
Characteristic 3 ◽  
Simple Quotient ◽  
Classical Lie Algebra

In this paper we calculate cohomology of a classical Lie algebra of type A2 over an algebraically field k of characteristic p = 3 with coefficients in simple modules. To describe their structure we will consider them as modules over an algebraic group SL3(k). In the case of characteristic p = 3, there are only two peculiar simple modules: a simple that module isomorphic to the quotient module of the adjoint module by the center, and a one-dimensional trivial module. The results on the cohomology of simple nontrivial module are used for calculating the cohomology of the adjoint module. We also calculate cohomology of the simple quotient algebra Lie of A2 by the center.

Sign in / Sign up

Export Citation Format

Share Document