scholarly journals FIELD OF INVARIANTS OF BORELEAN GROUP OF ADJOINT REPRESENTATION OF GL(n,K)

2017 ◽  
Vol 20 (3) ◽  
pp. 34-40
Author(s):  
K.A. Vyatkina
Keyword(s):  
Explicit Form ◽  
Lie Group ◽  
Invariant Theory ◽  
Algebraic Independence ◽  
Adjoint Representation ◽  
Unitriangular Group ◽  
Joint Representation

The paper is devoted to invariant theory problems, in particular to the problem of finding generators of invariant fields in an explicit form. The set of generators is given for invariant field of unitriangular group concerning the ad-joint representation of GL(n, K) group. Moreover, the set of generators of Borel group for the field of invariants is constructed and their algebraic independence is proved. Lie group;adjoint representation;field of invariant;generators of the field of invariants;Borel group;

scholarly journals Lie Supergroups Obtained from 3-Dimensional Lie Superalgebras Associated to the Adjoint Representation and Having a 2-Dimensional Derived Ideal

2008 ◽  
Vol 2008 ◽  
pp. 1-9
Author(s):  
I. Hernández ◽  
R. Peniche
Keyword(s):  
Lie Group ◽  
Lie Superalgebra ◽  
Adjoint Representation ◽  
Lie Superalgebras ◽  
Base Manifold ◽  
3 Dimensional ◽  
Lie Supergroups

We give the explicit multiplication law of the Lie supergroups for which the base manifold is a 3-dimensional Lie group and whose underlying Lie superalgebrag=g0⊕g1which satisfiesg1=g0,g0acts ong1via the adjoint representation andg0has a 2-dimensional derived ideal.

scholarly journals Low codimensional embeddings of Sp(n) and Su(n)

1984 ◽  
Vol 27 (1) ◽  
pp. 25-29 ◽  
Author(s):  
G. Walker ◽  
R. M. W. Wood
Keyword(s):  
Euclidean Space ◽  
Lie Group ◽  
Maximal Torus ◽  
Unitary Group ◽  
Normal Bundle ◽  
Symplectic Group ◽  
Adjoint Representation ◽  
General Technique ◽  
Special Cases ◽  
Connected Lie Group

In [4] Elmer Rees proves that the symplectic group Sp(n) can be smoothly embedded in Euclidean space with codimension 3n, and the unitary group U(n) with codimension n. These are special cases of a result he obtains for a compact connected Lie group G. The general technique is first to embed G/T, where T is a maximal torus, as a maximal orbit of the adjoint representation of G, and then to extendto an embedding of G by using a maximal orbit of a faithful representation of G. In thisnote, we observe that in the cases G = Sp(n) or SU(n) an improved result is obtained byusing the “symplectic torus” S3 x … x S3 in place of T = S1 x … x S1. As in Rees's construction, the normal bundle of the embedding of G is trivial.

scholarly journals Poincare series for the algebras of joint invariants and covariants of $n$ quadratic forms

2017 ◽  
Vol 9 (1) ◽  
pp. 57-62 ◽  
Author(s):  
N.B. Ilash
Keyword(s):  
Lie Group ◽  
Invariant Theory ◽  
Quadratic Forms ◽  
Laurent Series ◽  
Poincaré Series ◽  
Binary Forms ◽  
Poincare Series ◽  
Classical Invariant Theory ◽  
Joint Invariants ◽  
Classical Invariant

We consider one of the fundamental problems of classical invariant theory - the research of Poincare series for an algebra of invariants of Lie group $SL_2$. The first two terms of the Laurent series expansion of Poincare series at the point $z = 1$ give us important information about the structure of the algebra $\mathcal{I}_{d}.$ It was derived by Hilbert for the algebra ${\mathcal{I}_{d}=\mathbb{C}[V_d]^{\,SL_2}}$ of invariants for binary $d-$form (by $V_d$ denote the vector space over $\mathbb{C}$ consisting of all binary forms homogeneous of degree $d$). Springer got this result, using explicit formula for the Poincare series of this algebra. We consider this problem for the algebra of joint invariants $\mathcal{I}_{2n}=\mathbb{C}[\underbrace{V_2 \oplus V_2 \oplus \cdots \oplus V_2}_{\text{n times}}]^{SL_2}$ and the algebra of joint covariants $\mathcal{C}_{2n}=\mathbb{C}[\underbrace{V_2 {\oplus} V_2 {\oplus} \cdots {\oplus} V_2}_{\text{n times}}{\oplus}\mathbb{C}^2 ]^{SL_2}$ of $n$ quadratic forms. We express the Poincare series $\mathcal{P}(\mathcal{C}_{2n},z)=\sum_{j=0}^{\infty }\dim(\mathcal{C}_{2n})_{j}\, z^j$ and $\mathcal{P}(\mathcal{I}_{2n},z)=\sum_{j=0}^{\infty }\dim(\mathcal{I}_{2n})_{j}\, z^j$ of these algebras in terms of Narayana polynomials.   Also, for these algebras we calculate the degrees and asymptotic behavious of the degrees, using their Poincare series.

scholarly journals Symmetry of Dressed Photon

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1283
Author(s):  
Hiroyuki Ochiai
Keyword(s):  
Theoretical Model ◽  
Homogeneous Space ◽  
Lie Group ◽  
Invariant Theory ◽  
Grassmann Manifold ◽  
Vector Spaces ◽  
Classical Invariant Theory ◽  
Lie Group Actions ◽  
Prehomogeneous Vector Spaces ◽  
Classical Invariant

Motivated by describing the symmetry of a theoretical model of dressed photons, we introduce several spaces with Lie group actions and the morphisms between them depending on three integer parameters n≥r≥s on dimensions. We discuss the symmetry on these spaces using classical invariant theory, orbit decomposition of prehomogeneous vector spaces, and compact reductive homogeneous space such as Grassmann manifold and flag variety. Finally, we go back to the original dressed photon with n=4,r=2,s=1.

A homotopy construction of the adjoint representation for Lie groups

2002 ◽  
Vol 133 (3) ◽  
pp. 399-409 ◽  
Author(s):  
NATÀLIA CASTELLANA ◽  
NITU KITCHLOO
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Maximal Torus ◽  
Maximal Rank ◽  
Adjoint Representation ◽  
Simply Connected

Let G be a compact, simply-connected, simple Lie group and T ⊂ G a maximal torus. The purpose of this paper is to study the connection between various fibrations over BG (where G is a compact, simply-connected, simple Lie group) associated to the adjoint representation and homotopy colimits over poset categories [Cscr ], hocolim[Cscr ]BGI where GI are certain connected maximal rank subgroups of G.

scholarly journals Hom-Lie Algebras and Hom-Lie Groups, Integration and Differentiation

2020 ◽  
Author(s):  
Jun Jiang ◽  
◽  
Satyendra Kumar Mishra ◽  
Yunhe Sheng ◽  
◽  
...  
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Group Action ◽  
Lie Group ◽  
Smooth Manifold ◽  
Adjoint Representation ◽  
Lie Group Action

In this paper, we introduce the notion of a (regular) Hom-Lie group. We associate a Hom-Lie algebra to a Hom-Lie group and show that every regular Hom-Lie algebra is integrable. Then, we define a Hom-exponential (Hexp) map from the Hom-Lie algebra of a Hom-Lie group to the Hom-Lie group and discuss the universality of this Hexp map. We also describe a Hom-Lie group action on a smooth manifold. Subsequently, we give the notion of an adjoint representation of a Hom-Lie group on its Hom-Lie algebra. At last, we integrate the Hom-Lie algebra (gl(V),[.,.],Ad), and the derivation Hom-Lie algebra of a Hom-Lie algebra.

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