On the universal envelope of a Jordan triple system of n × n matrices

2021 ◽  
pp. 2250126
Author(s):  
Hader A. Elgendy
Keyword(s):  
Triple System ◽  
Product Formula ◽  
Triple Product ◽  
Irreducible Representations ◽  
Monomial Basis ◽  
Jordan Triple System ◽  
Finite Dimensional

We study the universal (associative) envelope of the Jordan triple system of all [Formula: see text] [Formula: see text] matrices with the triple product [Formula: see text] over a field of characteristic 0. We use the theory of non-commutative Gröbner–Shirshov bases to obtain the monomial basis and the center of the universal envelope. We also determine the decomposition of the universal envelope and show that there exist only five finite-dimensional inequivalent irreducible representations of the Jordan triple system of all [Formula: see text] matrices.

Representations of simple anti-Jordan triple systems of m × n matrices

2017 ◽  
Vol 16 (05) ◽  
pp. 1750093 ◽  
Author(s):  
Hader A. Elgendy
Keyword(s):  
Triple System ◽  
Closed Field ◽  
Monomial Basis ◽  
Triple Systems ◽  
Infinite Dimensional ◽  
Matrix Algebras ◽  
Jordan Triple System ◽  
Finite Dimensional ◽  
Jordan Triple Systems

We show that the universal associative envelope of the simple anti-Jordan triple system of all [Formula: see text] ([Formula: see text] is even, [Formula: see text]) matrices over an algebraically closed field of characteristic 0 is finite-dimensional. The monomial basis and the center of the universal envelope are determined. The explicit decomposition of the universal envelope into matrix algebras is given. The classification of finite-dimensional irreducible representations of an anti-Jordan triple system is obtained. The semi-simplicity of the universal envelope is shown. We also show that the universal associative envelope of the simple polarized anti-Jordan triple system of [Formula: see text] matrices is infinite-dimensional.

scholarly journals Description of unitary representations of the group of infinite 𝑝-adic integer matrices

2021 ◽  
Vol 25 (21) ◽  
pp. 606-643
Author(s):  
Yury Neretin
Keyword(s):  
Irreducible Representation ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Infinite Matrices ◽  
Natural Topology ◽  
Content Type ◽  
Residue Ring ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations

We classify irreducible unitary representations of the group of all infinite matrices over a p p -adic field ( p ≠ 2 p\ne 2 ) with integer elements equipped with a natural topology. Any irreducible representation passes through a group G L GL of infinite matrices over a residue ring modulo p k p^k . Irreducible representations of the latter group are induced from finite-dimensional representations of certain open subgroups.

scholarly journals Finite-Dimensional Representations of Yangians in Complex Rank

2019 ◽  
Vol 2020 (20) ◽  
pp. 6967-6998 ◽  
Author(s):  
Daniil Kalinov
Keyword(s):  
Irreducible Representations ◽  
Finite Dimensional

Abstract We classify the “finite-dimensional” irreducible representations of the Yangians $Y(\mathfrak{g}\mathfrak{l}_t)$ and $Y(\mathfrak{s}\mathfrak{l}_t)$. These are associative ind-algebras in the Deligne category $\textrm{Rep}(GL_t)$, which generalize the regular Yangians $Y(\mathfrak{g}\mathfrak{l}_n)$ and $Y(\mathfrak{s}\mathfrak{l}_n)$ to complex rank. They were first defined in the paper [14]. Here we solve [14, Problem 7.2]. We work with the Deligne category $\textrm{Rep}(GL_t)$ using the ultraproduct approach introduced in [7] and [16].

scholarly journals Parallel submanifolds of complex space forms II

1983 ◽  
Vol 91 ◽  
pp. 119-149 ◽  
Author(s):  
Hiroo Naitoh
Keyword(s):  
Lie Algebra ◽  
Jordan Algebra ◽  
Triple System ◽  
Complex Space ◽  
Space Forms ◽  
Jordan Triple System ◽  
Graded Lie Algebra ◽  
Complex Space Forms

This is a continuation of Part I, which appeared in this journal.In the previous paper I we have defined the following notions: orthogonal Jordan triple system (OJTS), orthogonal symmetric graded Lie algebra (OSGLA), orthogonal Jordan algebra (OJA), Hermitian symmetric graded Lie algebra (HSGLA). And we have shown that equivalent classes of OJTS naturally correspond to equivalent classes of OSGLA and through this correspondence we have naturally constructed HSGLA’s from the OJTS’s associated with OJA’s with unity.

Jacobi–Trudi type formula for a class of irreducible representations of 𝔤𝔩(m|n)

2020 ◽  
pp. 2250015
Author(s):  
Nguyên Luong Thái Bình
Keyword(s):  
Irreducible Representations ◽  
Fundamental Representation ◽  
Symmetric Powers ◽  
Type Formula ◽  
Finite Dimensional

We prove a determinantal type formula to compute the characters of a class of finite-dimensional irreducible representations of the general Lie super-algebra [Formula: see text] in terms of the characters of the symmetric powers of the fundamental representation and their duals. This formula, originally conjectured by van der Jeugt and Moens, can be regarded as a generalization of the well-known Jacobi–Trudi formula.

scholarly journals Canonical bases and standard monomials

1998 ◽  
Vol 41 (3) ◽  
pp. 611-623
Author(s):  
R. J. Marsh
Keyword(s):  
Lie Algebra ◽  
Canonical Basis ◽  
Monomial Basis ◽  
Fundamental Weight ◽  
Enveloping Algebra ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Standard Monomials ◽  
Standard Monomial ◽  
Quantized Enveloping Algebra

Let U be the quantized enveloping algebra associated to a simple Lie algebra g by Drinfel'd and Jimbo. Let λ be a classical fundamental weight for g, and ⋯(λ) the irreducible, finite-dimensional type 1 highest weight U-module with highest weight λ. We show that the canonical basis for ⋯(λ) (see Kashiwara [6, §0] and Lusztig [18, 14.4.12]) and the standard monomial basis (see [11, §§2.4 and 2.5]) for ⋯(λ) coincide.

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