scholarly journals The Index and Spectrum of Lie Poset Algebras of Types B, C, and D

10.37236/9558 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Vincent Coll ◽  
Nicholas W. Mayers ◽  
Nicholas Russoniello
Keyword(s):  
Lie Algebras ◽  
Primary Concern ◽  
Type B ◽  
Restricted Class ◽  
The Matrix ◽  
Index Formulas

We define posets of types B, C, and D. These posets encode the matrix forms of certain Lie algebras which lie between the algebras of upper-triangular and diagonal matrices. Our primary concern is the index and spectral theories of such type-B, C, and D Lie poset algebras. For an important restricted class, we develop combinatorial index formulas and, in particular, characterize posets corresponding to Frobenius Lie algebras. In this latter case we show that the spectrum is binary; that is, consists of an equal number of 0's and 1's. Interestingly, type-B, C, and D Lie poset algebras can be related to Reiner's notion of a parset.

Trace Forms on Lie Algebras

1962 ◽  
Vol 14 ◽  
pp. 553-564 ◽  
Author(s):  
Richard Block
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Trace Form ◽  
Finite Degree ◽  
Trace Forms ◽  
Matrix Products ◽  
The Matrix ◽  
Image Position

If L is a Lie algebra with a representation Δ a→aΔ (a in L) (of finite degree), then by the trace form f = fΔ of Δ is meant the symmetric bilinear form on L obtained by taking the trace of the matrix products:Then f is invariant, that is, f is symmetric and f(ab, c) — f(a, bc) for all a, b, c in L. By the Δ-radical L⊥ = L⊥ of L is meant the set of a in L such that f(a, b) = 0 for all b in L. Then L⊥ is an ideal and f induces a bilinear form , called a quotient trace form, on L/L⊥. Thus an algebra has a quotient trace form if and only if there exists a Lie algebra L with a representation Δ such that

scholarly journals Combinatorial index formulas for Lie algebras of seaweed type

2020 ◽  
Vol 48 (12) ◽  
pp. 5430-5454 ◽  
Author(s):  
Alex Cameron ◽  
Vincent E. Coll ◽  
Matt Hyatt
Keyword(s):  
Lie Algebras ◽  
Index Formulas

English Speakers’ Comprehension of Embedded Relative Clauses in L2 Japanese

2017 ◽  
Vol 7 (10) ◽  
pp. 841
Author(s):  
Shinichi Shoji
Keyword(s):  
Relative Clause ◽  
Relative Clauses ◽  
Sentence Type ◽  
Type A ◽  
Native English Speakers ◽  
English Speakers ◽  
Type B ◽  
The Matrix ◽  
Embedded Clauses ◽  
Accuracy Rates

This study investigated native English speakers’ comprehension of Japanese sentences in which relative clauses are embedded. Specifically, this study contrasted between (a) short-before-long sentences with center-embedded relative clauses and (b) long-before-short sentences with non-center-embedded relative clauses. Sentence-type (a) indicates a sentence that includes a short phrase before a long phrase and includes a relative clause that is embedded in the middle of the sentence, e.g., Onna-ga Ken-ga kiratteiru giin-o hometa ‘The woman praised the senator who Ken hated’. Sentence-type (b) indicates a sentence with a long phrase before a short phrase and includes a relative clause that is embedded peripherally, e.g., Ken-ga kiratteiru onna-ga giin-o hometa ‘The woman who Ken hated praised the senator’. Experiment 1 revealed that native English speakers, who are learners of Japanese, comprehended the type (b) sentences with long-before-short phrases and with non-center-embedded relative clauses more accurately than the type (a) sentences with short-before-long phrases with center-embedded relative clauses. The results indicate that the preference for the non-center-embedded clauses to center-embedded clauses is universal across languages, while the preference for short-before-long phrases is language-specific. However, Experiment 2 indicated that the different accuracy rates in comprehensions of (a) and (b) disappeared when the matrix subjects are marked by the topic-morpheme wa. The outcome indicated that the topic phrases are immediately interpreted as a part of main clauses.

COMPUTING THE LAW OF A FAMILY OF SOLVABLE LIE ALGEBRAS

2009 ◽  
Vol 19 (03) ◽  
pp. 337-345 ◽  
Author(s):  
JUAN C. BENJUMEA ◽  
JUAN NÚÑEZ ◽  
ÁNGEL F. TENORIO
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Computational Data ◽  
Solvable Lie Algebras ◽  
Final Output ◽  
The Matrix ◽  
The Law

This paper shows an algorithm which computes the law of the Lie algebra associated with the complex Lie group of n × n upper-triangular matrices with exponential elements in their main diagonal. For its implementation two procedures are used, respectively, to define a basis of the Lie algebra and the nonzero brackets in its law with respect to that basis. These brackets constitute the final output of the algorithm, whose unique input is the matrix order n. Besides, its complexity is proved to be polynomial and some complementary computational data relative to its implementation are also shown.

Enveloping Algebras of Semi-Simple Lie Algebras

1950 ◽  
Vol 2 ◽  
pp. 257-266 ◽  
Author(s):  
N. Jacobson
Keyword(s):  
Quantum Mechanics ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Finite Basis ◽  
Multiplication Table ◽  
Polynomial Equations ◽  
Systems Of Equations ◽  
Enveloping Algebras ◽  
Starting Point ◽  
The Matrix

In a recent paper we studied systems of equations of the form(1) (2) where as usual [a,b] = ab — ba and ϕ(λ) is a polynomial. Equations of this type have arisen in quantum mechanics. In our paper we gave a method of determining the matrix solutions of such equations. The starting point of our discussion was the observation that if the elements xi satisfy (1) then the elements xi, [xj,xk] satisfy the multiplication table of a certain basis of the Lie algebra of skew symmetric (n + 1) ⨯ (n + 1) matrices. We proved that if (2) is imposed as an added condition, then the algebra generated by the has a finite basis, and we obtained the structure of the most general associative algebra that is generated in this way.

scholarly journals Duality Identities for Moduli Functions of Generalized Melvin Solutions Related to Classical Lie Algebras of Rank 4

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
S. V. Bolokhov ◽  
V. D. Ivashchuk
Keyword(s):  
Lie Algebras ◽  
Toda Chain ◽  
Scalar Fields ◽  
Cartan Matrix ◽  
Radial Coordinate ◽  
Cylindrically Symmetric ◽  
Symmetry Properties ◽  
The Matrix ◽  
Chain Type ◽  
Internal Symmetries

We consider generalized Melvin-like solutions associated with nonexceptional Lie algebras of rank 4 (namely, A4, B4, C4, and D4) corresponding to certain internal symmetries of the solutions. The system under consideration is a static cylindrically symmetric gravitational configuration in D dimensions in presence of four Abelian 2-forms and four scalar fields. The solution is governed by four moduli functions Hs(z) (s=1,…,4) of squared radial coordinate z=ρ2 obeying four differential equations of the Toda chain type. These functions turn out to be polynomials of powers (n1,n2,n3,n4)=(4,6,6,4),(8,14,18,10),(7,12,15,16),(6,10,6,6) for Lie algebras A4, B4, C4, and D4, respectively. The asymptotic behaviour for the polynomials at large distances is governed by some integer-valued 4×4 matrix ν connected in a certain way with the inverse Cartan matrix of the Lie algebra and (in A4 case) the matrix representing a generator of the Z2-group of symmetry of the Dynkin diagram. The symmetry properties and duality identities for polynomials are obtained, as well as asymptotic relations for solutions at large distances. We also calculate 2-form flux integrals over 2-dimensional discs and corresponding Wilson loop factors over their boundaries.

scholarly journals On Trace Bilinear Forms on Lie-Algebras

1959 ◽  
Vol 4 (2) ◽  
pp. 62-72 ◽  
Author(s):  
Hans Zassenhaus
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Matrix Representation ◽  
Bilinear Forms ◽  
Finite Degree ◽  
Matrix Products ◽  
The Matrix ◽  
Image Position

To what extent is the structure of a Lie-algebra L over a field F determined by the bilinear formon L that is derived from a matrix representationof L with finite degree d(Δ) by forming the trace of the matrix productsSuch a bilinear form is a function with two arguments in L, values in F and the properties:

scholarly journals ELECTRON MICROSCOPY OF THE HUMAN SYNOVIAL MEMBRANE

1962 ◽  
Vol 14 (2) ◽  
pp. 207-220 ◽  
Author(s):  
Peter Barland ◽  
Alex B. Novikoff ◽  
David Hamerman
Keyword(s):  
Electron Microscopy ◽  
Golgi Apparatus ◽  
Synovial Membrane ◽  
Current Knowledge ◽  
Cell Types ◽  
Type B ◽  
Joint Cavity ◽  
The Matrix ◽  
And Function ◽  
Cytoplasmic Processes

The structure of the lining cells at the surface of the synovial membrane facing the joint cavity has been studied by electron microscopy. The long cytoplasmic processes of these cells appear to be oriented toward the surface of the membrane, where they overlap and intertwine. The matrix of the lining cells contains dense material but no fibers with the periodicity of collagen. The lining cells are divided into two cell types or states of activity on the basis of their cytoplasmic contents. Type A is more numerous and contains a prominent Golgi apparatus, numerous vacuoles (0.4 to 1.5 microns in diameter) containing varying amounts of a dense granular material, many filopodia, mitochondria, intracellular fibrils, and micropinocytotic-like vesicles. Type B contains large amounts of ergastoplasm with fewer large vacuoles, micropinocytotic-like vesicles, and mitochondria. The probable functions of these cells are discussed in the light of current knowledge of the metabolism and function of the synovial membrane.

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