The centralizer algebra of the diagonal action of the group GL n (ℂ) in a mixed tensor space

We find raising and lowering operators distinguishing the degenerate states for the Hamiltonian [Formula: see text] at x = ± 1 for spin 1 that was given by Happer et al.1,2 to interpret the curious degeneracies of the Zeeman effect for condensed vapor of 87 Rb . The operators obey Yangian commutation relations. We show that the curious degeneracies seem to verify the Yangian algebraic structure for quantum tensor space and are consistent with the representation theory of Y(sl(2)).

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Towards a classification of rigid product quotient varieties of Kodaira dimension 0

Bollettino dell Unione Matematica Italiana ◽

10.1007/s40574-021-00295-4 ◽

2021 ◽

Author(s):

Ingrid Bauer ◽

Christian Gleissner

Keyword(s):

Finite Group ◽

Finite Groups ◽

Structure Theorem ◽

Complete Classification ◽

Kodaira Dimension ◽

The Other ◽

Diagonal Action ◽

Quotient Varieties ◽

A Finite Group

AbstractIn this paper the authors study quotients of the product of elliptic curves by a rigid diagonal action of a finite group G. It is shown that only for $$G = {{\,\mathrm{He}\,}}(3), {\mathbb {Z}}_3^2$$ G = He ( 3 ) , Z 3 2 , and only for dimension $$\ge 4$$ ≥ 4 such an action can be free. A complete classification of the singular quotients in dimension 3 and the smooth quotients in dimension 4 is given. For the other finite groups a strong structure theorem for rigid quotients is proven.

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On the Vanishing of a (G, σ) Product in a (G, σ) Space

Canadian Journal of Mathematics ◽

10.4153/cjm-1970-044-4 ◽

1970 ◽

Vol 22 (2) ◽

pp. 363-371 ◽

Cited By ~ 6

Author(s):

K. Singh

Keyword(s):

Vector Space ◽

Multiplicative Group ◽

Cartesian Product ◽

Arbitrary Field ◽

Necessary And Sufficient Condition ◽

Tensor Space ◽

Linear Character ◽

Finite Dimensional Vector Space ◽

Finite Dimensional ◽

Necessary And Sufficient

In this paper, we shall construct a vector space, called the (G, σ) space, which generalizes the tensor space, the Grassman space, and the symmetric space. Then we shall determine a necessary and sufficient condition that the (G, σ) product of the vectors x1, x2, …, xn is zero.1. Let G be a permutation group on I = {1, 2, …, n} and F, an arbitrary field. Let σ be a linear character of G, i.e., σ is a homomorphism of G into the multiplicative group F* of F.For each i ∈ I, let Vi be a finite-dimensional vector space over F. Consider the Cartesian product W = V1 × V2 × … × Vn.1.1. Definition. W is called a G-set if and only if Vi = Vg(i) for all i ∊ I, and for all g ∊ G.

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Operator Algebras with Contractive Approximate Identities

Canadian Journal of Mathematics ◽

10.4153/cjm-1994-021-0 ◽

1994 ◽

Vol 46 (2) ◽

pp. 397-414 ◽

Cited By ~ 6

Author(s):

Yiu-Tung Poon ◽

Zhong-Jin Ruan

Keyword(s):

Operator Algebra ◽

Operator Algebras ◽

Banach Algebras ◽

Approximate Identity ◽

The Other ◽

Double Centralizer ◽

Approximate Identities ◽

Centralizer Algebras ◽

Matrix Norms ◽

Centralizer Algebra

AbstractWe study operator algebras with contractive approximate identities and their double centralizer algebras. These operator algebras can be characterized as L∞- Banach algebras with contractive approximate identities. We provide two examples, which show that given a non-unital operator algebra A with a contractive approximate identity, its double centralizer algebra M(A) may admit different operator algebra matrix norms, with which M(A) contains A as an M-ideal. On the other hand, we show that there is a unique operator algebra matrix norm on the unitalization algebra A1 of A such that A1 contains A as an M-ideal.

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THE SECOND FUNDAMENTAL THEOREM OF INVARIANT THEORY FOR THE ORTHOSYMPLECTIC SUPERGROUP

Nagoya Mathematical Journal ◽

10.1017/nmj.2019.25 ◽

2019 ◽

pp. 1-25 ◽

Cited By ~ 1

Author(s):

G. I. LEHRER ◽

R. B. ZHANG

Keyword(s):

Invariant Theory ◽

Fundamental Theorem ◽

Brauer Algebra ◽

Tensor Space ◽

General Linear Supergroup ◽

Special Cases ◽

Linear Description ◽

Tensor Representations ◽

Capelli Identities ◽

Fundamental Theorems

The first fundamental theorem of invariant theory for the orthosymplectic supergroup scheme $\text{OSp}(m|2n)$ states that there is a full functor from the Brauer category with parameter $m-2n$ to the category of tensor representations of $\text{OSp}(m|2n)$ . This has recently been proved using algebraic supergeometry to relate the problem to the invariant theory of the general linear supergroup. In this work, we use the same circle of ideas to prove the second fundamental theorem for the orthosymplectic supergroup. Specifically, we give a linear description of the kernel of the surjective homomorphism from the Brauer algebra to endomorphisms of tensor space, which commute with the orthosymplectic supergroup. The main result has a clear and succinct formulation in terms of Brauer diagrams. Our proof includes, as special cases, new proofs of the corresponding second fundamental theorems for the classical orthogonal and symplectic groups, as well as their quantum analogues, which are independent of the Capelli identities. The results of this paper have led to the result that the map from the Brauer algebra ${\mathcal{B}}_{r}(m-2n)$ to endomorphisms of $V^{\otimes r}$ is an isomorphism if and only if $r<(m+1)(n+1)$ .

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