A connection behind the Terwilliger algebras of \(H(D,2)\) and \(\frac{ 1}{ 2} H(D,2)\)

DOI10.1016/J.JALGEBRA.2023.07.019arXiv2210.15733OpenAlexW4385302640MaRDI QIDQ6052918

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Publication date: 25 September 2023

Published in: (Search for Journal in Brave)

Abstract: The universal enveloping algebra

U (m a t h f r a k {s l}_{2})

of

m a t h f r a k {s l}_{2}

is a unital associative algebra over

m a t h b b C

generated by

E, F, H

subject to the relations �egin{align*} [H,E]=2E, qquad [H,F]=-2F, qquad [E,F]=H. end{align*} The distinguished central element

Lambda=EF+FE+frac{H^2}{2}

is called the Casimir element of

U (m a t h f r a k {s l}_{2})

. The universal Hahn algebra

m a t h c a l H

is a unital associative algebra over

m a t h b b C

with generators

A, B, C

and the relations assert that

[A, B] = C

and each of �egin{align*} alpha=[C,A]+2A^2+B, qquad �eta=[B,C]+4BA+2C end{align*} is central in

m a t h c a l H

. The distinguished central element

Omega=4ABA+B^2-C^2-2�eta A+2(1-alpha)B

is called the Casimir element of

m a t h c a l H

. By investigating the relationship between the Terwilliger algebras of the hypercube and its halved graph, we discover the algebra homomorphism

a t u r a l : m a t h c a l H i g h t a r r o w U (m a t h f r a k {s l}_{2})

that sends �egin{eqnarray*} A &mapsto & frac{H}{4}, \ B & mapsto & frac{E^2+F^2+Lambda-1}{4}-frac{H^2}{8}, \ C & mapsto & frac{E^2-F^2}{4}. end{eqnarray*} We determine the image of

a t u r a l

and show that the kernel of

a t u r a l

is the two-sided ideal of

m a t h c a l H

generated by

and

16 O m e g a - 24 a l p h a + 3

. By pulling back via

a t u r a l

each

U (m a t h f r a k {s l}_{2})

-module can be regarded as an

m a t h c a l H

-module. For each integer

n g e q 0

there exists a unique

(n + 1)

-dimensional irreducible

U (m a t h f r a k {s l}_{2})

-module

L_{n}

up to isomorphism. We show that the

m a t h c a l H

-module

L_{n}

(

n g e q 1

) is a direct sum of two non-isomorphic irreducible

m a t h c a l H

-modules.

Full work available at URL: https://arxiv.org/abs/2210.15733

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