The standard generators of the tetrahedron algebra and their look-alikes
Jae-Ho Lee

TL;DR
This paper explores the structure of the tetrahedron algebra, introducing elements that resemble standard generators and establishing a basis where each element is similar to specific generators, revealing new insights into its algebraic structure.
Contribution
The paper defines a new class of elements called 'look-alikes' in the tetrahedron algebra and constructs a basis where each element is either a look-alike of certain standard generators.
Findings
Established a basis where each element is $x_{ij}$-like, $x_{jk}$-like, or $x_{ki}$-like.
Characterized the properties of these look-alike elements and their relations.
Provided multiple perspectives on the new basis within the algebra.
Abstract
The tetrahedron algebra is an infinite-dimensional Lie algebra defined by generators and some relations, including the Dolan-Grady relations. These twelve generators are called standard. We introduce a type of element in that "looks like" a standard generator. For mutually distinct , consider the standard generator of . An element is called -like whenever both (i) commutes with ; (ii) and satisfy a Dolan-Grady relation. Pick mutually distinct . In our main result, we find an attractive basis for with the property that every basis element is either -like or -like or -like. We discuss this basis from multiple points of view.
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Taxonomy
TopicsMathematics and Applications · Advanced Topics in Algebra · Algebraic and Geometric Analysis
