Introduction
The use of Lie algebra and its representation is especially widespread in particle physics. Gauge theories have been used to express descriptions of nature. Each description requires that fields be assigned to representations of compact Lie groups and their respective algebraic representations. Therefore, mass and interaction terms give rise to a need to compute tensor products within the Lie algebra groups. Sub-algebra decomposition is part of spontaneous symmetry breaking.
When considering E8 from a mathematical perspective, this value represents the exceptional simple Lie groups of the dimension 248. This connotes the same corresponding root lattice having the rank 8. This classification comes from the Cartan-Killing classifications for the simply Lie algebras falling into the infinite labels A, B, C and D. moreover, there are five exceptional cases which are labeled as E6, E7, E8, F4, and G2. By far, the E8 forms the largest and most complicated of the exceptional cases, thereby sparking interest of study. This paper discusses the applications of E8 algebra and its computations in general.
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Background Information
William Killing in his work throughout the nineteenth century discovered complex Lie algebra, despite his lack of proof that it existed. Later on, Elie Cartan determined that the simple Lie algebra type E8 could admit three real forms. In his work, Carlan determined that the Lie group has the dimension of 248, the only one of its kind which is compact. Later works have introduced the algebraic group of E8 over other groups and fields including finite fields that ultimately lead to infinite family groups in the Lie type.
In a brief description, the Lie Group value E8 has a dimension of 248. The rank represents the dimension of maximum torus and its value is 8. As a result, every vector in the root system with the eight-dimension space is described within the confines of the E8 value. Furthermore, the E8 has a Weyl group who order is represented by the value 21435527. The total value of the symmetries described by the E8 value then is equal to 696729600. These are all the groups for which the maximal torus can be induced through conjugation within the entire group. The E8 group is quite the unique group among the Lie groups. The smallest dimension in the representation is the E8 itself. Furthermore it has the trivial center, which is simply connected and laced as well as compact since all the roots have the same length with the value 8. Additionally, there is a Lie algebra value for every integer whose value is equal to or more than 3, whose dimension is infinite if the value is greater than 8.
Complex and Real Forms
Unique complex with the type E8 exists corresponding to complex groups within the dimension 248. Additionally, all groups within this dimension are considered complex alongside the group itself. Therefore, this E8 within the complex dimension of 248 is normally considered as a real Lie group of dimension 486. They are simply connected and have maximal compact subgroups in a compact form that will be shown below. Furthermore, the value has an outer automorphism of order two which can be generated through complex conjugation.
Aside from the complex group E8, there are three real forms with a trivial center with the real dimension of 248. Among the three, two have non-algebraic covers which produce real forms. They can be given as follows:
A compact form – this is a value having trivial auto morphism characteristics within its group and is simply connected supposing that no other information is given.
A split form – this has a maximal compact spin within the subgroups given at a value of [16/ (Z/2Z)]. There is the inclusion of the fundamental order 2, showing that the value has a double cover connected simply as a real Lie group and equally has an automorphic group.
EIX having the maximal compact of E7×SU (2)/ (−1, −1). This is a fundamental group within order 2, showing that the value has an outer cover as well as an auto morphism group.
Other simple Lie groups exist. Nonetheless, the focus of the paper remains on the E8 and its proportions. The E8 has also been considered as an algebraic group by the Chevalley basis. By virtue of this basis, the group is considered as an algebraic group over a number of integers forming a commutative ring in their field. This constitutes the spilt form of the value. In a closed field, the E8 value consists the only form. However, a larger field sees the E8 take many other forms and twists, otherwise known as the Galois cohomology where the perfect field of k value exists.
Where R exists, real components with the identity matching the algebraically twisted forms normally coincide with either one of the three groups mentioned above. Nonetheless, subtlety is normally experienced with the fundamental groups since all forms are connected with geometry within the E8 value. This ultimately means that a restriction is put on non-trivial coverings against admission. This E8 value, therefore, is non-compact and will not admit faithful finite representations. Within finite fields, it follows that the Lang-Steinberg theory will apply to remove any twisted forms from the E8 value.
The nature of finite representations within complex and real Lie algebra and groups are given through the Weyl character formula. At the 248-dimension, one gets the adjoint representation. Furthermore, there exist non-isomorphic representations of the dimension. The fundamental representations correspond to eight nodes within E8. Coefficients within character formulae depend on large square matrices which consist of polynomials. Values at 1 within this polynomial structure give the coefficient to the matrices which relate to standard representations which have irreducible representations. The Deligne-Lusztig theory gives the representations of E8 in finite fields.
Research has shown that it is possible to construct the E8 group from the corresponding algebraic expression. This kind has 120 dimensions alongside 128 new generators. These transform the Weyl-Majorana spinor at a rate of 16. Additionally, the real form of E8 can equally be constructed within geometry to give the compact EVIII within the Cartan classification. This value is also known as the octooctoionic plane which can be built by the use of tensor product algebra within the octoions. The construction of the magic square is an application of this geometrical application of the value. Further, the E8 has a root system that can handle specific configurations of vectors with a rank r. the roots must also have r-dimension properties in the Euclidean space to enable them to qualify. Considering the E8, one notes that it is a value with rank 8 containing 240 root vectors. The value cannot be built from smaller rank roots, showing that it is irreducible. The fact that all vectors within the E8 have the same length, it is possible that any number can be normalized to give the value of the square root of 2.
The Lie algebra value E8 contains other sub algebras, every exception Lie algebra alongside other important Lie algebra applications in physics and math. Therefore, the height of the Lie algebra is directly corresponding to its rank. A line in the algebra showing downward movement depicts a sub-algebra signifying the correspondence of different levels of the algebra. The application of this knowledge has been used in various fields including theoretical physics and supergravity as well as string theory. E8 multiplied by itself gives the gauge group in the heterotoric string group and will normally create an anomaly-free gauge group whose supergravity value is in the ten dimensions. In the split form, E8 gives the duality value in 8-torus. The symmetry breaking of the E8 is another way to incorporate the standard model into heterotoric string applications. This is because the E8 is normally broken to sub-algebra of its fitting, thereby allowing its application within the standard model. Moreover, the E8 lattice was used in 1982 to create a 4-manifold which lacks a smooth structure. Different research tries to show the full applications of an E8 structure, but the fundamentals of the operations still exist.
Conclusion
In conclusion, it is possible to see that the applications of the E8 are yet to become quantified in full, with recent research stretching back to only a decade ago. Of necessity then, is to understand the working of dimensional applications within the groups and determine possible uses extrapolated from current uses.
References
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