Investigating the existence of algebra and finding hidden symmetries in

physical systems is one of the most important aspects for understanding their

behavior and predicting their future. Expanding this unique method of study to

cosmic structures and combining past knowledge with new data can be very

interesting and lead to discovering new ways to analyze these systems. However,

studying black hole symmetries always presents many complications and sometimes

requires computational approximations. For example, checking the existence of

Killing vectors and then calculating them is not always an easy task. It

becomes much more difficult as the structure and geometry of the system become

more complex. In this work, we will show that if the wave equations with a

black hole background can be converted in the form of general Heun equation,

based on its structure and coefficients, the algebra of the system can be

easily studied, and computational and geometrical complications can be omitted.

For this purpose, we selected two $AdS_5$ black holes: Reissner-Nordstrom (R-N)

and Kerr, and analyzed the Klein-Gordon equation with the background of these

black holes. Based on this concept, we observed that the radial part of the R-N

black hole and both the radial and angular parts of the Kerr black hole could

be transformed into the general form of the Heun equation. As a result,

according to the algebraic structure that governs the Heun equation and its

coefficients, one can easily achieve generalized $sl(2)$ algebra.

## Understanding Algebra and Symmetries in Physical Systems

Investigating the existence of algebra and finding hidden symmetries in physical systems is crucial for understanding how these systems behave and predicting their future. By expanding this method of study to cosmic structures and combining past knowledge with new data, we can discover new ways to analyze these systems.

### Challenges and Opportunities

Studying black hole symmetries poses many complications and often requires computational approximations. Checking the existence of Killing vectors and calculating them is not always easy, especially as the complexity of the system increases.

### Roadmap: Using the Heun Equation for Studying Black Hole Symmetries

- Select two AdS
_{5}black holes: Reissner-Nordstrom (R-N) and Kerr. - Analyze the Klein-Gordon equation with the background of these black holes.
- Convert the wave equations with black hole backgrounds into the form of the general Heun equation.
- Study the algebra of the system based on the structure and coefficients of the Heun equation.
- Omit computational and geometrical complications by leveraging the algebraic structure of the Heun equation.
- Observe that the radial part of the R-N black hole and both the radial and angular parts of the Kerr black hole can be transformed into the general form of the Heun equation.
- Achieve generalized sl(2) algebra based on the algebraic structure governing the Heun equation and its coefficients.

By following this roadmap, researchers can gain a deeper understanding of black hole symmetries without relying on complex computational approximations. The use of the Heun equation allows for a more streamlined analysis of the algebraic structure governing these systems.