- #1
bob2
- 7
- 0
The Poincare algebra is given by isl(2, R) ~ sl(2,R) + R^3. What exactly does the i stand for?
Thanks a lot in advance!
Thanks a lot in advance!
Last edited:
I'm sorry, my statement was incorrect. You are right about the Poincare algebra. I meant the Poincare algebra in 3dim (not 4dim) space and not the dimension of the Lie algebra. by sl(2,R) + R^3 the direct sum of the special linear algebra sl(2, R) and R^3 is denotedfresh_42 said:I have never seen a notation ##isl(2,\mathbb{R})##. Can you give some more background? And what does ##sl_2 + \mathbb{R}^3## mean? And as far as I know, the Poincaré algebra is ten dimensional.
In this case my question is: ##sl(2,\mathbb{R}) \oplus \mathbb{R}^3## as a Lie algebra or simply the vector space? I assume ##isl(2,\mathbb{R})## is simply an abbreviation the author of your source uses for this (presumably Lie algebra) sum. And I further assume that it is not a direct product, but an indirect, i.e. I suppose ##sl(2,\mathbb{R})## to act non-trivially on ##\mathbb{R}^3##.bob2 said:I'm sorry, my statement was incorrect. You are right about the Poincare algebra. I meant the Poincare algebra in 3dim (not 4dim) space and not the dimension of the Lie algebra. by sl(2,R) + R^3 the direct sum of the special linear algebra sl(2, R) and R^3 is denoted
The 3dim Poincare Algebra - isl(2,R) is a mathematical structure that describes the symmetries of spacetime in three dimensions. It combines the Poincare group, which describes the symmetries of special relativity, with the special linear group in two dimensions (isl(2,R)). This algebra is important in theoretical physics for understanding the behavior of particles in three-dimensional space.
The components of 3dim Poincare Algebra - isl(2,R) are the generators of translations, rotations, and boosts in three-dimensional space, as well as the generators of special linear transformations in two dimensions. These components, when combined, form a Lie algebra with specific commutation relations that describe the symmetries of spacetime.
3dim Poincare Algebra - isl(2,R) is used in theoretical physics to describe the symmetries of spacetime in three dimensions. This algebra is essential for understanding the behavior of particles and fields in three-dimensional space, and it is also used in the study of quantum field theories and string theory.
The special linear group isl(2,R) plays a crucial role in 3dim Poincare Algebra as it represents the symmetries of two-dimensional space. This group is important in describing the behavior of particles and fields in two dimensions, and when combined with the Poincare group, it forms a complete description of the symmetries of spacetime in three dimensions.
While 3dim Poincare Algebra - isl(2,R) is primarily used in theoretical physics, it also has applications in other fields such as mathematics and engineering. This algebra is used in the study of Lie algebras and their representations, as well as in the development of geometric control systems and robotics. It also has applications in computer graphics and computer vision.