Subalgebras of Lie algebras and different classes of subalgebras of gl n;C 1. Ideals, Quotient Lie algebras, derived sub Lie algebras and direct sum 1. Isomorphism theorems, Killing form and some basic theorems 1. Derivation of Lie algebras 1. Representations of Lie algebras and Representations of sl 2;C 1.
Rootspace decomposition of semisimple Lie algebras 1. Root system in Euclidean spaces and Root diagrams 1. Coxeter graphs and Dynkin Diagrams 1. Cartan Matrices, Ranks and dimensions of simple Lie algebras 1. Weyl groups and Structure of Weyl groups of simple Lie algebras 1. Root systems of Classical simple Lie algebras and Highest long and short roots 1. Universal enveloping algebras of Lie algebras 1.
Representation theory of semisimple Lie algebras 1. Construction of semisimple Lie algebras by Generators and Relations 1.
Cartan-Weyl basis 1. Character of a finite dimensional representation and Weyl dimension formula 1. Lie algebras of vector fields 2. Kac - Moody algebras 2. Basic concepts in Kac-Moody algebras 2. Invariant bilinear forms 2. Coxeter Groups and Weyl Groups 2. Real and imaginary roots of Kac-Moody algebras 2. Weyl Groups of affine Lie algebras 2. Realization of Affine Lie algebras 2. Different classes of imaginary roots special imaginary roots, strictly imaginary roots, purely imaginary roots in Kac-Moody algebras 2.
Graded Lie algebras and Root multiplicities 3. Generalized Kac-Moody algebras 3. Dynkin Diagrams of GKM algebras 3. Root systems and Weyl groups of GKM algebras 3. Special imaginary roots in GKM algebras and their complete classifications 3. Strictly imaginary roots in GKM algebras and their complete classifications 3.
Purely imaginary roots in GKM algebras and their complete classifications 3. Representations of GKM algebras 3. Homology modules and root multiplicities in GKM algebras 4. Lie superalgebras 4. Basic concepts in Lie superalgebras with examples 1. Subsuperalgebras, ideals of Lie superalgebras, abelian Lie superalgebras, solvable and nilpotent Lie superalgebras 4. General linear Lie superalgebras 4. Simple and semisimple Lie superalgebras and bilinear forms 4.
Representations of Lie superalgebras 4. Different classes of classical Lie superalgebras 4.
Dimension Ind and mappings. Chapter 3: The covering dimension.
Basic properties of the dimension dim in normal spaces. Relations between the dimensions ind, Ind and dim. Characterizations of the dimension dim in normal spaces. Dimension dim and mappings. The compactification, universal space and Cartesian product theorems for the dimension dim. Dimension dim and inverse systems of compact spaces. Chapter 4: Dimension theory of metrizable spaces.
Basic properties of dimension in metrizable spaces. Characterizations of dimension in metrizable spaces. The universal space theorems. Dimension and mappings in metrizable spaces. Chapter 5: Countable-dimensional spaces. Definitions and characterizations of countable-dimensional and strongly countable-dimensional spaces. Basic properties of countable-dimensional and strongly countable-dimensional spaces. The compactification and universal space theorems for countable-dimensional and strongly countable-dimensional spaces.
Countable dimensionality and mappings.
Locally finite-dimensional spaces. Chapter 6: Weakly infinite-dimensional spaces. Definition and basic properties of weakly infinite-dimensional spaces. An example of a totally disconnected strongly infinite-dimensional space. Weak infinite dimensionality and mappings. Maybe in another universe there would be only black holes or only photons, only light.
Some universes could be very short-lived, lasting only a few milliseconds; others would be eternal. In fact, in this theory there are approximately 10 solutions, so there is almost an infinity of solutions. All the possible realizations of universes could be obtained in this kind of theory, but it is still very hypothetical. String theory does not yet have an exact equation that can be solved, just mathematical ideas that people try to put together with real physics. But success is still far away.
So where are these parallel universes if they exist? Well, it depends on the model, because there are many models and each model proposes a different kind of parallel universes. In some models, for instance in the inflationary universe model the ultra-fast expansion of the universe in its initial instants , you have a single universe but within this universe you have very large regions which in the past have been subjected to inflation, namely a specific process that greatly extends the size of space, creating a sort of bubbles of space-time, but with specific physical properties.
And if the model of what we call chaotic inflation did not occur in the same way in all the places in space and time, it would create many bubbles with different physical properties. In some sense, each of these bubbles can be called a universe. So now the question is whether two adjacent bubbles can collide or interact. This is only one example of the Multiverse.
In other models, the Multiverses are just in other dimensions. For instance, in string theory -with a space of 10 dimensions- we can consider that the Universe is just a three-dimensional section of a ten-dimensional space. And if you change the section —the cut-, you change the Universe. Each three-dimensional brain could correspond to a different universe. We know of the existence of at least two kinds of black holes. Stellar-mass black holes , which are formed by the gravitational collapse of single massive stars.
These black holes are not very big, just a few kilometers in size. And maybe in our galaxy there are several millions of stellar-mass black holes.
Some of them can be observed indirectly because they belong to binary systems. In a binary system, the black hole in fact attracts the gaseous envelope of the other star and the gas flows into the black hole before disappearing into it. It is heated at very high temperatures, emitting a lot of x-rays. And this is actually detected. Now —maybe most interestingly and spectacularly- we believe, and are almost sure —due to observations-, that in the center of each galaxy there is a huge black hole.
Ten dimensions are used to describe superstring theory , eleven dimensions can describe supergravity and M-theory , and the state-space of quantum mechanics is an infinite-dimensional function space. This provides a better definition as then the finite spherical universe still makes sense with respect to Mach's Principle and Einstein General Relativity. Root supermultiplicities of Borcherds superalgebras which are extensions of Kac-Moody Algebras and some combinatorial identities. Author index. Lemaitre developed a new cosmological theory. As our finite spherical universe is part of this infinite Space then this uniform distribution of matter on the large scale explains our 'flat' universe.
Not a stellar-mass black hole, but what we call a super-massive black hole with a mass more than a million times the solar mass, even in some cases seven billion solar masses. For instance, in the center of our own galaxy —the Milky Way- we measured indirectly the mass of a black hole with four million solar masses. With such masses, these black holes are able to capture full stars and destroy them by what we call tidal forces, namely gravitational differences that elongate the star so strongly that the star is destroyed.
Many years ago I worked on the process of tidal destruction of a star and I predicted the process of pancake stars, namely the star is flattened crossing a region around the big black hole -flattened into the orbital plane in the shape of a gaseous pancake-, it is strongly heated and it explodes and produces a flare.
So I called that the flambeed stellar pancake. And it has finally been observed.
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Now, with big telescopes, we can observe in other galaxies some flares of light, which are interpreted as tidal destructions of stars. So it is clear that very big black holes can destroy stars. And even bigger black holes —because in some galaxies we have recently measured black holes weighing more than 10 billion solar masses- which cannot destroy stars outside of them, but only inside.
Stars can just drop into this black hole and disappear without producing astronomical effects. So we do have some sort of monsters. It is a very interesting problem to try to explain the formation of such big black holes. If the black hole is not too massive —for instance, a few million solar masses, like the one in the center of our galaxy-, we can explain the formation as the growth of an initial small black hole formed by a star -during a million or a billion years-, that grows and grows in mass and size, but just by attracting matter.
But huge black holes -with for instance 10 billion solar masses- have no time to grow by this process the Universe is So this is one of the unsolved questions now. A possible solution is that the Big Bang could have produced a first generation of massive black holes, which would explain why we can observe very big massive black holes very early in the history of the Universe.