Lie Group Representations III

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In the other direction, the definition of a reductive Lie algebra means that it decomposes as a direct sum of ideals i. Some of these ideals will be one-dimensional and the rest are simple Lie algebras. Thus, a reductive Lie algebra is a direct sum of a commutative algebra and a semisimple algebra. In the physics literature, the tensor product with the identity operator is often suppressed in the notation, with the formula written as.

In the context of representations of the Lie algebra su 2 , the tensor product of representations goes under the name "addition of angular momentum.

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If we work in a basis, then the transpose in the above definition can be interpreted as the ordinary matrix transpose. The finite-dimensional irreducible representations are described by a theorem of the highest weight. Following an overview, the theory is described in increasing generality, starting with two simple cases that can be done "by hand" and then proceeding to the general result. The emphasis here is on the representation theory; for the geometric structures involving root systems needed to define the term "dominant integral element," follow the above link on weights in representation theory.

The first step amounts to analysis of hypothesized representations resulting in a tentative classification. The second step is actual realization of these representations. A real Lie algebra is usually complexified enabling analysis in an algebraically closed field. Working over the complex numbers in addition admits nicer bases.

The following theorem applies: A real-linear finite-dimensional representations of a real Lie algebra extends to a complex-linear representations of its complexification. The real-linear representation is irreducible if and only if the corresponding complex-linear representation is irreducible.

This means that every finite-dimensional representation decomposes as a direct sum of irreducible representations. The first step is to hypothesize the existence of irreducible representations. The properties of these hypothetical representations are investigated, [7] and conditions necessary for the existence of an irreducible representation are then established. The properties involve the weights of the representation.

Here is the simplest description. A partial ordering on the set of weights is defined, and the notion of highest weight in terms of this partial ordering is established for any set of weights.

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Using the structure on the Lie algebra, the notions dominant element and integral element are defined. One then shows that two irreducible representations with the same highest weight are isomorphic. Step One has the side benefit that the structure of the irreducible representations is better understood.

Representations decompose as direct sums of weight spaces , with the weight space corresponding to the highest weight one-dimensional. Repeated application of the representatives of certain elements of the Lie algebra called lowering operators yields a set of generators for the representation as a vector space. The application of one such operator on a vector with definite weight results either in zero or a vector with strictly lower weight. Raising operators work similarly, but results in a vector with strictly higher weight or zero.

The representatives of the Cartan subalgebra acts diagonally in a basis of weight vectors.

Step Two is concerned with constructing the representations that Step One allows for. The Lie algebra sl 2, C of the special linear group SL 2, C is the space of 2x2 trace-zero matrices with complex entries. The following elements form a basis:.

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Every finite-dimensional representation of sl 2, C decomposes as a direct sum of irreducible representations. This claim follows from the general result on complete reducibility of semisimple Lie algebras, [15] or from the fact that sl 2, C is the complexification of the Lie algebra of the simply connected compact group SU 2. That is to say, in this case, a "dominant integral element" is simply a non-negative integer. This analysis is described in detail in the representation theory of SU 2 from the point of the view of the complexified Lie algebra.

One can give a concrete realization of the representations Step Two in the overview above in either of two ways. There is a similar theory [20] classifying the irreducible representations of sl 3, C , which is the complexified Lie algebra of the group SU 3. The Lie algebra sl 3, C is eight dimensional. We may work with a basis consisting of the following two diagonal elements. That is to say, in this setting, a "dominant integral element" is precisely a pair of non-negative integers.

Unlike the representations of sl 2, C , the representation of sl 3, C cannot be described explicitly in general.

This can be done as follows. We will study representation theory and its role in the harmonic analysis on a Lie group. The classification of the irreducible representations of SU 2 will be studied. The final part of the course will be devoted to the classification of compact Lie algebras. Key words are: root system, finite reflection group, Cartan matrix, Dynkin diagram.

The course will be concluded with the formulation of the classification of irreducible representations by their highest weight and with the formulation of Weyl's character formula. A Lie group is a group with the additional structure of a differentiable manifold for which the group operation is differentiable. The name Lie group comes from the Norwegian mathematician M. Sophus Lie who was the first to study these groups systematically in the context of symmetries of partial differential equations.

Relationship between representations of Lie groups and Lie algebras - Mathematics Stack Exchange

The theory of Lie groups has developed vastly in the course of the previous century. It plays a vital role in the description of symmetries in Physics quantum physics, elementary particles , Geometry and Topology, and Number Theory automorphic forms. JavaScript is not enabled in your browser!

This is effected by taking the matrix exponential of the matrices of the Lie algebra representation, a topic which is investigated in some depth. The Lie correspondence gives results only for the connected component of the groups, and thus the components of the full Lorentz that contain the operations of time reversal and space inversion are treated separately, mostly from physical considerations, by defining representatives for the space inversion and time reversal matrices.

According to the general representation theory of Lie groups, one first looks for the representations of the complexification , so 3; 1 C of the Lie algebra so 3; 1 of the Lorentz group. A convenient basis for so 3; 1 is given by the three generators J i of rotations and the three generators K i of boosts. They are explicitly given in conventions and Lie algebra bases. Now complexify the Lie algebra, and then change basis to the components of [55]. One has the isomorphisms [57] [nb 16] String Module Error: function rep expects a number as second parameter, received ".

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The utility of these isomorphisms comes from the fact that all irreducible representations of su 2 are known. Every irreducible representation of su 2 is isomorphic to one of the highest weight representations. In A1 , all isomorphisms are C -linear the last is just a defining equality. The most important part of the manipulations below is that the R -linear irreducible representations of a real or complex Lie algebra are in one-to-one correspondence with C -linear irreducible representation of its complexification.

The manipulations to obtain representations of a non-compact algebra here so 3; 1 , and subsequently the non-compact group itself, from qualitative knowledge about unitary representations of a compact group here SU 2 is a variant of Weyl's so-called unitarian trick. The trick specialized to SL 2, C can be summarized concisely. Let V be a finite-dimensional complex vector space. The following statements are equivalent, in the sense that if one of them holds, then there is a uniquely determined modulo choice of basis for V corresponding representation either via given Lie algebra isomorphisms, or via complexification of Lie algebras per above, or via restriction to real forms, or via the exponential mapping to be introduced , or, finally, via a standard mechanism also to be introduced for obtaining Lie algebra representations given group representations of the appropriate type for the other groups and Lie algebras:.

If one representation is irreducible, then all of them are. In this list, direct products groups or direct sums Lie algebras may be introduced if done consistently. The essence of the trick is that the starting point in the above list is immaterial. Both qualitative knowledge like existence theorems for one item on the list and concrete realizations for one item on the list will translate and propagate, respectively, to the others. Here, the latter interpretation is intended. The representations for all Lie algebras and groups involved in the unitarian trick can now be obtained. The real linear representations for sl 2, C and so 3; 1 follow here assuming the complex linear representations of sl 2, C are known.

Explicit realizations and group representations are given later. The linearity properties follow from the canonical injection, the far right in A1 , of sl 2, C into its complexification. Here the tensor product is interpreted in the former sense of A0. These representations are concretely realized below. It's worth noting that the representations of so 3; 1 obtained this way are real linear and not complex or conjugate linear because the algebra is not closed upon conjugation, but they are still irreducible.

The notation D m , n is usually reserved for the group representations. These are, up to a similarity transformation , uniquely given by [nb 18] String Module Error: function rep expects a number as second parameter, received ".

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These are explicitly given as [64]. The approach in this section is based on theorems that, in turn, are based on the fundamental Lie correspondence. It is one-to-one in a neighborhood of the identity. The Lie correspondence and some results based on it needed here and below are stated for reference. The Lie correspondence reads in modern language as follows:. A linear Lie group is one that has at least one faithful finite-dimensional representation.

Using the above theorem it is always possible to pass from a representation of a Lie group G to a representation of its Lie algebra g. It is explicitly computed using [nb 25] String Module Error: function rep expects a number as second parameter, received ". This, of course, holds for the Lorentz group in particular, but not all Lie algebra representations arise this way because their corresponding group representations may not exist as proper representations, i.