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The group elements are in principle abstract objects defined by the way they act on some structure. For example, the rotation group in three dimensions is formed by elements that rotate coordinate systems in some appropriate way. In order to make things easier to understand and visualize we assign linear representations, i.e. matrices to the elements of these groups. All elements of a Lie group in a given $d$-dimensional representation are then mapped to matrices of dimension $d$.

Among all elements of a Lie group there are special ones that can be used to generate any other. These are called generators of the group and they satisfy a particular structure, called Lie algebra. For example, the group $\mathrm{SU}(2)$ has a Lie algebra $\mathfrak{su}(2)$ whose generators are $T_a$, $a=1,2,3$, satisfying $$[T_a,T_b]=i\epsilon_{abc}T_c.$$ A representation $R$ of these abstract elements has to preserve this structure, i.e., $$[R(T_a),R(T_b)]=i\epsilon_{abc}R(T_c),$$ where $R(T)$ shall be understood as a $d$-dimensional matrix.

From the Lie algebra one can obtain all possible representations. For instance, the $\mathfrak{su}(2)$ algebra has $d$-dimensional representations for any integer $d$. The singlet is the one dimensional representation, i.e. they are only numbers. Notice that the only numbers can satisfy a non-trivial algebra is that all of them is zero. The doublet is two dimensional representation and the matrices representing the generators $T_a$ are just the Pauli matrices. The group obtained, as the exponential of the algebra is then $\mathrm{SU}(2)$. The triplet has dimension three, the same as the number of the generators of the algebra. This representation is called the adjoint and the explicit matrix form of the generators is given by the $l=1$ angular momentum matrices. There is although some subtleties. Starting from a given Lie algebra and assigning a given representation one can ends up with different Lie groups. So for adjoint representation of $\mathfrak{su}(2)$ the group generated turns out to be $\mathrm{SO}(3)$ instead of $\mathrm{SU}(2)$.

Regarding the physical meaning of a doublet or a triplet, it depends actually on the physics you want to describe. For instance, the spin in quantum mechanics is associated to a Lie algebra $\mathfrak{su}(2)$. If you are going to describe the spin of an electron you need the two dimensional representation of this algebra, since the electron has two states of spin, $\pm 1/2$. On the other hand if you interested in the spin of a $\rho$ meson, then you shall describe it through a triplet, since this vector meson has three states of spin, $0,\, \pm1$.

The group elements are in principle abstract objects defined by the way they act on some structure. For example, the rotation group in three dimensions is formed by elements that rotate coordinate systems in some appropriate way. In order to make things easier to understand and visualize we assign linear representations, i.e. matrices to the elements of these groups. All elements of a Lie group in a given $d$-dimensional representation are then mapped to matrices of dimension $d$.

Among all elements of a Lie group there are special ones that can be used to generate any other. These are called generators of the group and they satisfy a particular structure, called Lie algebra. For example, the group $\mathrm{SU}(2)$ has a Lie algebra $\mathfrak{su}(2)$ whose generators are $T_a$, $a=1,2,3$, satisfying $$[T_a,T_b]=i\epsilon_{abc}T_c.$$ A representation $R$ of these abstract elements has to preserve this structure, i.e., $$[R(T_a),R(T_b)]=i\epsilon_{abc}R(T_c),$$ where $R(T)$ shall be understood as a $d$-dimensional matrix.

From the Lie algebra one can obtain all possible representations. For instance, the $\mathfrak{su}(2)$ algebra has $d$-dimensional representations for any integer $d$. The singlet is the one dimensional representation, i.e. they are only numbers. Notice that the only possibility numbers can satisfy a non-trivial algebra is that they are all zero. The doublet is a two dimensional representation and the matrices representing the generators $T_a$ are just the Pauli matrices. We normally call the $n$-dimensional representation of $\mathfrak{su}(n)$ the defining representation. The group obtained, as the exponential of the algebra, is then $\mathrm{SU}(2)$. The triplet has dimension three which equals the number of generators of the algebra. This representation is called the adjoint representation and the explicit matrix form of the generators is given by the $l=1$ angular momentum matrices. There is although some subtleties. Starting from a given Lie algebra and assigning a given representation one can ends up with different Lie groups. So for adjoint representation of $\mathfrak{su}(2)$ the group generated turns out to be $\mathrm{SO}(3)$ instead of $\mathrm{SU}(2)$.

Regarding the physical meaning of a doublet or a triplet, it depends actually on the physics you want to describe. For instance, the spin in quantum mechanics is associated to a Lie algebra $\mathfrak{su}(2)$. If you are going to describe the spin of an electron you need the two dimensional representation of this algebra, since the electron has two states of spin, $\pm 1/2$. On the other hand if you interested in the spin of a $\rho$ meson, then you shall describe it through a triplet, since this vector meson has three states of spin, $0,\, \pm1$.

The group elements are in principle abstract objects defined by the way they act on some structure. For example, the rotation group in three dimensions is formed by elements that rotate coordinate systems in some appropriate way. In order to make things easier to understand and visualize we assign linear representations, i.e. matrices to the elements of these groups. All elements of a Lie group in a given $d$-dimensional representation are then mapped to matrices of dimension $d$.

Among all elements of a Lie group there are special ones that can be used to generate any other. These are called generators of the group and they satisfy a particular structure, called Lie algebra. For example, the group $\mathrm{SU}(2)$ has a Lie algebra $\mathfrak{su}(2)$ whose generators are $T_a$, $a=1,2,3$, satisfying $$[T_a,T_b]=i\epsilon_{abc}T_c.$$ A representation $R$ of these abstract elements has to preserve this structure, i.e., $$[R(T_a),R(T_b)]=i\epsilon_{abc}R(T_c),$$ where $R(T)$ shall be understood as a $d$-dimensional matrix.

From the Lie algebra one can obtain all possible representations. For instance, the $\mathfrak{su}(2)$ algebra has $d$-dimensional representations for any integer $d$. The singlet is the one dimensional representation, i.e. they are only numbers. Notice that the only numbers can satisfy a non-trivial algebra is that all of them is zero. The doublet is two dimensional representation and the matrices representing the generators $T_a$ are just the Pauli matrices. The group obtained, as the exponential of the algebra is then $\mathrm{SU}(2)$. The triplet has dimension three, the same as the number of the generators of the algebra. This representation is called the adjoint and the explicit matrix form of the generators is given by the $l=1$ angular momentum matrices. There is although some subtleties. Starting from a given Lie algebra and assigning a given representation one can ends up with different Lie groups. So for adjoint representation of $\mathfrak{su}(2)$ the group generated turns out to be $\mathrm{SO}(3)$ instead of $\mathrm{SU}(2)$.

The group elements are in principle abstract objects defined by the way they act on some structure. For example, the rotation group in three dimensions is formed by elements that rotate coordinate systems in some appropriate way. In order to make things easier to understand and visualize we assign linear representations, i.e. matrices to the elements of these groups. All elements of a Lie group in a given $d$-dimensional representation are then mapped to matrices of dimension $d$.

Among all elements of a Lie group there are special ones that can be used to generate any other. These are called generators of the group and they satisfy a particular structure, called Lie algebra. For example, the group $\mathrm{SU}(2)$ has a Lie algebra $\mathfrak{su}(2)$ whose generators are $T_a$, $a=1,2,3$, satisfying $$[T_a,T_b]=i\epsilon_{abc}T_c.$$ A representation $R$ of these abstract elements has to preserve this structure, i.e., $$[R(T_a),R(T_b)]=i\epsilon_{abc}R(T_c),$$ where $R(T)$ shall be understood as a $d$-dimensional matrix.

From the Lie algebra one can obtain all possible representations. For instance, the $\mathfrak{su}(2)$ algebra has $d$-dimensional representations for any integer $d$. The singlet is the one dimensional representation, i.e. they are only numbers. Notice that the only numbers can satisfy a non-trivial algebra is that all of them is zero. The doublet is two dimensional representation and the matrices representing the generators $T_a$ are just the Pauli matrices. The group obtained, as the exponential of the algebra is then $\mathrm{SU}(2)$. The triplet has dimension three, the same as the number of the generators of the algebra. This representation is called the adjoint and the explicit matrix form of the generators is given by the $l=1$ angular momentum matrices. There is although some subtleties. Starting from a given Lie algebra and assigning a given representation one can ends up with different Lie groups. So for adjoint representation of $\mathfrak{su}(2)$ the group generated turns out to be $\mathrm{SO}(3)$ instead of $\mathrm{SU}(2)$.

Regarding the physical meaning of a doublet or a triplet, it depends actually on the physics you want to describe. For instance, the spin in quantum mechanics is associated to a Lie algebra $\mathfrak{su}(2)$. If you are going to describe the spin of an electron you need the two dimensional representation of this algebra, since the electron has two states of spin, $\pm 1/2$. On the other hand if you interested in the spin of a $\rho$ meson, then you shall describe it through a triplet, since this vector meson has three states of spin, $0,\, \pm1$.

The group elements are in principle abstract objects defined by the way they act on some structure. For example, the rotation group in three dimensions is formed by elements that rotate coordinate systems in some appropriate way. In order to make things easier to understand and visualize we assign linear representations, i.e. matrices to the elements of these groups. All elements of a Lie group in a given $d$-dimensional representation are then mapped to matrices of dimension $d$.

Among all elements of a Lie group there are special ones that can be used to generate any other. These are called generators of the group and they satisfy a particular structure, called Lie algebra. For example, the group $SU(2)$ has a Lie algebra $\mathfrak{su}(2)$ whose generators are $T_a$, $a=1,2,3$, satisfying $$[T_a,T_b]=i\epsilon_{abc}T_c.$$ A representation $R$ of these abstract elements has to preserve this structure, i.e., $$[R(T_a),R(T_b)]=i\epsilon_{abc}R(T_c),$$ where $R(T)$ shall be understood as a $d$-dimensional matrix.

From the Lie algebra one can obtain all possible representations. For instance, the $\mathfrak{su}(2)$ algebra has $d$-dimensional representations for any integer $d$. The singlet is the one dimensional representation, i.e. they are only numbers. Notice that the only numbers can satisfy a non-trivial algebra is that all of them is zero. The doublet is two dimensional representation and the matrices representing the generators $T_a$ are just the Pauli matrices. The group obtained, as the exponential of the algebra is then $\mathrm{SU}(2)$. The triplet has dimension three, the same as the number of the generators of the algebra. This representation is called the adjoint and the explicit matrix form of the generators is given by the angular momentum generators. There is although some subtleties. Starting from a given Lie algebra and assigning a given representation one can ends up with different Lie groups. So for adjoint representation of $\mathfrak{su}(2)$ the group generated turns out to be $\mathrm{SO}(3)$ instead of $\mathrm{SU}(2)$.

The group elements are in principle abstract objects defined by the way they act on some structure. For example, the rotation group in three dimensions is formed by elements that rotate coordinate systems in some appropriate way. In order to make things easier to understand and visualize we assign linear representations, i.e. matrices to the elements of these groups. All elements of a Lie group in a given $d$-dimensional representation are then mapped to matrices of dimension $d$.

Among all elements of a Lie group there are special ones that can be used to generate any other. These are called generators of the group and they satisfy a particular structure, called Lie algebra. For example, the group $\mathrm{SU}(2)$ has a Lie algebra $\mathfrak{su}(2)$ whose generators are $T_a$, $a=1,2,3$, satisfying $$[T_a,T_b]=i\epsilon_{abc}T_c.$$ A representation $R$ of these abstract elements has to preserve this structure, i.e., $$[R(T_a),R(T_b)]=i\epsilon_{abc}R(T_c),$$ where $R(T)$ shall be understood as a $d$-dimensional matrix.

From the Lie algebra one can obtain all possible representations. For instance, the $\mathfrak{su}(2)$ algebra has $d$-dimensional representations for any integer $d$. The singlet is the one dimensional representation, i.e. they are only numbers. Notice that the only numbers can satisfy a non-trivial algebra is that all of them is zero. The doublet is two dimensional representation and the matrices representing the generators $T_a$ are just the Pauli matrices. The group obtained, as the exponential of the algebra is then $\mathrm{SU}(2)$. The triplet has dimension three, the same as the number of the generators of the algebra. This representation is called the adjoint and the explicit matrix form of the generators is given by the $l=1$ angular momentum matrices. There is although some subtleties. Starting from a given Lie algebra and assigning a given representation one can ends up with different Lie groups. So for adjoint representation of $\mathfrak{su}(2)$ the group generated turns out to be $\mathrm{SO}(3)$ instead of $\mathrm{SU}(2)$.