On the Homomorphisms of the Lie Groups 𝑆𝑈(2) and 𝑆³

Abstract

We first construct all the homomorphisms from the Heisenberg group to the 3-sphere. Also, defining a topology on these homomorphisms, we regard the set of these homomorphisms as a topological space. Next, using the kernels of homomorphisms, we define an equivalence relation on this topological space. We finally show that the quotient space is a topological group which is isomorphic to 𝕊¹.

1. Introduction

Discrete and continuous forms of the Heisenberg group have been studied in mathematics and physics such as analysis [1–3], geometry [4–6], topology [3, 7], and quantum physics [8–14]. An introductory review can be also found in [15].

In [16–18], it was shown that the Heisenberg group ℍ is nilpotent, and any arbitrary nilpotent subgroup of 𝑆𝑈(2) is conjugate to a subgroup of 𝑈(1), which is identified with the set of diagonal matrices in 𝑆𝑈(2). Since groups can be considered as metric spaces, it leads us to examine if there exists any geometry in these groups.

It is known that the matrices 1𝑥01,𝑥∈ℝ (1) form a linear group which is isomorphic to ℝ¹. The Lie groups ℝ¹ and 𝕊¹ are related, for the mapping 𝜌∶ℝ¹→𝕊¹ defined by 𝜌(𝑥)=𝑒^{2𝜋𝑖𝑥} is a continuous homomorphism from ℝ¹ onto 𝕊¹ [10]. The Lie algebras ℝ¹ and 𝕊¹ are trivially isomorphic. In this work, we search if there is a similar relationship between the only other sphere Lie group 𝕊³ and the linear group of matrices which is diffeomorphic to ℝ³ [17].

The subgroup ℍ of 𝔾𝕃(3,ℝ) formed by the matrices of type 1𝑥𝑧01𝑦001,𝑥,𝑦,𝑧∈ℝ (2) is called three-dimensional Heisenberg group. It is convenient to denote the elements of this group by three-tuples of numbers. Using this convention, that is, ℍ={(𝑥,𝑦,𝑧)∶𝑥,𝑦,𝑧∈ℝ}, the multiplicative operation of elements can be expressed as 𝑥,𝑦,𝑧⋅𝑥^′,𝑦^′,𝑧^′=𝑥+𝑥^′,𝑦+𝑦^′,𝑧+𝑧^′+𝑥𝑦^′. (3)

The identity element of this group is (0,0,0), and the inverse of an element (𝑥,𝑦,𝑥) is (−𝑥,−𝑦,−𝑧).

This Lie group is diffeomorphic to ℝ³ [17].

Moreover, the Lie groups 𝑆𝑈(2) and 𝕊³ are isomorphic, and they are diffeomorphic as manifolds. Since the respective Lie algebras ℎ and 𝑠𝑢(2) of ℍ and 𝑆𝑈(2) are three-dimensional real vector spaces, they are isomorphic as real vector spaces. But they are not isomorphic as Lie algebras, because there is no Lie algebra isomorphism between a compact and a noncompact Lie algebra. However, there may be a Lie algebra homomorphism between them. We try to find the relationship between ℎ and 𝑠𝑢(2) as Lie algebras.

Consider the following matrices: 𝑈₁1=−20𝑖𝑖0,𝑈₂=120−110,𝑈₃=12−𝑖00𝑖 (4) which span the Lie algebra 𝑠𝑢(2) of the Lie group 𝑆𝑈(2), and also consider the following matrices: 𝑉₁=010000000,𝑉₂=000001000,𝑉₃=001000000 (5) which span the Lie algebra ℎ of the Heisenberg group ℍ.

Using the exponential map from ℎ to ℍ we may convert our equations in ℎ to equations in ℍ. Letting 𝑉_𝑖=exp(𝑉_𝑖) for each 𝑖, we see that 𝑉₁=,110010001𝑉₂=,100011001𝑉₃=.101010001 (6)

The standard generating set for the Heisenberg group is {𝑉₁,𝑉₂,𝑉₁⁻¹,𝑉₂⁻¹}, and the group has the relation 𝑉₁𝑉₂𝑉₁⁻¹𝑉₂⁻¹=𝑉₂𝑉₁⁻¹𝑉₂⁻¹𝑉₁=𝑉₁⁻¹𝑉₂⁻¹𝑉₁𝑉₂=𝑉₂⁻¹𝑉₁𝑉₂𝑉₁⁻¹=𝑉₃. (7)

In the above, the notations 𝑉₁⁻¹ and 𝑉₂⁻¹ denote the inverses of the elements 𝑉₁ and 𝑉₂ in the Heisenberg group.

From (4) and (5), we find that 𝑈₁,𝑈₂=𝑈₃,𝑈₂,𝑈₃=𝑈₁,𝑈₃,𝑈₁=𝑈₂,𝑉₁,𝑉₂=𝑉₃,𝑉₂,𝑉₃=𝑉₃,𝑉₁=0. (8)

It was shown in ([13, 16]) that there is no nontrivial Lie algebra homomorphism from 𝑠𝑢(2) to ℎ. Here, we study Lie algebra homomorphisms from ℎ to 𝑠𝑢(2). We observe that there exists a nontrivial Lie algebra homomorphism from ℎ to 𝑠𝑢(2). Moreover, we describe all Lie algebra homomorphisms from ℎ to 𝑠𝑢(2).

2. Homomorphisms from ℍ to 𝑆𝑈(2)

Let 𝜑∶ℎ→𝑠𝑢(2) be a Lie algebra homomorphism. Suppose that 𝜑𝑉_𝑖=³_𝑗=1𝛼_𝑖𝑗𝑈_𝑗,𝑖=1,2,3 (9) for some real numbers 𝛼_𝑖𝑗.

By using (8) and the following commutators: 𝜑𝑉₁,𝑉₂=𝜑𝑉₁,𝜑𝑉₂,𝜑𝑉₃=𝜑𝑉₁,𝜑𝑉₂,𝜑𝑉₁,𝑉₃=𝜑𝑉₁,𝜑𝑉₃,𝜑𝑉₁,𝑉₃𝜑𝑉=0,₂,𝑉₃=𝜑𝑉₂,𝜑𝑉₃,𝜑𝑉₂,𝑉₃=0, (10) we obtain the system of equations of coefficients in the real constants 𝛼_𝑖𝑗's 𝛼₁₂𝛼₂₃−𝛼₁₃𝛼₂₂=𝛼₃₁,𝛼₁₃𝛼₂₁−𝛼₁₁𝛼₂₃=𝛼₃₂,𝛼₁₁𝛼₂₂−𝛼₁₂𝛼₂₁=𝛼₃₃,𝛼₁₂𝛼₃₃−𝛼₁₃𝛼₃₂𝛼=0,₁₃𝛼₃₁−𝛼₁₁𝛼₃₃=0,𝛼₁₁𝛼₃₂−𝛼₁₂𝛼₃₁𝛼=0,₂₁𝛼₃₂−𝛼₂₂𝛼₃₁=0,𝛼₂₃𝛼₃₁−𝛼₂₁𝛼₃₃𝛼=0,₂₂𝛼₃₃−𝛼₂₃𝛼₃₂=0. (11)

By using the Mathematica program, we may solve the system (11). We write below only two of the non-trival solutions, because a group homomorphism induced by any of the other solutions will be equal to a group homomorphism induced by one of the solutions given below. Here, a nontrivial solution means a solution in which at least one of the constants 𝛼_𝑖𝑗 is nonzero.(1)𝛼₂₂=𝛼₁₂𝛼₂₁/𝛼₁₁,𝛼₂₃=𝛼₁₃𝛼₂₁/𝛼₁₁,𝛼₃₁=𝛼₃₂=𝛼₃₃=0,  𝛼₁₁≠0, 𝛼₁₂, 𝛼₁₃, 𝛼₂₁ arbitrary.(2)𝛼₂₃=𝛼₁₃𝛼₂₂/𝛼₁₂,𝛼₁₁=𝛼₂₁=𝛼₃₁=𝛼₃₂=𝛼₃₃=0,  𝛼₁₂≠0,  𝛼₁₃, 𝛼₂₂ arbitrary.

Also, in the system of (11), if 𝛼₁₁≠0 and 𝛼₁₂=0, then the first set of solutions is obtained. If 𝛼₁₂≠0 and 𝛼₁₁=0, then the second set of solutions is obtained. Moreover, if 𝛼₁₁ and 𝛼₁₂ are both 0, then (11) has trivial solution.

In the set of solutions, we consider 𝛼₂₂ and 𝛼₂₃ in terms of the other arbitrary constants 𝛼₁₁, 𝛼₁₂, 𝛼₁₃, and 𝛼₂₁ and obtain that the group homomorphism is spanned by elements {𝑈₁,𝑈₂,𝑈₃} or {𝑈₂,𝑈₃}. If another configuration, say 𝛼₁₁, 𝛼₁₃, was chosen, then we see that the homomorphism would be spanned by the same elements.

Since 𝜑 is a Lie algebra homomorphism, we observe from solution sets 1 and 2 that the subalgebra of 𝑠𝑢(2) is only generated by 𝑎₁𝑈₁+𝑎₂𝑈₂+𝑎₃𝑈₃ or 𝑎𝑈₂+𝑏𝑈₃ and not by 𝑎₁𝑈₁+𝑎₂𝑈₃.

As we will use the constants 𝛼_𝑖𝑗 frequently, to simplify the notations we put 𝑎₁, 𝑎₂, 𝑎₃, and 𝑏 for the coefficients 𝛼₁₁, 𝛼₁₂, 𝛼₁₃, and 𝛼₂₁, respectively, and, 𝑎, 𝑏, and 𝑐 for 𝛼₁₂, 𝛼₁₃, and 𝛼₂₂.

Hence, for the first set of solutions, 𝜑 has the following form: 𝜑𝑉₁=𝑎₁𝑈₁+𝑎₂𝑈₂+𝑎₃𝑈₃,𝜑𝑉₂=𝑏𝑎₁𝜑𝑉₁,𝑎₁𝜑𝑉≠0,₃=0. (12)

We note that the rank of 𝜑 is one, and thus, 𝜑(ℎ) is a one-dimensional Lie subalgebra of 𝑠𝑢(2) generated by 𝑎₁𝑈₁+𝑎₂𝑈₂+a₃𝑈₃.

For the second set of the solutions, 𝜑 is of the following form: 𝜑𝑉₁=𝑎𝑈₂+𝑏𝑈₃,𝜑𝑉₂=𝑐𝑎𝜑𝑉₁𝜑𝑉,𝑎≠0,₃=0. (13)

Here, we note again that the rank of 𝜑 is one, and thus, 𝜑(ℎ) is a one-dimensional Lie subalgebra of 𝑠𝑢(2) generated by 𝑎𝑈₂+𝑏𝑈₃.

It is known from ([8, 18]) that if Φ∶ℍ→𝑆𝑈(2) is a homomorphism, then the following diagram commutes: (14) where 𝑑Φ is the differential of Φ, and it is a Lie algebra homomorphism. It is also known that for the matrix groups the exponential map is given by the exponentiation of matrices. In our notation, 𝜑=𝑑Φ for some Φ, and we will determine Φ.

By using (5), for any element of  𝑉=𝑐₁𝑉₁+𝑐₂𝑉₂+𝑐₃𝑉₃∈ℎ we have 𝑉=0𝑐₁𝑐₃00𝑐₂000,𝑉²=00𝑐₁𝑐₂,𝑉000000₃=000000000,𝑐₁,𝑐₂,𝑐₃∈ℝ. (15)

By using (15), we obtain exp(𝑉) as (𝑉)1exp=1+𝑉+2𝑉²+1𝑉3!³+⋯=1𝑐₁𝑐₃+𝑐₁𝑐₂201𝑐₂.001 (16)

To guarantee that exp(𝑉)==1𝑥𝑧01𝑦001𝑥,𝑦,𝑧 (17) we must take 𝑉=𝑥𝑉₁+𝑦𝑉₂+(𝑧−𝑥𝑦/2)𝑉₃∈ℎ. Here, we put (𝑥,𝑦,𝑧) for the matrix (2) for any real 𝑥, 𝑦, and 𝑧.

For the first set of solutions, we obtain 𝜑(𝑉)=𝜑𝑥𝑉₁+𝑦𝑉₂+𝑧−𝑥𝑦2𝑉₃=𝑏𝑥+𝑎₁𝑦𝑎₁𝑈₁+𝑎₂𝑈₂+𝑎₃𝑈₃. (18)

Thus, the kernel of 𝜑 is the plane 𝑥+(𝑏/𝑎₁)𝑦=0 in ℝ³.

For 𝑉∉Ker𝜑, we obtain (𝑉)𝑏exp𝜑=𝐼cos𝐴𝑥+𝑎₁𝑦+1𝐴𝑥+𝑏/𝑎₁𝑦𝐴𝑏sin𝑥+𝑎₁𝑦𝜑(𝑉), (19) where 𝐼 is the 2×2 identity matrix, and 𝐴=(1/2)𝑎₁²+𝑎₂²+𝑎₃².

In this case, Φ is of the following form: 𝑏𝑥,𝑦,𝑧⟶𝐼cos𝐴𝑥+𝑎₁𝑦+1𝐴𝐴𝑏sin𝑥+𝑎₁𝑦𝑎₁𝑈₁+𝑎₂𝑈₂+𝑎₃𝑈₃. (20)

For the second set of solutions, we obtain 𝜑(𝑉)=𝑐𝑥+𝑎𝑦𝑎𝑈₂+𝑏𝑈₃. (21)

From (21), the kernel of 𝜑 is the plane 𝑥+(𝑐/𝑎)𝑦=0 in ℝ³.

For 𝑉∉Ker𝜑, we have (𝑉)𝑐exp𝜑=𝐼cos𝐴𝑥+𝑎𝑦+1𝐴()𝑦𝐴𝑐𝑥+𝑐/𝑎sin𝑥+𝑎𝑦𝜑(𝑉), (22) where 𝐼 is the 2×2 identity matrix, and 𝐴=(1/2)𝑎²+𝑏².

In this case, Φ is of the form 𝑐𝑥,𝑦,𝑧⟶𝐼cos𝐴𝑥+𝑎𝑦+1𝐴𝐴𝑐sin𝑥+𝑎𝑦𝑎𝑈₂+𝑏𝑈₃. (23)

Hence, we can state the following theorem.

Theorem 1. Any nontrivial homomorphism from the Heisenberg group to the 3-sphere is one of (20) and (23).

We now observe the following properties for the first type of homomorphisms. It can be shown that same observations are valid for the second set of solutions.(a) By considering Φ as a map from ℝ³ to ℝ⁴, we can write Φ∶x,𝑦,𝑧⟶𝑢₁,𝑢₂,𝑢₃,𝑢₄, (24) where 𝑢₁=cos𝐴𝑡,𝑢₂𝑎=−³2𝐴sin𝐴𝑡,𝑢₃𝑎=−²𝑢2𝐴sin𝐴𝑡,₄𝑎=−¹𝑏2𝐴sin𝐴𝑡,𝑡=𝑥+𝑎₁𝑦. (25) Then, we find that the image of the planes 𝑃_𝑡 in ℍ, with the equation 𝑥+𝑏/𝑎₁=𝑡, is a periodic (closed) curve in 𝕊³.(b) In the first case, every homomorphism Φ from ℍ to 𝑆𝑈(2) depends on four parameters, namely, 𝑎₁, 𝑎₂, 𝑎₃, and 𝑏 with 𝑎₁≠0. We now concentrate on the parameters 𝑎₁ and 𝑏, since they are involved in the kernel of Φ. To each kernel 𝑥+(𝑏/𝑎₁)𝑦=0, we associate a point (𝑏,−𝑎₁) in the 𝑥𝑦-plane. Furthermore, we normalize the vector (𝑏,−𝑎₁) as (𝑏/𝑏²+𝑎²₁,−𝑎₁/𝑏²+𝑎²₁) which can also be considered as a point of 𝕊¹ in the 𝑥𝑦-plane. By considering the principal value of arctan(−𝑏/𝑎₁), we define a topology on the set ℍ^′ of all homomorphisms from ℍ to 𝑆𝑈(2) as follows. For any Φ₁(𝑎₁,𝑎₂,𝑎₃,𝑏)∈ℍ^′, Φ₂(𝑎^′₁,𝑎^′₂,𝑎^′₃,𝑏^′)∈ℍ^′ is in the 𝜖-neighborhood of Φ₁, if and only if |arctan(−𝑏/𝑎₁)−arctan(−𝑏^′/𝑎^′₁)|<𝜖, |𝑎₂−𝑎^′₂|<𝜖, and |𝑎₃−𝑎^′₃|<𝜖. Now let us define an equivalence relation on ℍ^′. For Φ₁, Φ₂∈ℍ^′, Φ₁ is equivalent to Φ₂ if and only if Φ₁ and Φ₂ have the same kernel. Denote the set of equivalence classes by ℍ. We define a multiplication on ℍ such that, for any Φ₁ and Φ₂ in ℍ, Φ₁⋅Φ₂ denotes the element of ℍ whose kernel is the plane obtained by the product of the elements of 𝕊¹ corresponding to Φ₁ and Φ₂. This multiplication makes ℍ into a group which is isomorphic to 𝕊¹.(c) The set of the kernels of all the homomorphisms from ℍ to 𝕊³ is a subset of the Grassmann manifold of 2 planes in ℝ³. It is known that the Grassmann manifold of 2 planes in ℝ³ is diffeomorphic to 𝕊¹. Any point 𝑝 of 𝕊¹ corresponds to the plane which is orthogonal to the normal of 𝕊¹ at 𝑝 and which contains the 𝑧-axis. Hence, there exists a 1-1 correspondence between the set of the kernels of all the homomorphisms from ℍ to 𝕊³ and the equator 𝕊¹ of 𝕊².

Thus, we state the following theorem.

Theorem 2. The set of all homomorphisms from the Heisenberg group to the 3-sphere is isomorphic (up to a certain equivalence relation concerning kernels) with the topological group 𝕊¹.

3. Conclusion

In this paper, our aim is to construct all homomorphisms between the Heisenberg group and the 3-sphere which is isomorphic to 𝑆𝑈(2). In the literature, it has been shown that no nontrivial Lie algebra homomorphism from 𝑠𝑢(2) to ℎ exists ([8, 11, 17]). So, a natural question arises: is there a homomorphism from ℎ to 𝑠𝑢(2)? Here, we answer this question completely.

We here observe that there are nontrivial homomorphisms from the Heisenberg group to the 3-sphere, and they can be only in the form of (20) or (23).

Also, we use these maps to define a topology in order to construct an equivalence relation, and we show that quotient space is isomorphic to 𝕊¹.

Acknowledgment

The authors would like to thank the referee for his/her helpful comments and suggestions.