, (2.4)

where. The matrix V has the following entries:

, , ,.

The singularity of V (expressed as proportionality of its rows) is equivalent to the existence of a certain (not necessarily real) number q to satisfy the equality of the second rows in (2.3):

, and .

In other words, the matrix g(Z) is as follows:


Since the matrix (2.5) has to be an element of U(2), v = 0 holds. It means that is on the torus T and that (2.1) and (2.3) hold. Now, the equality of the first rows in (2.3) is equivalent to the singularity of the matrix

. (2.6)

This last matrix has entries

, , ,.

Similarly, the singularity of the matrix (2.6) is equivalent to the existence of a certain (not necessarily real) number p to satisfy the equality of the first rows in (2.3). Again, g(Z) has to be on the torus T, and the entire (2.3) has to hold. Equality of the second rows in (2.3) forces the matrix to be singular. Lemma is proven.

Let us now proceed with

Theorem 2. Let be in GF and let U be in F. The image is defined if and only if is not on the torus T.

Proof. Let exists. Then according to [6] , Theorem 1,

. (2.7)

This implies being off the torus T, since (as it has been mentioned in Section 1) W is one-to-one between F and D\T.

If is not on the torus T, then according to Lemma, is non- singular (which means that is defined). Theorem 2 is proven.

Remark 2. Several examples of transformations are presented below. It turns out that the “is on/off the torus T” condition is easier to verify than to determine whether the is zero or not.

3. Explicit Description of Singularities of a Transformation

For a given, let us denote by the totality of all matrices U in F = U(1,1) where is undefined. Let g, satisfy (1.5).

If a matrix K is the image of Z under g, then the equality


holds. Clearly, (3.1) is equivalent to

. (3.2)

Additionally, let a matrix K be of the form (1.11). Since g is a bijection of D, the matrix is non-degenerate. Hence

. (3.3)

For a matrix U in, the matrix is of the form (1.11). Hence, the set of all matrices Z (which satisfy (3.3)) is defined by the ranges of parameters p, q in (1.11). Now, exclude those matrices Z which have zeros as (both) entries on the main diagonal and denote the remaining set by Y. In other words, exclude those matrices Z which belong to the torus T. We have thus proven the following

Theorem 3. (Description of singularities of).The set is the image of the above set Y under the map (1.10):

. (3.4)

Let us continue to discuss (including examples―see below) the set (3.4) properties (in other words, to discuss a domain of a transformation). On the basis of (3.4), the next statement holds true.

Corollary 1. is diffeomorphic to a subset of a (two-dimensional) torus.

As the first example, consider the following one-parameter subgroup in G: each g is determined by blocks

. (3.5)

Here c = ch(t/2), s = sh(t/2)―hyperbolic cosine and sine of a real parameter t. Assume that t is not zero (that is, g is not an identical map). Interestingly, the matrix is the same asg. It is an important example (see [2] , p. 85) since the isometry sub-algebra and the infinitesimal generator of the subgroup (3.5) generate the entire (15-dimensional) Lie algebra su(2,2). This holds both for the D-case, as well as for the F-case. Recall that each of the isometry groups is determined by the totality of all block-diagonal matrices: (1.6) for D, and its analogue for F. In [1] , Theorem 9, it was proven that these isometries of F act without singularities on it.

Proposition 1. Each matrix Z in (3.3) is of the form


The proof reduces to a (3.5)-based direct computation. Notice that for any (admissible in these circumstances) choice of parameters p, q, t, the expression () in (3.6) is never zero.

Hence, the following statement holds.

Corollary 2. The set of all singular points of a transformation (3.5) is W-diffeomor- phic to a set which is a (two-dimensional) torus with acircle cut off it: this circle is determined by the equation pq = 1 in (3.6).

Recall [1] , Section 6, where it has been shown that transformations (3.5) are singular in F. The example from there corresponds to a choice pq = −1 in (3.6).

Corollary 3. is contained in the subgroup SU(1,1) of the group F = U(1,1).

Proof. Applying Theorem 3, compute W(Z), where Z is an element of the set Y. One gets W(Z) as the product RS where


, ,. From (3.7) it follows that the determinant of the matrix W(Z) equals 1. Corollary 3 is thus proven.

As the second example, consider a two-parameter group А in G which is an (Abelian) subgroupА from the Iwasawa decomposition G = КAN. An arbitrary element min А is of the form


where the blocks (1.6) of the matrix g are as follows:

, , ,. (3.9)

Here c = cht1, s = sht1, hyperbolic cosine and sine of a real parameter t1. The blocks (1.6) of the matrix f are as follows:

, , ,. (3.10)

Here c = cht2, s = sht2, hyperbolic cosine and sine of a real parameter t2.

The following statement can be proven by a direct computation:

Proposition 2. For an arbitrary element m of the form (3.8), the matrix Z in (3.3) belongs to the torus T, given by (1.11). Namely,


where c1 = cht1, s1 = sht1, с2 = cht2, s2 = sht2. Notice that none of the denominators in (3.11) vanishes since.

In other words, restriction onto T of a transformation (3.8) is a bijection of T. The mapping W is inapplicable to matrices (3.11).That is why (according to Theorem 3) the set is an empty one. We have thus proven

Corollary 4. Each transformation is everywhere defined on U(1,1).

Remark 3. Corollary 4 can be proven on the basis of Theorem 2: in this case each transformation m of the form (3.8) is a bijection of the set D\T onto itself. In other words, none of the matrices is an element of the torus T.

Remark 4. Corollary 4 is coherent to the matrix being a block-diagonal one (compare to [1] , Theorem 9).

4. Conclusion

The action (1.8) of GF on F = U(1,1) has been introduced in [1] where it has been detected that this action has singularities. In [6] the fundamental relationship (1.12) between the action of G on D = U(2) and of GF on F has been determined. However, in [6] , it has not been explored when (that is, for which U in F) the right side of (1.12) was defined. Our Theorem 2 provides these singularities’ general (“geometric”) description. Theorem 3 and examples (in Section 3) indicate that the description is quite a working one. In particular, it is now guaranteed that the new analysis of space-time bundles (based on U(1,1) as the parallelizing group) is mathematically possible.

Cite this paper

Levichev, A. (2016) A Contribution to the DLF-Theory: On Sin- gularities of the SU(2,2)-Action in U(1,1). Journal of Modern Physics, 7, 1963-1971. http://dx.doi.org/10.4236/jmp.2016.715174


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