On a Class of Unitary Representations of the Braid Groups B3 and B4 Sergio Albeverio, Slavik Rabanovich
no. 327
Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungsgemeinschaft getragenen Sonderforschungsbereiches 611 an der Universität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, April 2007
On a class of unitary representations of the braid groups B3 and B4 Sergio Albeverio Institut f¨ ur Angewandte Mathematik, Universit¨ at Bonn, Wegelerstr. 6, D-53115 Bonn, Germany; SFB 611, IZKS, Bonn, BiBoS, Bielefeld–Bonn, Germany; CERFIM, Locarno; Accademia di Architettura, USI, Mendrisio, Switzerland E-mail:
[email protected]
Slavik Rabanovich Institute of Mathematics, Ukrainian National Academy of Sciences, 3 Tereshchenkivs’ka, Kyiv, 01601, Ukraine,
Abstract We describe a class of irreducible unitary representations of the braid group B3 in every dimension. Moreover using Burau unitarisable representation, we present a class of nontrivial unitary representations also for the braid group B4 in the case where the dimension of the space is a multiple of 3. Key words: Braid group, unitary representation, tensor product, conjugate class 1991 MSC: 20F36, (20C08,57M25)
1
Introduction
In this paper we shall work with Artin’s braid groups Bk , k ∈ N. It has a standard presentation in generators and relations which first appeared in [2]: σi σi+1 σi = σi+1 σi σi+1 , i σ1 , . . . , σk−1 σi σj = σi σj , |i − j| > 1
*
Bk =
= 1, . . . , k − 2,
+
There are various representations of Bk (see, for example, [4]). The description of finite dimensional unitary representations connects with constrains on the spectra of unitary matrices. Among the results let us mention the finding
all such representations for B3 in small dimensions [13], and the study of representations with a small number of points in the spectrum of the matrix corresponding to σ1 (see, for example, [7] and [10]). Also let us mention several interesting connections have been found between the representations of the braid groups and representations of other objects such as quantum groups and Kac-Moody algebras as well as with R-matrices and solutions of quantum Yang-Baxter equations, see e.g., [3]. The braid group B3 can be generated by two elements S and J and the relation S 2 = J 3 (see Preliminaries for precise formulas). For an irreducible unitary representation π of B3 , this means that π(S) can be defined by one of its eigenvalues and the corresponding eigenspace projection and π(J) can be defined by two of its eigenvalues and the two corresponding eigenspaces projections. So the unitary representations of B3 are directly connected with ∗-representations of the algebra generated by the three projections q1 , q2 and q3 and the relation q2 q3 = q3 q2 = 0. It was shown in [9] that such an algebra has as many ∗-representations as some matrix algebra over the free group with two generators. Therefore the classification of all unitary representations of B3 up to unitary equivalence seems to be a problem which is too complicated to solve by the pure means (see also [8]). Nevertheless in view of certain applications it is interesting to find various classes of non equivalent irreducible representations of B3 . Recently it has been found a class of representations of B3 in every dimension n depending on n parameters [1]. The authors use a deformation of Pascal’s triangle connected with q-shifted factorials to obtain the representations, and thus generalize the results from [14] and some results from [6]. In section 3 we present a class of unitary irreducible representations of B3 by n × n matrices for every n ≥ 3. Also, using a tensor product of the reduced Burau representation [7] and the unitary representations of B3 , we find a class of nontrivial irreducible unitary representations for B4 , see the section 4. In the last section we describe a procedure for obtaining polynomials in 2n variables, each of them being an invariant of a class of conjugate elements of B3 or B4 .
2
Preliminaries
Consider the product of generators of Bn : J = σ1 σ2 . . . σk−1 . It was proved in [2] that J k lies in the center Z(Bk ) of Bk . It is generated by the only one generator J k . Using the diagrams for braid groups, it is easy to 2
show that σi = Jσi−1 J −1 = J i−1 σi−1 J 1−i , i < k.
(1)
Moreover, following [2], if S := σ1 J, then S k−1 = J k .
(2)
Thus S k−1 is also a generator of Z(Bk ). Note that the group Bk can be generated by σ1 and J as well as by J and S. It follows from the braid relation that σ1 Jσ1 J −1 σ1 = Jσ1 J −1 σ1 J 1 σ1 J −1 or, in S and J terms, after multiplying both sides of the above equation by J: S 2 J −2 S = JSJ −2 S 2 J −2 .
(3)
Now let k = 3. Then S 2 = J 3 commutes with every element of Bk and the equation (3) turns into an identity. This means that B3 is generated by two elements J and S and only one relation S 2 = J 3 . (If k > 3, then beside the equation 3 there exist also relations deriving from the commutation relation between the generators of Bk ).
3
Irreducible representations of B3
In this section we shall use the notations In and 0n for the n × n identity and zero matrices and the notation Ei,j for the matrix unit with 1 in the (i,j) position and 0 in the other positions. A diagonal matrix with entries a1 , a2 , . . . , am will be denoted by diag (a1 , a2 , . . . , am ). Suppose we have an irreducible unitary representation π of Bk by n × n matrices. Then the image of Z(Bk ) is the set of all scalar unitary matrices. So π(S 2 ) = π(J 3 ) = uI, u¯ u = 1 with u ∈ C, u¯ the conjugator of u. To describe the representation it suffices to consider the case u = 1. Note that every such representation will be a unitary representation of the unimodular group P SL2 (Z) (see [12]). 3
Let U and V be 2n + m × 2n + m block matrices of the form.
U=
A − In /2 2 B∗
B ∗
C
−1
∗
−1
B A B − In /2
B A C
C ∗ A−1 B
C ∗ A−1 C − Im /2
C∗
,
V = diag (In , βIn , β 2 Im ), √ where β = 3 1 is a primitive root, 1 ≤ m ≤ n, A and B are n × n matrices and C is an n × m matrix. Theorem 1 For appropriate A, B and C the representation π, π(S) = U, π(J) = V defines an irreducible unitary representation of B3 . PROOF. We have V 3 = I2n+m . At the same time, if A∗ = A and BB ∗ + CC ∗ = A − A2 , then U ∗ = U and U 2 = I2n+m . So π is a unitary representation of B3 . If in addition A, B are invertible, rank C = m, B ∗ B is a diagonal matrix with simple spectrum and every entry ai,i+1 of A is nonzero for i = 1, . . . , n, then π will be irreducible. We shall show this using the irreducibility of the corresponding ∗-representation π ˜ of the group algebra C hB3 i defined on the generators S and J by the formulas π ˜ (S) = U , π ˜ (J) = V . Let us denote by A the algebra π ˜ (C hB3 i). Our goal is to show that Ei,j ∈ A for every i, j. Since V is a block diagonal matrix with different scalar matrices in the blocks, there exists a polynomial P11 such that E11 (In ) = diag (In , 0n , 0n ) = P11 (V ) which yields that E11 (In ) belongs to A. Similarly we see that E22 (In ) ∈ A and E33 (Im ) ∈ A. The matrix BB ∗ is diagonal with simple spectrum, so similar arguments for the diagonal matrix E11 (In )U E22 (In )U E11 (In ) lead to Ei,i ∈ A for i = 1, . . . , n. On the other hand, ai, i+1 6= 0. So Ei,i U Ei+1, i+1 ∈ A and Ei, i+1 ∈ A. Recalling that π ˜ is a star representation, we have Ei,∗ i+1 ∈ A. It is now easy to show that diag (W1 , 0n , 0n ) ∈ A for every n × n matrix W1 . ˜ = diag (B −1 , 0n , 0n ), we obtain Putting B
0n
E12 (In ) = 0n
In 0nm
0n 0nm
˜ E2,2 (In ) ∈ A, hence E21 (In ) ∈ A, = BU
0mn 0mn 0m
4
where 0nm is an n × m zero matrix. So diag (W2 , 0m ) ∈ A
(4)
for every 2n × 2n matrix W2 . At the same time, the matrix C has rank m. Therefore there exists an n × n matrix Cˆ such that we have for the transposed matrix
1 0 0 ... 0 0 ... 0
DT =
0 0 .. .
2 0 ... 0 0 ... 0 ˆ T 0 3 ... 0 0 ... 0 = (CC) . .. .. . . .. .. .. . . . . . .
0 0 0 ... m 0 ... 0
This leads to the inclusion
˜= D
0n
0n
0n
0n 0nm
D =
0mn 0mn 0m
ˆ C
0n
0n 0nm
U E33 (Im ) ∈ A. 0n 0nm
0mn 0mn 0m
Using once again the fact that A is a ∗-algebra, we obtain that ˜ ∗D ˜ = diag (0n , 0n , 1, 4, . . . , m2 ) ∈ A, hence Eii ∈ A D
for 2n < i ≤ 2n + m. Thus every Eii ∈ A, 1 ≤ i ≤ 2n + m. Beside this, we have
0 0 ... 0 1 ... 1 1 1/2 . . . 1/m 0 . . . 0 0 0 ... 0 0 ... 0 0 0 ... 0 0 ... 0 ˜ = . D . . . . .. .. .. .. .. .. . . .. .. . . . . . . . .
0 0 ...
0
0 ... 0
0 0 ... 0 0 ... 0
5
Now (4) yields
T =
..
1 ... 1 0 ... 0 ..
..
∈A
0 ... 0
and we achieve our goal to show Ei,j ∈ A from the equalities E1,i = T Ei,i ∗ and Ei,j = E1,i E1,j . Therefore A contains every Ei,j , i, j = 1, 2, . . . , 3n, and as a result A is the matrix algebra of all 2n + m × 2n + m complex matrices. This proves the irreducibility of the representation π ˜ . Since a reducible representation of a group leads to a reducible representation of the corresponding group algebra, we conclude that π is an irreducible unitary representation of B3 . 2 Corollary 2 If one consider two unitary irreducible representations π1 and π2 from Theorem 1 with matrices A1 , B1 and C1 and A2 , B2 and C2 respectively, then if A1 is not similar to A2 or B1 B1∗ is not similar to B2 B2∗ , then the representations π1 and π2 are not equivalent. PROOF. It follows from the proof of the Theorem that there exist polynomials P11 and P22 such that Pii (˜ π1 (J)) = Pii (˜ π2 (J)) = Eii (In ). Also direct calculations show that
E11 (In )˜ πi (S)E11 (In ) =
Ai 0n 0nm 0n
0n 0nm
0mn 0mn 0m
and
∗ Bi Bi
E11 (In )˜ πi (S)E22 (In )˜ πi (S)E11 (In ) =
0n
0n 0nm
, 0n 0nm
i = 1, 2
0mn 0mn 0m
So, defining equivalence of representations by ∼ and similarity of matrices by ≈, if π1 ∼ π2 , then both A1 ≈ A2 and B1 B1∗ ≈ B2 B2∗ . 2. Remark 3 At the begin of the proof of Theorem 1, instead of the set Ω = {(i, i + 1) | i = 1, . . . , n} for which ai,j 6= 0, (i, j) ∈ Ω, one can choose 6
˜ having the property that the set of the matrix units any set of indices Ω ˜ generates the algebra of all n × n complex E1,1 , . . . , En,n , Ei,j , Ej,i , (i, j) ∈ Ω matrices.
4
Irreducible representations of B4
To construct nontrivial irreducible representations of B4 we use the notion of tensor products of matrices. Let D and G be d × d and l × l matrices respectively. Then the ld × ld matrix
g11 Id . . . g1l Id . .. .. diag (D, D, . . . , D) .. . .
gl1 Id . . . gll Id
will be the tensor product D ⊗ G of D and G. Suppose we have two irreducible unitary representations π1 and π2 of B4 . Then π1 ⊗ π2 is a unitary representations of B4 too. Now putting π1 (σ1 ) = π1 (σ3 ) = π(σ1 ) and π1 (σ2 ) = π(σ2 ) for the representation π from the previous section, we obtain trivially a representation of B4 . For the representation π2 we use reduced Burau representation (see [7]) written in the base where every matrix π2 (σi ) is unitary:
π2 (σ1 ) = diag (u, 1, 1),
(u − 1)α1 + 1 q
π2 (σ2 ) = (u − 1) α1 − α12
0
(1 − u)α1 + u 0 0 1
π2 (σ3 ) =
q
(u − 1) α1 − α12 0
1 0
0
0
(u − 1)α2 + 1
q
q
0 (u − 1) α2 − α22
(u − 1)
, α2 − α22
(1 − u)α2 + u
where u¯ u = 1, α1 = −u/(u−1)2 , α2 = α1 /(1−α1 ). We remark that since both numbers α1 and α2 have to be positive and less then 1, we have to assume that the real part of u is less then 0. Theorem 4 The representation π1 ⊗ π2 is an irreducible unitary representation of B4 by 3n × 3n matrices and the natural restriction of it to a representation of B3 is reducible. 7
PROOF. To minimize the number of additional symbols we shall use the same notation πi for the ∗-representation of the group algebra C hB4 i. Let A = π˜1 ⊗ π2 (C hB4 i). As in the proof of Theorem 1 we shall show that Em,k ⊗ Ei,j ∈ A. We remark that 3
[π1 (σ1 ) ⊗ π2 (σ1 ) · π1 (σ2 ) ⊗ π2 (σ2 )] = I
3
u ⊗ 0
0 0
. u3 0 0 0 0
This implies that
P =I
1 ⊗ 0
0 0
∈ A. 1 0 000
Consider the element
(π1 (σ1 ) ⊗ π2 (σ1 ))∗ =
¯ u π1 (σ1−1 ) ⊗ 0
0 0
. 1 0 001
(5)
It lies in A and we can multiply it with another element of A, π1 (σ3 )⊗π2 (σ3 ) = π1 (σ1 ) ⊗ π2 (σ3 ) from the right. Using also the projection P constructed above, one has
P (π1 (σ1 ) ⊗ π2 (σ1 ))∗ π1 (σ3 ) ⊗ π2 (σ3 )P = I
¯ u ⊗0
0
0
∈ A.(6) (u − 1)α2 + 1 0 0 0 0
Since u¯ 6= (u−1)α2 +1 by the definitions of u and α2 , we have I ⊗Ei,i ∈ A. Note that by construction of π2 every entry of π2 (σ2 ) and π2 (σ3 ) which depends on u, α1 or α2 is different from zero. Therefore, I ⊗ E1,2 is a scalar multiple of I ⊗ E1,1 · π1 (σ2∗ ) ⊗ π2 (σ2∗ ) · I ⊗ E1,1 · π1 (σ2 ) ⊗ π2 (σ2 ) · I ⊗ E2,2 and I ⊗ E2,3 is a scalar multiple of I ⊗ E2,2 · π1 (σ3∗ ) ⊗ π2 (σ3∗ ) · I ⊗ E2,2 · π1 (σ3 ) ⊗ π2 (σ3 ) · I ⊗ E3,3 . 8
Thus since A is a ∗-algebra, I⊗Ei,j ∈ A for every i, j. This leads to I⊗π2 (σi∗ ) ∈ A and I ⊗ π2 (σi∗ ) · π1 (σi ) ⊗ π2 (σi ) = π1 (σi ) ⊗ I3 ∈ A for every i. Using the irreducibility of π1 , we conclude that Em,k ⊗I3 ∈ A and, hence Em,k ⊗Ei,j ∈ A. 2
Remark 5 One can consider any irreducible unitary representation π ˆ of B4 instead of π1 in the Theorem with the restriction of π ˆ to B3 generated by σ1 and σ2 being irreducible. The proof is similar if one defines an orthogonal projection P as in the proof of the Theorem 4 and uses the inclusion (P π1 (σ3 ) ⊗ π2 (σ3 )P )∗ π1 (σ3 ) ⊗ π2 (σ3 )P ∈ A
to show that I ⊗ Ei,i ∈ A. Remark 6 Although every Burau representation is unitary with respect to some Hermitian form[11] it is a curious fact that there are some restrictions on the spectra of unitary dilations to be the images of the standard generators of Bn for an irreducible unitary representation of Bn . More discussions on this theme can be found in [15].
5
Invariant polynomials of conjugacy classes in B3 and B4
The results obtained in the previous sections provide many representations of B3 and B4 . For example, for m = n, one can find no less than n(n−1)/8−1 real parameters ci , ci ∈ [ai , bi ], ai 6= bi ∈ R yielding non equivalent representations of B3 by 3n × 3n matrices. For the purpose of applications, it is convenient to suggest that some parameters are simply fixed real numbers. Let us consider one such construction. Let t0 , t1 , . . . , tn , t ∈ R, 0 = t0 < t1 < . . . < tn < t < 1/4. Suppose xi is a variable taking values in (ti−1 , ti ). The matrices B and C are defined by the following formulas B = diag (x1 , . . . , xn ),
q
q
C = diag ( t − x21 , . . . , t − x2n ).
Note that BB ∗ + CC ∗ = diag (t, . . . , t). 9
Let α and β be the roots of the equation x − x2 − t = 0 and consider the idempotent n × n matrix 1 1 . Q = .. n 1
... .. .
1 .. . ,
... 1
We then define A = αQ + βIn .
This leads to the simple form for A−1 : A−1 = ((α + β)−1 − β −1 )Q + β −1 In .
Thus by Theorem 1, we have a simple construction of representations π = π(x1 , . . . , xn ) of the group B3 which are continuous in x1 , . . . , xn . Let a ∈ B3 . Set P (xq 1 , . . . , xn ) = tr π(x1 , . . . , xn )(a). This is a polynomial in the variables xi and t − x2i . Since similar matrices have the same trace, we obtain the following proposition. Proposition 7 If the elements a and b are conjugate in B3 , then P (x1 , . . . , xn )(a) = P (x1 , . . . , xn )(b). Remark 8 In the above construction one can replace the numerical matrix Q with any orthogonal projection depending on some variables ci . But not every such representation will be irreducible. Remark 9 Polynomials similar to the ones in Proposition 7 can be constructed for elements of B4 using the representation described in section 4. We hope that the obtained representations will help in the solutions of the word and conjugacy problems in Bk [5] and in discussing knot theoretical problems [7]. Acknowledgements. The second author would like to thank the Institute of Applied Mathematics, University of Bonn for the hospitality. The partial financial support by the DFG project 436 UKR 113/87 is gratefully acknowledged. 10
References
[1] S. Albeverio and A. Kosyak, q-Pascal’s triangle and irreducible representations of the braid group B3 in abitrary dimension. 2007, in preparation. [2] E. Artin, Theorie der Z¨ opfe, Abh. Math. Sem. Hamburg. Univ. 4 (1926) 47–72. [3] J. S. Birman, New points of view in knot theory. Bull. of Amer. Math. Soc. 28 (1993) 253–287. [4] J. S. Birman and T. E. Brendle, Braids: A survey. In Handbook of Knot Theory (Ed. W. Menasko and M. Thistlethwaite), Elsevier, 2005. [5] J. S. Birman, K. H. Ko and S. J. Lee, A new approach to the word and conjugacy problems in the braid groups. Advances in Math. 139 (1998) 322–353. [6] S. P. Humphries, Some linear representations of braid groups. J. Knot Theory and Its Ramifications 9(3) (2000) 3415–366. [7] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials. Annals of Math. 126 (1987) 335–388. [8] S. A. Kruglyak and Yu. S. Samoˇilenko, On the complexity of description of representations of ∗-algebras generated by idempotents. Proc. Amer. Math. Soc. 128 (2000), 1655-1664. [9] S. A. Kruglyak and Yu. S. Samoˇilenko, On unitary equivalence of collections of selfajoint operators. Functional Analysis and Appl. 14, no. 1 (1980) 60–62 (Russian). [10] I. Marin, On the representation theory of Braid groups, Cornell Univ. Math. e-preprint arhiv RT/0502118. V3. (2006), 1–49. [11] C. Squier, The Burau representation is unitary, Proc. Amer. Math. Soc. 90, (1984), 199–202. [12] J. P. Serre, J. P., Cours d’arithm´etique, P.U.F. , 1970. [13] I. Tuba, Low-dimensional unitary representations of B3 , Proc. Amer. Math. Soc. 129 (2001), 2597–2606. [14] I. Tuba and H. Wenzl, Representations of the braid group B3 and of SL(2, Z), Pacific J. Math. 197 (2001), no. 2, 491-510. [15] H. Wenzl, On sequences of projections, C.R. Math. Rep. Acad. Sci. Canada 9 (1987), 5-9.
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Verzeichnis der erschienenen Preprints ab No. 310
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324. Albeverio, Sergio; Ayupov, Sh. A.; Kudaybergenov, K. K.: Derivations on the Algebra of τ-Compact Operators Affiliated with a Type I von Neumann Algebra 325. Castaño Díez, Daniel; Gunzburger, Max; Kunoth, Angela: An Adaptive Wavelet Viscosity Method for Hyperbolic Conservation Laws 326. Albeverio, Sergio; Ayupov, Sh. A.; Omirov, B. A.; Khudoyberdiyev, A. Kh: n-Dimensional Filiform Leibniz Algebras of Length (n-1) and Their Derivations 327. Albeverio, Sergio; Rabanovich, Slavik: On a Class of Unitary Representations of the Braid Groups B3 and B4
