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Special Unitary Group

The special unitary group SU_n(q) is the set of n×n unitary matrices with determinant +1 (having n^2-1 independent parameters). SU(2) is homeomorphic with the orthogonal group O_3^+(2). It is also called the unitary unimodular group and is a Lie group.

Special unitary groups can be represented by matrices

U(a,b)=[a b; -b^_ a^_],
(1)

where a^_a+b^_b=1 and a,b are the Cayley-Klein parameters. The special unitary group may also be represented by matrices

U(xi,eta,zeta)=[e^(ixi)coseta e^(izeta)sineta; -e^(-izeta)sineta e^(-ixi)coseta],
(2)

or the matrices

U_x(1/2phi)=[cos(1/2phi) isin(1/2phi); isin(1/2phi) cos(1/2phi)]
(3)
U_y(1/2beta)=[cos(1/2beta) sin(1/2beta); -sin(1/2beta) cos(1/2beta)]
(4)
U_z(xi)=[e^(ixi) 0; 0 e^(-ixi)].
(5)

The order 2j+1 representation is

U_(p,q)^((j))(alpha,beta,gamma)=sum_(m)((-1)^(m-q-p)sqrt((j+p)!(j-p)!(j+q)!(j-q)!))/((j-p-m)!(j+q-m)!(m+p-q)!m!) ×e^(iqalpha)cos^(2j+q-p-2m)(1/2beta)sin^(p+2m-q)(1/2beta)e^(ipgamma).
(6)

The summation is terminated by putting 1/(-N)!=0. The group character is given by

chi^((j))(alpha)={1+2cosalpha+...+2cos(jalpha) ; 2[cos(1/2alpha)+cos(3/2alpha)+...+cos(jalpha)]
(7)
={(sin[(j+1/2)alpha])/(sin(1/2alpha)) for j=0,1,2,...; (sin[(j+1/2)alpha])/(sin(1/2alpha)) for j=1/2,3/2,....
(8)

See also

Orthogonal Group, Special Linear Group, Special Orthogonal Group

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References

Arfken, G. "Special Unitary Group, SU(2) and SU(2)-O_3^+ Homomorphism." Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 253-259, 1985.Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. "The Groups GU_n(q), SU_n(q), PGU_n(q), and PSU_n(q)=U_n(q)." §2.2 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, p. x, 1985.

Referenced on Wolfram|Alpha

Special Unitary Group

Cite this as:

Weisstein, Eric W. "Special Unitary Group." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SpecialUnitaryGroup.html

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