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A note on the special unitary group of a division algebra

If D is a division algebra with its center a number field K and with an involution of the second kind, it is unknown if the group SU(1, D)/[U(1, d), U(1, D)] is trivial. We show that, by contrast, if K is a function field in one variable over a number field, and if D is an algebra with center K and with an involution of the second kind, the group SU(1, D)/[U(1, d), U(1, D)] can be infinite in general. We give an infinite class of examples.

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