Alexander Schneider made a touching and long overdue return last night to the Coolidge Auditorium of the Library of Congress, where he played a solo recital with Peter Serkin. It had been virtually 20 years since "Sasha" Schneider had set foot on the stage where, as second violin of the Budapest String Quartet, he played a major role in setting 20th-century chamber music standards. The Budapest had been in residence there for the previous 24 years, playing about 300 concerts and making many of the countless recordings from which the younger generation learned much of its chamber music.

It became clear in the emotional speeches preceding last night's recital that there was more of a story than many realized behind Schneider's long absence. Schneider, who is now 73, recalled the Budapest's last concert and the moment before it when "two men from the library came and told us our services were no longer desired." He dedicated the concert to, among others, violinist Joseph Roisman, the Budapest's founder, and violist Boris Kroyt, both of them now dead. Whatever the depth of the rift, it now seems repaired. Music division chief Don Levitt declared, "Welcome home, Sasha," announced that Schneider has donated his papers to the library, and the evening proceeded.

The program was vintage Schneider, the powerful Bach G-major sonata, some Schubert, the A-minor sonata and the bucolic A-major duo, plus one of the most scintillating Mozart sonatas, the F-major.

Schneider's fingers do not move as well as in the old days, but the musical mind is just as incisive and the spirit just as wonderfully jaunty. Serkin, of course, is the consummate chamber music pianist. Schneider recalled, by the way, that he and Serkin first played together 22 years ago; the pianist was just 12 then.

There was one especially grand gesture. Schneider was asked to play Fritz Kreisler's priceless violin made by Guarneri del Gesu and left to the library by Kreisler.Just last night's exposure showed that the matchless dark sound Kreisler cultivated in the lower strings owed as much to Guarneri as to Kreisler. But, wouldn't you know it, in the last movement of the Schubert sonata one of the strings started to slip, and Schneider had to stop and fix it. Schneider smiled. The audience laughed.