Ilyumzhinov has presided over FIDE since 1995. After an internal revolt following a congress in Moscow — at one point during the assembly, the former world champion Anatoly Karpov was supposedly threatened with having his legs broken — Ilyumzhinov emerged as a compromise choice to take over the presidency. He was backed by Karpov then and has been re-elected ever since, despite what his many critics say is an erratic, even scandalous tenure that has undermined the sport’s international prestige. Ilyumzhinov blurred the lines between the presidencies of Kalmykia and FIDE, traveling the world to promote chess and the republic simultaneously. He befriended Saddam Hussein in the 1990s, for example, and one of his first acts as FIDE’s president was to schedule the 1996 world championship in Baghdad at a time when Iraq faced punishing international sanctions. The United States warned one contestant, Gata Kamsky, then a resident, that playing a match there might lead to civil and criminal charges against him. Under enormous pressure, Ilyumzhinov moved the match to Kalmykia’s small capital, Elista. Envisioning a new era of international attention for Kalmykia, Ilyumzhinov then built a Chess City on Elista’s outskirts, including a glass-domed palace intended as a venue for matches and tournaments that he had almost solitary power to organize.

Ilyumzhinov’s forays into international affairs as FIDE’s president — and his contacts with some of the world’s most reviled leaders in times of crisis — have raised the question of whether he serves as an envoy for the Kremlin. In June 2011, he arrived unexpectedly in Libya’s capital, Tripoli, and met Qaddafi nearly three months into the NATO air war there. (In 2004, he placed FIDE’s championship in Tripoli, where he introduced a controversial new knockout format; only five of the world’s top 20 players attended.) The two men sat down for a game of chess, broadcast on state television as evidence that Qaddafi was still in charge. Ilyumzhinov offered him a draw. He told me that Qaddafi asked him to deliver messages to Angela Merkel, Nicolas Sarkozy and other leaders of NATO nations, offering to hold a constitutional referendum as a compromise to end the war. It was Qaddafi’s last public appearance before he was captured by rebels and killed in October 2011.

“Everybody who supports chess is my friend,” Ilyumzhinov said, showing me photographs to make his point: the Dalai Lama and the pope, as well as Syria’s embattled president, Bashar al-Assad, whom he visited in May 2012, in the midst of the country’s civil war. According to Ilyumzhinov, he and Assad opened a chess school together that was later bombed by the Syrian rebels. “Chess is beyond politics,” he said, not very convincingly. That, he added, “is why Kasparov is so dangerous. This is why it’s necessary to fight him. This is what the chess world was afraid of: Kasparov started mixing chess with politics.”

In the world of chess — and it is bigger than you might think, with an estimated 600 million regular players, according to a survey commissioned by FIDE — Kasparov is a living legend, whose fame eclipses that of the reigning champion, Magnus Carlsen. I heard him repeatedly referred to in this way by players, delegates and even Ilyumzhinov. Kasparov became the sport’s youngest world champion in 1985, when he was 22 (a mark that has since been lowered), a woolly-haired challenger up against the Soviet Union’s chess establishment and its reigning champion for the previous decade, Anatoly Karpov. Among those watching the tense finale on Soviet television on Nov. 9, 1985, was Vladimir Putin, a young major in the K.G.B., stationed in a small intelligence outpost in the East German city of Dresden. According to a memoir written by one of Putin’s colleagues, the entire cadre of Soviet agents and military men stationed there rooted for Karpov, thinking even then that Kasparov was “an extremely impudent upstart,” which was exactly how the propaganda machine of the Soviet Union described him. The author thought Putin showed “dangerous sympathy” for Kasparov. Times have changed, clearly.

Chess has always occupied a disproportionate place in the Russian psyche — as much as any other game or sport, it forms part of the national identity. The Bolsheviks, who overthrew the last czar in 1917, initially disparaged the game as bourgeois, but some were ardent players, Lenin among them. According to “The Immortal Game,” by David Shenk, the Soviets saw the political and ideological value of the sport, “turning the popular but ragtag nature of public chess play into one of the self-identifying marks of emerging Soviet culture.” The Soviets created a system of academies that resulted in their domination of international chess for the second half of the 20th century. Only Bobby Fischer’s upset victory over Boris Spassky in 1972 — a televised Cold War drama — interrupted the Soviet Union’s reign until the state itself collapsed, and the sport, like nearly everything else in Russia, lurched through a decade of crises.

Kasparov was a product of the Soviet sports machine and, almost from the beginning, a rebel against it. He began playing seriously at 7 at the Young Pioneers Palace in Baku, the capital of Azerbaijan. By the time he was 13, he was already traveling abroad, representing the Soviet Union internationally. In his memoirs — and he is a prolific author of autobiographies and books on chess — Kasparov describes his rise through the chess ranks as a series of struggles against a sclerotic Soviet chess bureaucracy and against FIDE, which, he wrote in “Unlimited Challenge,” published in the United States in 1990, had “legalized tyranny in the world of chess.” He was convinced that the Soviet chess federation conspired with FIDE to block his ultimately successful challenge against Karpov, whom he described as “just the man for a system which elevated to the skies everything that helped to affirm its own ideological fetishes, even in sport.”

Admired in Russia not just as an intellectual but also as an athlete, whose sturdy build has diminished only slightly over time, Kasparov had the kind of fame that allowed him to immerse himself in the roiling politics of the time. He strongly, and publicly, backed Mikhail Gorbachev’s efforts to open the Soviet Union in the 1980s — and then later criticized him for backsliding. After the abortive coup in August 1991 hastened the collapse of the Soviet Union, Kasparov threw his support behind Boris Yeltsin and Russia’s new generation of democrats. By 1996, though, he had broken with Yeltsin and, at least early on, supported one of his challengers in that year’s election.