## Archive for **September 2009**

## The death of love

Arguably the greatest long poem (*mahakavya*) in classical Sanskrit is Kalidasa’s *Kumarasambhava* (or “The Birth of Kumara”), which tells the tale of how the warrior god Skanda (the “Kumara” of the title) came to be born. Once, the gods were suffering greatly from the attacks of the demon Taraka. Unable to defeat Taraka, the gods approached the creator-god Brahma to ask for his help. Specifically, the gods asked for a general who could lead them to victory against Taraka. Brahma told the gods that they must find a way to convince Shiva the Destroyer to marry Parvati, the daughter of the mountain god. The child of Shiva and Parvati would be the general that they were looking for. However, Shiva was deeply absorbed in meditation and would not easily be tempted into marriage. The gods approached Kamadeva, god of love (armed, like his Greco-Roman counterpart, with bow and arrow), and requested that he use his unique abilities to make Shiva fall in love with Parvati. Kamadeva agreed, and went to the mountaintop where the god Shiva was engaged in meditation. Shiva, however, sensed the intrusion of the love god. Kalidasa describes what happened next (the Clay Sanskrit Library edition 3.69-72):

Then Three-eyed Shiva,

through his self-control

powerfully suppressing

the disturbance of his senses,

wished to see the cause

of his mind’s disturbance

and sent his gaze in all directions.

He saw Self-born Love ready to attack,

his lovely bow drawn right back

to form a circle,

his fist resting

at the corner of his right eye,

shoulder hunched,

left foot arched.

Enraged by the violation of his penance,

his frown made his face

dreadful to behold,

and from his third eye

a sparkling, blazing fire

suddenly flew forth.

“Lord, hold back your anger,

hold back!”—

even as the cries of the wind-gods

crossed the sky,

that fire born from the eye of Shiva who is Being,

reduced to ashes Intoxicating Love.

His corporeal form having been disintegrated by the fire emanating from the mystical third eye of Shiva the Destroyer, Kamadeva, the god of love, is henceforth known as Ananga, the Bodiless God.

## The General Burnside Problem

In 1902 William Burnside posed the following question concerning finitely generated groups:

**Bounded Burnside Problem**. If is a finitely generated group and there is an integer such that for every , then must be finite?

The problem also has the following variant:

**General Burnside Problem**. If is a finitely generated group and every element of has finite order, then must be finite?

The answer to both questions was expected to be “yes”’; the solution to the General Burnside Problem was therefore anticipated to be somewhat harder than that of the Bounded Burnside Problem. However, the answer to both problems turned out to be “no”. A counterexample to the General Burnside problem was given by a beautiful and elegant construction of Golod and Shafarevich in 1964; and a counterexample to the Bounded Burnside Problem was given by Novikov and Adjan in 1968.

The Burnside Problem has the following ring-theoretic analogue:

**Kurosh’s Problem**. If is a finitely generated algebra over a field and every element of is nilpotent, then must be nilpotent?

(A element is nilpotent if for some ; and is itself nilpotent if there is an such that for all .) The Golod-Shafarevich construction also provides a counter-example to Kurosh’s Problem. We sketch the construction below.

**The Golod-Shafarevich Theorem
**

Let be a field and let be the free non-commutative algebra over generated by the variables . Let denote the subspace of consisting of all linear combinations of monomials of degree . The elements of are the homogeneous elements of degree . Let be a two-sided ideal of generated by a set of homogeneous elements, each of degree at least 2. Suppose that has at most elements of degree for . The following (which we do not prove here) is the main computational result of the Golod-Shafarevich construction:

**Theorem**. The quotient algebra is infinite dimensional over if the coefficients in the power series expansion of are nonnegative.

Using this theorem, one constructs a counterexample to Kurosh’s Problem as follows.

**Counterexample to Kurosh’s Problem**

Let be the free algebra over a countable field . Let be the ideal of consisting of all elements of without constant term. Enumerate the elements of as . Choose an integer and write , where each . Choose another positive integer sufficiently large so that for some . Continue in this way for sufficiently large powers of . Now let be the ideal generated by the defined in the process above. Consider the quotient . The construction of guarantees that each element of is nilpotent; but the theorem above ensures that is infinite dimensional over (and hence not nilpotent). Thus is a counterexample to Kurosh’s Problem. From this construction, we can in turn build a counterexample to the General Burnside Problem.

**Counterexample to the General Burnside Problem**

Let us suppose now that is a prime number and is the field with elements. Let and be as defined above. Let be the elements of the quotient respectively. Let be the multiplicative semigroup in generated by the elements , , and . An element of has the form for some . The element is nilpotent by the construction of , and so for sufficiently large , we have . Since we are in characteristic we have . It follows that has an inverse, whence is a group. Moreover every element of has finite order (indeed order a power of ). Thus satisfies the conditions of the General Burnside Problem. It remains to show that is infinite. If were finite, then the linear combinations of its elements would form a finite dimensional algebra over . Moreover, since both 1 and are in , the linear combination is in . Thus, are all in . But suffice to generate , which was previously shown to be infinite dimensional. The algebra is therefore also infinite dimensional, a contradiction. Thus is infinite, as required.