I think the title says it. I always read that alleles are gene variants at a given locus, which confuses me. Thanks!
EDIT: Short answer : Yes: "allele" is a synonym for "gene variant", irrespective of location (locus).
Original answer: From About Education I found this definition for an allele:
An allele is an alternative form of a gene (one member of a pair) that is located at a specific position on a specific chromosome.
Hence, two alleles are located at the same locus, but are situated on a different chromosome.
ADDITION EDIT: Given the many high-quality comments (and the downvote) I erased the concluding sentence of my first attempt and I will elaborate below:
Feero et al. (2010) gives the following more elaborate definition of "allele" (text relevant to the question is bolded out):
Specific locations in the human genome where differences between individual people are found are generally referred to as variations, and the term “normal” or “wild type” is often used to refer to the most common variant at a location in a given population group. In its simplest form, the variation has two different spellings, referred to as “alleles.” If the frequency of the minor allele is greater than 1%, such variants are called polymorphisms. The word “mutation” is generally reserved for changes in DNA that are believed or known to be pathologic [...] or for changes that are [very] recent (e.g., not inherited DNA changes).
Hence, from my understanding, the original quoted definition from About Education still stands.
As to the critical comments placed by the PO, @fileunderwater and @WYSIWYG:
[...] what is a copy or a variant of a gene at a different locus on the same chromosome? - That is a very interesting question, and I think Feero et al. (2010) deals with that simply by saying
In its simplest form, the variation has two different spellings, referred to as “alleles.”
In other words - perhaps the gene duplication is a step too far for this terminology. And in fact, although I regret having to cite wikipedia here, it does provide a very enlightening sentence:
[...] [A]t the gene locus for the ABO blood type [...] classical genetics recognizes three alleles, IA, IB, and i, that determine compatibility of blood transfusions. Any individual has one of six possible genotypes (IAIA, IAi, IBIB, IBi, IAIB, and ii) that produce one of four possible phenotypes [...].
Hence, what this implicitly tells us is that an allele is, as Feero et al. (2010) also says, simply a gene variation.
This means that it basically boils down to what is a "chromosome locus"?
An article by Proulx & Phillips, 2006 entitled "ALLELIC DIVERGENCE PRECEDES AND PROMOTES GENE DUPLICATION" mentions:
[...] [H]eterozygotes at a single locus are broken up by segregation, whereas for duplicate loci an individual can carry copies of alternative alleles at separate loci.
Hence, what the authors implicitly say is that an allele is a gene variation and that a locus is a fixed position on a chromosome. Hence, a gene duplication means 2 loci that can carry the same or a different allele. This should answer
Are the alleles at the two duplicated loci then named differently, even if their DNA sequences are identical (as well as the protein they might be coding for), namely no, an allele has a name (e.g. IA) that remains unchanged when duplicated. The locus is then different for the duplicate, but the name of the allele stays the same (as an allele is, by definition, simply a gene variant wherever on the chromosome).
[...] two versions of a gene at two different loci are alleles (not paralogs)?: yes
What about the same identical version at two different loci: same allele or not? yes, same allele, allele has nothing to do with locus. Just a gene variant. Where ever.
Feero et al. N Engl J Med 2010;362:2001-11
Proulx & Phillips. Evolution 2006;60:881-892