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Over the years questions have arisen about T-ideals of (noncommutative) polynomials. But when evaluating a noncentral polynomial in subalgebras of matrices, one often has little control in determining the specific evaluations of the polynomial. One way of overcoming this difficulty in characteristic 0, is to reduce to multilinear polynomials and to utilize the representation theory of the symmetric group. But this technique is unavailable in characteristic p > 0. An alternative method, which succeeds, is the process of "hiking" a polynomial, in which one specializes its indeterminates in several stages, to obtain a polynomial in which Capelli polynomial is embedded, in order to get control on its evaluations. This method was utilized on homogeneous polynomials in the proof of Specht's conjecture for affine algebras over fields of positive characteristic. In this paper, we develop hiking further to nonhomogeneous polynomials, to apply to the "representability question." Kemer proved in 1988 that every affine relatively free PI algebra over an infinite field, is representable. In 2010, the first author of this paper proved more generally that every affine relatively free PI algebra over any commutative Noetherian unital ring is representable [A. Belov, Local finite basis property and local representability of varieties of associative rings, Izv. Russian Acad. Sci. (1) (2010) 3–134. English Translation Izv. Math. 74(1) (2010) 1–126]. We present a different, complete, proof, based on hiking nonhomogeneous polynomials, over finite fields. We then obtain the full result over a Noetherian commutative ring, using Noetherian induction on T-ideals. The bulk of the proof is for the case of a base field of positive characteristic. Here, whereas the usage of hiking is more direct than in proving Specht's conjecture, one must consider nonhomogeneous polynomials when the base ring is finite, which entails certain difficulties to be overcome. In Appendix A, we show how hiking can be adapted to prove the involutory versions, as well as various graded and nonassociative theorems. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if R is an affine algebra of dimension d ≥ 4 over ¯ p and 1 / (d − 1) ! ∈ R , then any unimodular row over R of length d can be mapped to a factorial row by elementary transformations. [ABSTRACT FROM AUTHOR]

Two presentations of quantum affine superalgebras were introduced by Yamane in [On defining relations of affine Lie superalgebras and affine quantized universal enveloping superalgebras, Publ. Res. Inst. Math. Sci. 35 (1999) 321–390], which were called Drinfeld–Jimbo realization and Drinfeld realization. Drinfeld realization contains infinite sequences of generators and relations. In this paper, we consider the Drinfeld realization of quantum affine superalgebra q ( ̂ m | n) associated to type m | n and define a simple algebra 0 ( ̂ m | n) generated by only a finite part of these sequences of quantum affine superalgebra q ( ̂ m | n). We show that the algebra 0 ( ̂ m | n) is isomorphic to the quantum affine superalgebra q ( ̂ m | n). Using the above isomorphism, we prove there exists an isomorphism between the two realizations. [ABSTRACT FROM AUTHOR]

We construct the basic representation of the double affine Hecke algebra at critical level q = 1 associated to an irreducible reduced affine root system R with a reduced gradient root system. For R of untwisted type such a representation was studied by Oblomkov [A. Oblomkov, Double affine Hecke algebras and Calogero–Moser spaces, Represent. Theory 8(2004) 243–266] and further detailed by Gehles [K. E. Gehles, Properties of Cherednik algebras and graded Hecke algebras, PhD thesis, University of Glasgow (2006)] in the presence of minuscule weights. [ABSTRACT FROM AUTHOR]