• 대한수학회지
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• 제39권1호
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• pp.1-11
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• 2002

A Schur algebra was generalized to projective Schur algebra by admitting twisted group algebra. A Schur algebra is a projective Schur algebra with trivial 2-cocycle. In this paper we study situations that Schur algebra is a projective Schur algebra with nontrivial cocycle, and we find a criterion for a projective Schur algebra to be a Schur algebra.

• 대한수학회보
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• 제42권3호
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• pp.453-467
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• 2005

Let K be an algebraic number field. If k is the maximal cyclotomic subextension in K then the Schur K-group S(K) is obtained from the Schur k-group S(k) by scalar extension. In the paper we study projective Schur group PS(K) which is a generalization of Schur group, and prove that a projective Schur K-algebra is obtained by scalar extension of a projective Schur k-algebra where k is the maximal radical extension in K with mild condition.

• 대한수학회보
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• 제38권4호
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• pp.803-814
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• 2001

A projective Schur algebra associated with a partition of finite group G can be constructed explicitly by defining linear transformations of G. We will consider various linear transformations and count the number of equivalent classes in a finite group. Then we construct projective Schur algebra dimension is determined by the number of classes.

• 대한수학회지
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• 제38권1호
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• pp.77-85
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• 2001

If an Azumaya algebra A is a homomorphic image of a finite group ring RG where G is a direct product of subgroups then A can be decomposed into subalgebras A(sub)i which are homomorphic images of subgroup rings of RG. This result is extended to projective Schur algebras, and in this case behaviors of 2-cocycles will play major role. Moreover considering the situation that A is represented by Azumaya group ring RG, we study relationships between the representing groups for A and A(sub)i.

• 대한수학회보
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• 제36권3호
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• pp.503-512
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• 1999

Over a field k, every schur k-algebra is a cyclotomic algebra due to Brauer-Witt theorem. Similarly every projective Schur k-division algebra is itself a radical algebra by Aljadeff-Sonn theorem. We study the two theorems over a certain commutative ring, and prove similar results over regular domain containing a field.

• 대한수학회지
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• 제44권1호
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• pp.151-167
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• 2007

If k is a subfield of $\mathbb{Q}(\varepsilon_m)$ then the cohomology group $H^2(k(\varepsilon_n)/k)$ is isomorphic to $H^2(k(\varepsilon_{n'})/k)$ with gcd(m, n') = 1. This enables us to reduce a cyclotomic k-algebra over $k(\varepsilon_n)$ to the one over $k(\varepsilon_{n'})$. A radical extension in projective Schur algebra theory is regarded as an analog of cyclotomic extension in Schur algebra theory. We will study a reduction of cohomology group of radical extension and show that a Galois cohomology group of a radical extension is isomorphic to that of a certain subextension of radical extension. We then draw a cohomological characterization of radical group.